Geometric Complexity Theory VI: the flip via saturated and positive
  integer programming in representation theory and algebraic geometry
Geometric Complexity Theory VI: the flip via
saturated and positive integer programming in
representation theory and algebraic geometry
Dedicated to Sri Ramakrishna
Ketan D. Mulmuley
The University of Chicago
http://ramakrishnadas.cs.uchicago.edu
(Technical report TR 2007-04, Comp. Sci. Dept.,
The University of Chicago, May, 2007)
Revised version
October 23, 2018
http://arxiv.org/abs/0704.0229v4
http://ramakrishnadas.cs.uchicago.edu
Abstract
This article belongs to a series on geometric complexity theory (GCT), an
approach to the P vs. NP and related problems through algebraic geometry
and representation theory. The basic principle behind this approach is called
the flip. In essence, it reduces the negative hypothesis in complexity theory
(the lower bound problems), such as the P vs. NP problem in characteristic
zero, to the positive hypothesis in complexity theory (the upper bound prob-
lems): specifically, to showing that the problems of deciding nonvanishing of
the fundamental structural constants in representation theory and algebraic
geometry, such as the well known plethysm constants [Mc, FH], belong to the
complexity class P . In this article, we suggest a plan for implementing the
flip, i.e., for showing that these decision problems belong to P . This is based
on the reduction of the preceding complexity-theoretic positive hypotheses to
mathematical positivity hypotheses: specifically, to showing that there exist
positive formulae–i.e. formulae with nonnegative coefficients–for the struc-
tural constants under consideration and certain functions associated with
them. These turn out be intimately related to the similar positivity proper-
ties of the Kazhdan-Lusztig polynomials [KL1, KL2] and the multiplicative
structural constants of the canonical (global crystal) bases [Kas2, Lu2] in
the theory of Drinfeld-Jimbo quantum groups. The known proofs of these
positivity properties depend on the Riemann hypothesis over finite fields
(Weil conjectures proved in [Dl]) and the related results [BBD]. Thus the
reduction here, in conjunction with the flip, in essence, says that the validity
of the P 6= NP conjecture in characteristic zero is intimately linked to the
Riemann hypothesis over finite fields and related problems.
The main ingradients of this reduction are as follows.
First, we formulate a general paradigm of saturated, and more strongly,
positive integer programming, and show that it has a polynomial time al-
gorithm, extending and building on the techniques in [DM2, GCT3, GCT5,
GLS, KB, KTT, Ki, KT1].
Second, building on the work of Boutot [Bou] and Brion (cf. [Dh]), we
show that the stretching functions associated with the structural constants
under consideration are quasipolynomials, generalizing the known result that
the stretching function associated with the Littlewood-Richardson coefficient
is a polynomial for type A [Der, Ki] and a quasi-polynomial for general types
[BZ, Dh]. In particular, this proves Kirillov’s conjecture [Ki] for the plethysm
constants.
Third, using these stretching quasi-polynomials, we formulate the math-
ematical saturation and positivity hypotheses for the plethysm and other
structural constants under consideration, which generalize the known sat-
uration and conjectural positivity properties of the Littlewood-Richardson
coefficients [KT1, DM2, KTT]. Assuming these hypotheses, it follows that
the problem of deciding nonvanishing of any of these structural constants,
modulo a small relaxation, can be transformed in polynomial time into a
saturated, and more strongly, positive integer programming problem, and
hence, can be solved in polynomial time.
Fourth, we give theoretical and experimental results in support of these
hypotheses.
Finally, we suggest an approach to prove these positivity hypotheses
motivated by the works on Kazhdan-Lusztig bases for Hecke algebras [KL1,
KL2] and the canonical (global crystal) bases of Kashiwara and Lusztig [Lu2,
Lu4, Kas2] for representations of Drinfeld-Jimbo quantum groups [Dri, Ji].
Steps in this direction are taken [GCT4, GCT7, GCT8].
Specifically, in [GCT4, GCT7] are constructed nonstandard quantum
groups, with compact real forms, which are generalizations of the Drinfeld-
Jimbo quantum group, and also associated nonstandard algebras, whose re-
lationship with the nonstandard quantum groups is conjecturally similar
to the relationship of the Hecke algebra with the Drinfeld-Jimbo quantum
group. The article [GCT8] gives conjecturally correct algorithms to con-
struct canonical bases of the matrix coordinate rings of the nonstandard
quantum groups and of nonstandard algebras that have conjectural posi-
tivity properties analogous to those of the canonical (global crystal) bases,
as per Kashiwara and Lusztig, of the coordinate ring of the Drinfeld-Jimbo
quantum group, and the Kazhdan-Lusztig basis of the Hecke algebra. These
positivity conjectures (hypotheses) lie at the heart of this approach. In view
of [KL2, Lu2], their validity is intimately linked to the Riemann hypothesis
over finite fields and the related works mentioned above.
Contents
1 Introduction 4
1.1 The decision problems . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Deciding nonvanishing of Littlewood-Richardson coefficients . 12
1.3 Back to the general decision problems . . . . . . . . . . . . . 16
1.4 Saturated and positive integer programming . . . . . . . . . . 16
1.5 Quasi-polynomiality, positivity hypotheses, and the canonical
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 The plethysm problem . . . . . . . . . . . . . . . . . . . . . . 20
1.7 Towards PH1, SH, PH2,PH3 via canonial bases and canonical
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 Basic plan for implementing the flip . . . . . . . . . . . . . . 29
1.9 Organization of the paper . . . . . . . . . . . . . . . . . . . . 30
1.10 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Preliminaries in complexity theory 34
2.1 Standard complexity classes . . . . . . . . . . . . . . . . . . . 34
2.1.1 Example: Littlewood-Richardson coefficients . . . . . 35
2.2 Convex #P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.1 Littlewood-Richardson coefficients . . . . . . . . . . . 38
2.2.2 Littlewood-Richardson cone . . . . . . . . . . . . . . . 38
2.2.3 Eigenvalues of Hermitian matrices . . . . . . . . . . . 39
2.3 Separation oracle . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Saturation and positivity 41
3.1 Saturated and positive integer programming . . . . . . . . . . 41
3.1.1 A general estimate for the saturation index . . . . . . 45
3.1.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.3 Is there a simpler algorithm? . . . . . . . . . . . . . . 47
3.2 Littlewood-Richardson coefficients again . . . . . . . . . . . . 47
3.3 The saturation and positivity hypotheses . . . . . . . . . . . 49
3.4 The subgroup restriction problem . . . . . . . . . . . . . . . . 52
3.4.1 Explicit polynomial homomorphism . . . . . . . . . . 53
3.4.2 Input specification and bitlengths . . . . . . . . . . . . 55
3.4.3 Stretching function and quasipolynomiality . . . . . . 57
3.5 The decision problem in geometric invariant theory . . . . . . 58
3.5.1 Reduction from Problem 1.1.3 to Problem 1.1.4 . . . . 59
3.5.2 Input specification . . . . . . . . . . . . . . . . . . . . 59
3.5.3 Stretching function and quasi-polynomiality . . . . . . 60
3.5.4 Positivity hypotheses . . . . . . . . . . . . . . . . . . . 61
3.5.5 G/P and Schubert varieties . . . . . . . . . . . . . . . 62
3.6 PH3 and existence of a simpler algorithm . . . . . . . . . . . 63
3.7 Other structural constants . . . . . . . . . . . . . . . . . . . . 63
4 Quasi-polynomiality and canonical models 65
4.1 Quasi-polynomiality . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.1 The minimal positive form and modular index . . . . 68
4.1.2 The rings associated with a structural constant . . . . 69
4.2 Canonical models . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 From PH0 to PH1,3 . . . . . . . . . . . . . . . . . . . 70
4.2.2 On PH0 in general . . . . . . . . . . . . . . . . . . . . 72
4.3 Nonstandard quantum group for the Kronecker and the plethysm
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 The cone associated with the subgroup restriction problem . 75
4.5 Elementary proof of rationality . . . . . . . . . . . . . . . . . 78
5 Parallel and PSPACE algorithms 84
5.1 Complex semisimple Lie group . . . . . . . . . . . . . . . . . 85
5.2 Symmetric group . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 General linear group over a finite field . . . . . . . . . . . . . 92
5.3.1 Tensor product problem . . . . . . . . . . . . . . . . . 93
5.4 Finite simple groups of Lie type . . . . . . . . . . . . . . . . . 94
6 Experimental evidence for positivity 95
6.1 Littlewood-Richardson problem . . . . . . . . . . . . . . . . . 95
6.2 Kronecker problem, n = 2 . . . . . . . . . . . . . . . . . . . . 95
6.3 G/P and Schubert varieties . . . . . . . . . . . . . . . . . . . 96
6.4 The ring of symmetric functions . . . . . . . . . . . . . . . . 97
7 On verification and discovery of obstructions 111
7.1 Obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Decision problems . . . . . . . . . . . . . . . . . . . . . . . . 113
7.3 Verification of obstructions . . . . . . . . . . . . . . . . . . . 113
7.4 Robust obstruction . . . . . . . . . . . . . . . . . . . . . . . . 115
7.5 Verification of robust obstructions . . . . . . . . . . . . . . . 116
7.6 Arithemetic version of the P#P vs. NC problem in charac-
teristric zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.6.1 Class varieties . . . . . . . . . . . . . . . . . . . . . . . 117
7.6.2 Obstructions . . . . . . . . . . . . . . . . . . . . . . . 118
7.6.3 Robust obstructions . . . . . . . . . . . . . . . . . . . 119
7.6.4 Verification of robust obstructions . . . . . . . . . . . 121
7.6.5 On explicit construction of obstructions . . . . . . . . 122
7.6.6 Why should robust obstructions exist? . . . . . . . . . 123
7.6.7 On discovery of robust obstructions . . . . . . . . . . 124
7.7 Arithmetic form of the P vs NP problem in characteristic zero126
Chapter 1
Introduction
This article belongs to a series of papers, [GCT1] to [GCT11], on geomet-
ric complexity theory (GCT), which is an approach to the P vs. NP and
related problems in complexity theory through algebraic geometry and rep-
resentation theory. We assume here that the underlying field of computation
is of characteristic zero. The usual P vs. NP problem is over a finite field.
The characteristic zero version is its weaker, formal implication, and philo-
sophically, the crux.
The basic principle underlying GCT is called the flip [GCTflip]. The
flip, in essence, reduces the negative hypotheses (lower bound problems) in
complexity theory, such as the P 6=?NP problem in characteristic zero, to
positive hypotheses in complexity theory (upper bound problems): specifi-
cally, to the problem of showing that a series of decision problems in rep-
resentation theory and algebraic geometry belong to the complexity class
P . Each of these decision problem is of the form: Given a nonnegative
structural constant in representation theory or geometric invariant theory,
such as the well known plethysm constant, decide if it is nonzero (nonvan-
ishing), or rather, if is nonzero after a small relaxation. This flip from the
negative to the positive may be considered to be a nonrelativizable form of
the flip–from the undecidable to the decidable–that underlies the proof of
Gödel’s incompleteness theorem. But the classical diagonalization technique
in Gödel’s result is relativizable [BGS], and hence, not applicable to the P
vs. NP problem. The flip, in contrast, is nonrelativizable. It is furthermore
nonnaturalizable [GCT10]); i.e., it crosses the natural proof barrier [RR]
that any approach to the P vs. NP problem must cross.
We suggest here a plan for implementating the flip; i.e., for showing that
the decision problems above belong to P . This is based on the reduction
in this paper of the complexity-theoretic positivity hypotheses mentioned
above to mathematical positivity hypotheses: specifically, to showing that
there exist positive formulae for the structural constants under consideration
and certain functions associated with them. We also give theoretical and
experimental evidence in support of the latter hypotheses.
Here we say that a formula is positive if its coefficients are nonegative.
The problem finding the positive formulae as above turns out be intimately
related to the analogous problem for the Kazhdan-Lusztig polynomials [KL1]
and the multiplicative structural constants of the canonical (global crystal)
bases [Kas2, Lu2] in the theory of Drinfeld-Jimbo quantum groups. The
known solution to the latter problem [KL2, Lu2] depends on the Riemann
hypothesis over finite fields, proved in [Dl], and the related results in [BBD].
Thus the flip and the reduction here together roughly say that the valid-
ity of the P 6= NP conjecture in characteristic zero is intimately linked
to the Riemann hypothesis over finite fields and related problems. This is
illustrated in Figure 1.1; the question marks there indicate unsolved prob-
lems. It seems that substantial extension of the techniques related to the
Riemann hypothesis over finite fields may be needed to prove the required
mathematical positivity hypotheses here. We do not have the necessary
mathematical expertize for this task. But it is our hope that the experts in
algebraic geometry and representation theory will have something to say on
this matter.
It may be conjectured that the flip paradigm would also work in the
context of the usual P vs. NP problem over F2 (the boolean field) or the
finite field Fp. But implementation of the flip over a finite field is expected
to be much harder than in characteristic zero. That is why we focus on
characteristic zero here, deferring discussion of the problems that arise over
finite field to [GCT11].
Now we turn to a more detailed exposition of the main results in this
paper and of Figure 1.1.
Acknoledgements
We are grateful to the authors of [BOR] for pointing out an error in the
saturation hypothesis (SH) in the earlier version of this paper. It has been
corrected in this version with appropriate relaxation without affecting the
overall approach of GCT (cf. Section 1.6 and also [GCT6erratum]). We
are also grateful to Peter Littelmann for bringing the reference [Dh] to our
Complexity theoretic negative hypotheses (lower bound problems)
The flip
Complexity theoretic positive hypotheses (upper bound problems)
The reduction in this paper|
Mathematical positivity hypotheses |
(?) The Riemann hypothesis over finite fields, related problems and their extensions
Figure 1.1: Pictorial depiction of the basic plan for implementing the flip
attention, to H. Narayanan for suggesting the use of [KB] in the proof of
Theorem 3.1.1 and bringing the positivity conjecture in [DM2] to our atten-
tion, and to Madhav Nori for a helpful discussion. The experimental results
in Chapter 6 were obtained using Latte [DHHH].
1.1 The decision problems
We now describe the relevant decision problems in representation theory
and algebraic geometry. The actual decision problems that arise in the flip
(cf. the second box in Figure 1.1) are relaxed versions of these problems
described later (cf. Hypothesis 1.1.6).
Problem 1.1.1 (Decision version of the Kronecker problem)
Given partitions λ, µ, π, decide nonvanishing of the Kronecker coefficient
. This is the multiplicity of the irreducible representation (Specht mod-
ule) Sπ of the symmetric group Sn in the tensor product Sλ ⊗ Sµ.
Equivalently [FH], let H = GLn(C) × GLn(C) and ρ : H → G =
GL(Cn ⊗ Cn) = GLn2(C) the natural embedding. Then k
is the multi-
plicity of the H-module Vλ(GLn(C))⊗Vµ(GLn(C)) in the G-module Vπ(G),
considered as an H-module via the embedding ρ.
Here Vλ(GLn(C)) denotes the irreducible representation (Weyl module)
of GLn(C) corresponding to the partition λ; Vπ(G) is the Weyl module of
G = GLn2(C).
Problem 1.1.1 is a special case of the following generalized plethysm
problem.
Problem 1.1.2 (Decision version of the plethysm problem)
Given partitions λ, µ, π, decide nonvanishing of the plethysm constant
. This is the multiplicity of the irreducible representation Vπ(H) of H =
GLn(C) in the irreducible representation Vλ(G) of G = GL(Vµ), where Vµ =
Vµ(H) is an irreducible representation H. Here Vλ(G) is considered an H-
module via the representation map ρ : H → G = GL(Vµ).
(Decision version of the generalized plethysm problem)
The same as above, allowing H to be any connected reductive group.
This is a special case of the following fundamental problem of represen-
tation theory (characteristic zero):
Problem 1.1.3 (Decision version of the subgroup restriction problem)
Let G be connected reductive group, H a reductive group, possibly discon-
nected, and ρ : H → G an explicit, polynomial homomorphism (as defined
in Section 3.4). Here H will generally be a subgroup of G, and ρ its em-
bedding. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an
irreducible representation of G. Here π and λ denote the classifying labels
of the irreducible representations Vπ(H) and Vλ(G), respectively. Let m
the multiplicity of Vπ(H) in Vλ(G), considered as an H-module via ρ.
Given specifications of the embedding ρ and the labels λ, π, as described
in Section 3.4, decide nonvanishing of the multiplicity mπ
All reductive groups in this paper are over C. The reductive groups
that arise in GCT in characteristic zero are: the general and special linear
groups GLn(C) and SLn(C), algebraic tori, the symmetric group Sn, and
the groups formed from these by (semidirect) products. The reader may
wish to focus on just these concrete cases, since all main ideas in this paper
are illustrated therein.
Problem 1.1.3 is, in turn, a special case of the following most general
problem.
Problem 1.1.4 (Decision problem in geometric invariant theory)
Let H be a reductive group, possibly disconnected, X a projective H-
variety (H-scheme), i.e., a variety with H-action. Let ρ denote this H-
action. Let R = ⊕dRd be the homogeneous coordinate ring of X. Assume
that the singularities of spec(R) are rational.
We assume that X and ρ have special properties (as described in Sec-
tion 3.5), so that, in particular, they have short specifications. Let Vπ(H)
be an irreducible representation of H. Let sπd be the multiplicity of Vπ(H) in
Rd, considered as an H-module via the action ρ.
Given d, π, the specifications of X and ρ, decide nonvanishing of the
multiplicity sπ
This last problem is hopeless for general X. Indeed the usual specifi-
cation of X, say in terms of the generators of the ideal of its appropriate
embedding, is so large as to make this problem meaningless for a general
X. But the instances of this decision problem that arise in GCT are for the
following very special kinds of projective H-varieties X, which, in particular,
have small specifications (Section 3.5):
1. G/P , where G is a connected, reductive group, P ⊆ G its parabolic
subgroup, and H ⊆ G a reductive subgroup with an explicit polyno-
mial embedding. Problem 1.1.3 reduces to this special case of Prob-
lem 1.1.4; cf. Section 3.5.
2. Class varieties [GCT1, GCT2], which are associated with the funda-
mental complexity classes such as P and NP . They are very special
like G/P , with conjecturally rational singularities [GCT10]. Each class
variety is specified by the complexity class and the parameters of the
lower bound problem under consideration. Briefly, the P vs. NP prob-
lem in characteristic zero is reduced in [GCT1, GCT2] to showing that
the class variety corresponding to the complexity class NP and the pa-
rameters of the lower bound problem (such as the input size) cannot
be embedded in the class variety corresponding to the complexity class
P and the same parameters. Efficient criteria for the decision prob-
lems stated above are needed to construct explicit obstructions [GCT2]
to such embeddings, thereby proving their nonexistence. Specifically,
Problems 1.1.3 and 1.1.4 are the decision problems associated with
Problems 2.5 and 2.6 in [GCT2], respectively. See Sections 7.6-7.7 for
a brief review of this story.
For these varieties Problem 1.1.4 turns out to be qualitatively similar to
Problem 1.1.3 (cf. Section 3.5 and [GCT2, GCT10]). For this reason, the
Kronecker and the plethysm problems, which lie at the heart of the subgroup
restriction problem, can be taken as the main prototypes of the decision
problems that arise here.
One can now ask:
Question 1.1.5 Do the decision problems above (Problems 1.1.1-1.1.3 and
Problem1.1.4, when X therein is G/P or a class variety) belong to P? That
is, can the nonvanishing of any of structural constants in these problems
be decided in poly(〈x〉) time, where x denotes the input-specification of the
structural constant and 〈x〉 its bitlength?
For Problem 1.1.2, the input specification for the plethysm constant aπλ,µ
is given in the form of a triple x = (λ, µ, π). Here the partition λ is specified
as a sequence of positive integers λ1 ≥ λ2 ≥ · · ·λk > 0 (the zero parts of
the partition are suppressed); k is called the height or length of λ, and is
denoted by ht(λ). The bitlength 〈λ〉 is defined to be the total bitlength of
the integers λr’s. The bitlength 〈x〉 is defined to be 〈λ〉 + 〈µ〉 + 〈π〉. A
detailed specification of the input specification x and its bitlength 〈x〉 for
the other problems is given in Section 3.3.
For the reasons described in Section 1.6, Question 1.1.5 may not have
an affirmative answer in general; i.e., these problems may not be in P in
their strict form stated above. The following main conjectural complexity-
theoretic positivity hypothesis governing the flip says that the relaxed forms
of these decision problems described in Section 3.3 belong to P . As we shall
see in Chapter 7, these relaxed forms suffice for the purposes of the flip.
Hypothesis 1.1.6 (PHflip) The relaxed forms (cf. Section 3.3) of Prob-
lems 1.1.1, 1.1.2, 1.1.3, and the special cases of Problem 1.1.4, when X
therein is G/P or a class variety–which together include all decision prob-
lems that arise in the flip–belong to the complexity class P .
This means nonvanishing of any of these structural constants, modulo a
small relaxation (as described in Section 3.3), can be decided in poly(〈x〉)
time, where x denotes the input-specification of the structural constant and
〈x〉 its bitlength.
The phrase “modulo a small relaxation” in the relaxed form of the
plethysm problem means the following:
(a) Let h = dimG + htλ + htπ, where dim(G) is the dimension of the
group G in Problem 1.1.2. Then there exist absolute nonnegative constants
c and c′, independent of λ, µ and π, such that nonvanishing of the relaxed
(stretched) plethysm constant abπ
bλ,bµ
, for any positive integral relaxation
parameter b > chc
, can be decided in O(poly(〈λ〉, 〈µ〉, 〈π〉, 〈b〉)) time, where
〈b〉 denotes the bitlength b. The notation poly(〈λ〉, 〈µ〉, 〈π〉, 〈b〉) here means
bounded by a polynomial of constant degree in 〈λ〉, 〈µ〉, 〈π〉 and 〈b〉. In
particular, the time is O(poly(〈λ〉, 〈µ〉, 〈π〉) if the relaxation parameter b is
small; i.e. if its bitlength 〈b〉 is O(poly(〈λ〉, 〈µ〉, 〈π〉)). (Observe that the bit
length of h is O(poly(〈λ〉, 〈µ〉, 〈π〉)).)
(b) There exists a polynomial time algorithm for deciding nonvanishing of
, which works correctly on almost all λ, µ and π. Here polynomial time
means O(poly(〈λ〉, 〈µ〉, 〈π〉) time. The meaning of “correctly on almost all”
is specified in Hypothesis 1.6.5 below.
A detailed specification of the relaxation, i.e., the meaning of the phrase
“modulo a small relaxation” for the other problems is given in Section 3.3.
The structural constants in Problems 1.1.1-1.1.3 are of fundamental im-
portance in representation theory. The kronecker and the plethysm con-
stants in Problems 1.1.1 and 1.1.2, in particular, have been studied inten-
sively; see [FH, Mc, St4] for their significance. There are many known
formulae for these structural constants based on on the character formulae
in representation theory. Several formulae for the characters of connected,
reductive groups are known by now [FH], starting with the Weyl character
formula. For the symmetric group, there is the Frobenius character formula
[FH], for the general linear group over a finite field, Green’s formula [Mc],
and for finite simple groups of Lie type, the character formula of Deligne-
Lusztig [DL], and Lusztig [Lu1]. (Finite simple groups of Lie type, other
than GLn(Fq), are not needed in GCT.)
One obvious method for deciding nonvanishing of the structural con-
stants in Problems 1.1.1-1.1.4 is to compute them exactly. But all known al-
gorithms for exact computation of the structural constants in Problems 1.1.1-
1.1.3 take exponential time. This is expected, since this problem is #P -
complete. In fact, even the problem of exact computation of a Kostka
number, which is a very special case of these structural constants, is #P -
complete [N]. This means there is no polynomial time algorithm for com-
puting any of them, assuming P 6= NP .
Of course, there are #P -complete quantities–e.g. the permanent of a
nonnegative matrix [V]–whose nonvanishing can still be decided in polyno-
mial time [Sc]. But the decision problems above are of a totally different
kind and, at the surface, appear to have inherently exponential complexity.
This is because the dimensions of the irreducible representations that occur
in their statements can be exponential in the ranks of the groups involved
and the bit lengths of the classifying labels of these representations. For
example, the dimension of the Weyl module Vλ(GLn(C)) can be exponen-
tial in n and the bit length of the partition λ. Furthermore, the number
of terms in any of the preceding character formulae is also exponential. All
these decisions problems ask if one exponential dimensional representation
can occur within another exponential dimensional representation. To solve
them, it may seem necessary to take a detailed look into these representa-
tions and/or the character formulae of exponential complexity. Hence, it
seemed hard to believe that nonvanishing of these structural constants can,
nevertheless, be decided in polynomial time (modulo a small relaxation).
This constituted the main philosophical obstacle in the course of GCT.
1.2 Deciding nonvanishing of Littlewood-Richardson
coefficients
The first result, which indicated that this obstacle may be removable, came
in the wake of the saturation theorem of Knutson and Tao [KT1]. This
concerns the following special case of Problem 1.1.3, with G = H ×H, the
embedding ρ : H → G being diagonal.
Problem 1.2.1 (Littlewood-Richardson problem)
Given a complex semisimple, simply connected Lie group H, and its
dominant weights α, β, λ, decide nonvanishing of a generalized Littelwood-
Richardson coefficient cλ
. This is the multiplicity of the irreducible repre-
sentation Vλ(H) of H in the tensor product Vα(H)⊗ Vβ(H).
It was shown in [GCT3, KT2, DM2] independently that nonvanishing of
the Littlewood-Richardson coefficient of type A can be decided in polyno-
mial time; i.e., polynomial in the bit lengths of α, β, λ. Furthermore, the
algorithm in [GCT3] works in strongly polynomial time in the terminology
of [GLS]; cf. Section 2.1. The three main ingradients in this result are:
1. PH1: The Littlewood-Richarson rule, which goes back to 1940’s, and
whose most important feature is that it is positive–i.e., it involves no
alternating signs as in character-based formulae–and its strengthening
in [BZ], which gives a positive, polyhedral formula for the Littlewood-
Richardson coefficient as the number of integer points in a polytope;
this can be the BZ-polytope [BZ] or the hive polytope [KT1]. We
shall refer to this positivity property as the first positivity hypothesis
(PH1).
2. The polynomial and strongly polynomial time algorithms for linear
programming [Kh, Ta], and
3. SH: The saturation theorem of Knutson and Tao [KT1]. This says
that cλ
is nonzero if cnλ
nα,nβ
is nonzero for any n ≥ 1. We shall refer
to this saturation property as the saturation hypothesis (SH).
Brion [Z] observed that the verbatim translation of the saturation prop-
erty in [KT1] fails to hold for the the generalized Littlewood-Richardson
coefficients of types B, C, D (it also fails for the Kronecker coefficients, as
well as the plethysm constants [Ki]). Hence, the algorithms in [GCT3, KT2,
DM2] do not work in types B, C and D. Fortunately, this situation can
be remedied. It is shown in [GCT5] that nonvanishing of the generalized
Littewood-Richardson coefficient cλα,β of arbitrary type can be decided in
(strongly) polynomial time, assuming the positivity conjecture of De Loera
and McAllister [DM2]. This conjectural hypothesis, based on considerable
experimental evidence, is as follows. Let
c̃λα,β(n) = c
nα,nβ (1.1)
be the stretching function associated with the Littlewood-Richardson co-
efficient cλα,β . It is known to be a polynomial in type A [Der, Ki], and a
quasi-polynomial, in general [BZ, Dh, DM2]. Recall that a fuction f(n) is
called a quasi-polynomial if there exist l polynomials fj(n), 1 ≤ j ≤ l, such
that f(n) = fj(n) if n = j mod l. Here l is supposed to be the smallest such
integer, and is called the period of f(n). The period of c̃λ
(n) for types
B,C,D is either 1 or 2 [DM2]. In general, it is bounded by a fixed constant
depending on the types of the simple factors the Lie algebra.
Definition 1.2.2 We say that the quasi-polynomial f(n) is strictly posi-
tive, if all coefficients of fj(n), for all j, are nonnegative; i.e., the nonzero
coefficients are positive. In general, we define the positivity index p(f) of
f to be the smallest nonnegative integer such that f(n + p(f)) is strictly
positive. We also say that f(n) is positive with index p(f).
Thus f(n) is strictly positive, iff its positivity index is zero.
With this terminology, the hypothesis mentioned above is the following.
We say a connected reductive group H is classical, if each simple factor of
its Lie algebra H is of type A,B,C or D. We also say that the type of H
or H is classical.
Hypothesis 1.2.3 (PH2): [KTT, DM2] Assume that H in Problem 1.2.1
is classical. Then the Littlewood-Richardson stretching quasi-polynomial
(n) is strictly positive.
We shall refer to this as the second positivity hypothesis (PH2). This
was conjectured by King, Tollu and Toumazet [KTT] for type A, and De
Loera and McAllister for types B,C,D. Since the stretching function above
is a polynomial in type A, the positivity conjecture of King et al clearly
implies the saturation theorem of Knutson and Tao. That is, PH2 implies
SH for type A.
We can formulate an analogue of SH for a Lie algerbra of arbitrary clas-
sical type so that PH2 implies SH for an arbitrary type. For this, we need
to formulate the notion of a saturated quasi-polynomial, which is not con-
tradicted by the counterexamples, mentioned above, to verbatim translation
of the saturation property in [KT1, Ki] to the setting of quasi-polynomials.
Specifically, the notion of saturation in [KT1, Ki] works well if the stretching
function is a polynomial, but not so if it is a quasipolynomial. Let f(n) be
a quasi-polynomial with period l. Let fj(n), 1 ≤ j ≤ l, be the polynomials
such that f(n) = fj(n) if n = j mod l. The index of f , index(f), is defined
to be the smallest j such that the polynomial fj(n) is not identically zero.
If f(n) is identically zero, we let index(f) = 0. If f(1) 6= 0, then clearly
index(f) = 1.
Definition 1.2.4 We say that f(n) is strictly saturated if for any i: fi(n) >
0 for every n ≥ 1 whenever fi(n) is not identically zero. The saturation in-
dex s(f) of f is defined to be the smallest nonnegative integer such that
f(n + s(f)) is strictly saturated. We also say that f(n) is saturated with
index s(f).
Thus f(n) is strictly saturated iff its saturation index is zero. Clearly
the saturation idex is bounded above by the positivity index. Thus if f(n)
is strictly positive, it is strictly saturated. Hence, PH2 (Hypothesis 1.2.3)
implies:
Hypothesis 1.2.5 (SH): The Littlewood-Richardson stretching quasi-polynomial
(n) of arbitary classical type is strictly saturated.
The polynomial time algorithm in [GCT5] works assuming SH as well.
For the Littlewood-Richardson coefficient of type A, the notion of strict
saturation here coincides with the notion of saturation in [KT1] since cλα,β(n)
is a polynomial in that case. Knutson and Tao [KT1] also conjectured
a generalized saturation property for arbitrary types. But that property,
unlike the one defined above, is only conjectured to be sufficient, but not
claimed to be, or expected to be necessary. For this reason, it cannot be
used in the complexity-theoretic applications in this paper.
There is another positivity conjecture for Littlewood-Richardson coeffi-
cients that also implies the saturation theorem of Knutson and Tao. For
this consider the generating function
Cλα,β(t) =
c̃λα,β(n)t
n. (1.2)
It is a rational function since c̃λ
(n) is a quasi-polynomial [St1]. For type
A, if c̃λ
(n) is not identically zero, then Cλ
(t) is a rational function of
d + · · ·+ h0
(1− t)d+1
, (1.3)
since c̃λα,β(n) is a polynomial [St1]. It is conjectured in [KTT] that:
Hypothesis 1.2.6 (PH3:) The coefficients hi’s in eq.(1.3) are nonnegative
(and h0 = 1).
We shall call this the third positivity hypothesis (PH3). It clearly implies SH
for Littlewood-Richardson coefficients of type A. To describe its analogue
for arbitrary classical type we need a definition.
Let F (t) =
n f(n)t
n be the generating function associated with the
quasi-polynomial f(n). It is a rational function [St1].
Definition 1.2.7 We say that F (t) has a positive form, if, when f(n) is
not identically zero, it can be expressed in the form
F (t) =
d + · · ·+ h0
i=0(1− t
ai)di
, (1.4)
where (1) h0 = 1, and hi’s are nonnegative integers, (2) ai’s and di’s are
positive integers, (3)
i di = d+ 1, where d = max deg(fj(n)) is the degree
of f(n).
We define the modular index of this positive form to be max{ai}.
If F (t) has a positive form with a0 = 1, then f(n) is strictly saturated
(Definition 1.2.4); this easily follows from the power series expansion of the
right hand side of eq.(1.4).
The analogue of Hypothesis 1.2.6 for arbitrary classical type is:
Hypothesis 1.2.8 (PH3:) The rational function Cλ
(t) has a positive
form, with a0 = 1, of modular index bounded by a constant depending only
on the types of the simple factors of the Lie algebra of H.
This too implies SH for arbitrary classical type. For types B,C,D, the
constant above is 2. Experimental evidence for this hypothesis is given in
Section 6.1.
The analogue of the PH3, even in the more general q-setting, is known to
hold for the generating function of the Kostant partition function of type A,
and more generally, for a parabolic Kostant partition function; cf. Kirillov
[Ki]. This also gives a support for the PH3 above, given a close relationship
between Littlewood-Richardson coefficients and Kostant partition functions
[FH].
1.3 Back to the general decision problems
It may be remarked that the Littlewood-Richardson problem actually never
arises in the flip. It is only used as a simplest proptotype of the actual (much
harder) problems that arise–namely relaxed forms of Problems 1.1.1-1.1.4.
Now we turn to these problems. The goal is to generalize the preced-
ing results and hypotheses for the Littlewood-Richardson coefficients to the
structural constants that arise in these problems. The problem of finding a
positive, combinatorial formula for the plethysm constant (Problem 1.1.2),
akin to the positive Littlewood-Richardson rule, has already been recog-
nized as an outstanding, classical problem in representation theory [St4]–
the known formulae based on character theory mentioned in Section 1.1
are not positive, because they involve alternating signs. Indeed, existence
of such a formula is a part of the first positivity hypothesis (PH1) below
for the plethysm constant, and this problem is the main focus of the work
in [GCT4, GCT7, GCT8, GCT9]. In view of the intensive work on the
plethym constant in the literature, it has now become clear that the com-
plexity of the plethysm problem (Problem 1.1.2) is far higher than that of
the Littlewood-Richardson problem (Problem 1.2.1). This gap in the com-
plexity is the main source of difficulties that has to be addressed. We now
state the main ingradients in the plan in this paper to show that the relaxed
forms of Problems 1.1.1, 1.1.2, 1.1.3, and 1.1.4, with X = G/P or a class
variety, belong to P .
1.4 Saturated and positive integer programming
First, we formulate a general algorithmic paradigm of saturated and positive
integer programming that can be applied in the context of these problems.
Let A be an m×n integer matrix, and b an integral m-vector. An integer
programming problem asks if the polytope P : Ax ≤ b contains an integer
point. In general, it is NP-complete. We want to define its relaxed version,
which will turn out to have a polynomial time algorithm.
We allow m, the number of constraints, to be exponential in n. Hence,
we cannot assume that A and b are explicitly specified. Rather, it is assumed
that the polytope P is specified in the form of a (polynomial-time) separation
oracle in the spirit of Grötschel, Lovász and Schrijver [GLS]; cf. Section 2.3.
Given a point x ∈ Rn, the separation oracle tells if x ∈ P , and if not, gives
a hyperplane that separates x from P .
Let fP (n) be the Ehrhart quasi-polynomial of P [St1]. By definition,
fP (n) is the number of integer points in the dilated polytope nP .
An integer programming problem is called saturated, if
1. The specification of P also contains a number sie(P ), called the sat-
uration index estimate, with the guarantee that the saturation in-
dex s(fP ) ≤ sie(P ); cf. Definition 1.2.4. In particular, this means
fP (n+ sie(P )) is strictly saturated.
2. the goal of the problem is to give an efficient algorithm to decide if,
given an integral relaxation parameter c > sie(P ), if cP contains an
integer point.
The algorithm has to work only for relaxation parameters c > sie(P ). In
particular, if sie(P ) ≥ 1, the algorithm problem does not have to determine
if P contains an integer point.
An integer programming problem is called positive, if
1. the specification of P also contains a number pie(P ), called the pos-
itivity index estimate, with the guarantee that the positivity index
p(fP ) ≤ pie(P ); cf. Definition 1.2.2. In particular, this means fP (n+
pie(P )) is strictly positive.
2. the goal of the problem is to give an efficient algorithm to decide if,
given an integral relaxation parameter c > pie(P ), if cP contains an
integer point.
Again, the algorithm has to work only for relaxation parameters c > pie(P ).
Since s(fP ) ≤ p(fP ), a positive integer programming problem is also satu-
rated.
The following is the main complexity-theoretic result in this paper.
Theorem 1.4.1 (cf. Section 3.1)
1. Index of the Ehrhart quasi-polynomial fP (n) of a polytope P presented
by a separation oracle can be computed in oracle-polynomial time, and
hence, in polynomial time, assuming that the oracle works in polyno-
mial time.
2. A saturated, and hence positive, integer programming problem has a
polynomial time algorithm.
3. Suppose the polytopes P ’s that arise in a specific decision problem have
the following property: whenever P is nonempty, the Ehrhart quasi-
polynomial fP (n) is “almost always” strictly saturated. Then there
exists a polynomial time algorithm for deciding if P contains an integer
point that works correctly “almost always”.
The meaning of the phrase “almost always” in the context of the decision
problems in this paper will be specified later (cf. Theorem 3.1.1).
It may be remarked that the index as well as the period of the Ehrhart
quasi-polynomial can be exponential in the bit length of the specification
of P . In contrast to the polynomial time algorithm above to compute the
index, the known algorithms to compute the period (e.g. [W]) take time
that is exponential in the dimension of P . It may be conjectured that one
cannot do much better: i.e., the period, unlike the index here, cannot be
computed in polynomial time, in fact, even in 2o(dim(P )) time.
The algorithm in Theorem 1.4.1 is based on the separation-oracle-based
linear programming algorithm of Grötschel, Lovász and Schrijver [GLS], and
a polynomial time algorithm for computing the Smith normal form [KB].
The paradigm of saturated integer programming is useful when one
knows, a priori, a good estimate for the saturation index of the polytope
under consideration, or when the saturation index is almost always zero.
For example, if P is the hive polypolype for the Littlewood-Richardson co-
efficient (type A), then sie(P ) = 0, by the saturation theorem [KT1], and
pie(P ) = 0, by PH2 (Hypothesis 1.2.3). For the polytopes P that would
arise in this paper, sie(P ) and pie(P ) would in general be nonzero, but con-
jecturally always small, and sie(P ) would conjecturally be almost always
zero.
1.5 Quasi-polynomiality, positivity hypotheses, and
the canonical models
The basic goal now is to use Theorem 1.4.1 to get polynomial time algorithms
to decide nonvanishing, modulo small relaxation, of the structural constants
in Problems 1.1.1, 1.1.2, 1.1.3 and 1.1.4, with X = G/P or a class variety.
The main results in this paper which go towards this goal are as follows.
Quasi-polynomiality
We associate stretching functions with the structural constants in Prob-
lems 1.1.1-1.1.4, akin to the stretching function c̃λα,β(n) in eq.(1.1) associ-
ated with the Littlewwod-Richardson coefficient, and show that they are
quasipolynomials; cf. Chapter 4. (But their periods need not be constants,
as in the case of Littlewood-Richardson coefficients; in fact, they may be
exponential in general.) In particular, this proves Kirillov’s conjecture [Ki]
for the plethysm constants. The proof is an extension of Brion’s remarkable
proof (cf. [Dh]) of quasi-polynomiality of the stretching function associ-
ated with the Littlewood-Richardson coefficient. The main ingradient in
the proof is Boutot’s result [Bou] that singularities of the quotient of an
affine variety with rational singularities with respect to the action of a re-
ductive group are also rational. This is a generalization of an earlier result
of Hochster and Roberts [Ho] in the theory of Cohen-Macauley rings.
Saturation and positivity hypotheses
Using the stretching quasipolynomials above, we formulate (cf. Section 3.3)
analogues of the saturation and positivity hypotheses SH, PH1,PH2,PH3 in
Section 1.2 for the structural constants in Problems 1.1.1-1.1.3 and Prob-
lem 1.1.4, with X = G/P or a class variety. As for Littlewood-Richardson
coefficients, it turns out that PH2 implies SH. The hypotheses PH1 and SH
(more strongly, PH2) together imply that the problem of deciding nonvan-
ishing of the structural constant in any of these problems, modulo a small
relaxation, can be transformed in polynomial time into a saturated (more
strongly, positive) integer programming problem, and hence, can be solved
in polynomial time by Theorem 1.4.1. In particular, this shows that all
the relaxed decision problems that arise in flip (cf. Hypothesis 1.1.6) have
polynomial time algorithms, assuming these positivity hypotheses. Though
these algorithms are elementary, the positivity hypotheses on which their
correctness depends turn out to be nonelementary. They are intimately
linked to the fundamental phenomena in algebraic geometry and the theory
of quantum groups, as we shall see.
We also give theoretical and experimental results in support of these
hypotheses; cf. Chapter 4-6.
Canonical models
The proofs of quasi-polynomiality mentioned above also associate with each
structural constant under consideration a projective scheme, called the canon-
ical model, whose Hilbert function coicides with the stretching quasi-polynomial
associated with that structural constant, akin to the model associated by
Brion [Dh] with the Littlewood-Richardson coefficient. These canonical
models play a crucial role in the approach to the posivity hypotheses sug-
gested in Section 1.7.
1.6 The plethysm problem
We now give precise statements of these results and hypotheses for the
plethysm problem (Problem 1.1.2). It is the main prototype in this paper,
which illustrates the basic ideas. Precise statements for the more general
Problems 1.1.3 and 1.1.4 appear in Section 3.3.
As for the Littlewood-Richardson coefficients (cf.(1.1)), Kirillov [Ki] as-
sociates with a plethysm constant aπ
a stretching function
ãπλ,µ(n) = a
nλ,µ, (1.5)
and a generating function
Aπλ,µ(t) =
anπnλ,µt
(Note that µ is not stretched in these definitions.)
He conjectured that Aπ
(t) is a rational function. This is verified here
in a stronger form:
Theorem 1.6.1 (a) (Rationality) The generating function Aπλ,µ(t) is ratio-
(b) (Quasi-polynomiality) The stretching function ãπ
(n) is a quasi-polynomial
function of n. This is equivalent to saying that all poles of Aπλ,µ(t) are roots
of unity, and the degree of the numerator of Aπλ,µ(t) is strictly smaller than
that of the denominator.
(c) There exist graded, normal C-algebras S = S(aπ
) = ⊕nSn, and T =
T (aπ
) = ⊕nTn such that:
1. The schemes spec(S) and spec(T ) are normal and have rational singu-
larities.
2. T = SH , the subring of H-invariants in S, where H = GLn(C) as in
Problem 1.1.2,
3. The quasi-polynomial ãπλ,µ(n) is the Hilbert function of T . In other
words, it is the Hilbert function of the homogeneous coordinate ring of
the projective scheme Proj(T ).
(d) (Positivity) The rational function Aπλ,µ(t) can be expressed in a positive
form:
Aπλ,µ(t) =
h0 + h1t+ · · ·+ hdt
j(1− t
a(j))d(j)
, (1.6)
where a(j)’s and d(j)’s are positive integers,
j d(j) = d + 1, where d is
the degree of the quasi-polynomial ãπ
(n), h0 = 1, and hi’s are nonnegative
integers.
The specific rings S(aπλ,µ) and T (a
λ,µ) constructed in the proof of The-
orem 1.6.1 are very special. We call them canonical rings associated with
the plethysm constant aπ
. We call Y (aπ
) = Proj(S(aπ
)), and Z(aπ
Proj(T (aπ
)) the canonical models associated with aπ
. The canonical rings
are their homogenous coordinate rings.
It may be remarked that the analogue of Theorem 1.6.1 (b) for Littlewood-
Richardson coefficients has an elementary polyhedral proof. Specifically, the
Littlewood-Richardson stretching function c̃λα,β(n) of any type is a quasi-
polynomial since it coincides with the Ehrhart quasi-polynomial of the BZ-
polytope [BZ]. Similarly, the analogue of Theorem 1.6.1 (d) for Littlewood-
Richardson coefficients follows from Stanley’s positivity theorem for the
Ehrhart series of a rational polytope (which is implicit in [St3]). These
polyhderal proofs cannot be extended to the plethysm constant at this point,
since no polyhedral expression for them is known so far–in fact, this is a part
of the conjectural positivity hypothesis PH1 below. In contrast, Brion’s
proof in [Dh] of quasi-polynomiality of c̃λα,β(n) can be extended to prove
Theorem 1.6.1 since it does not need a polyhedral interpretation for aπλ,µ.
But Boutot’s result [Bou] that it relies on is nonelementary (because it needs
resolution of singularities in characteristic zero [Hi], among other things).
We also give an elementary (nonpolyhedral proof) for Theorem 1.6.1 (a) (ra-
tionality). But this does not extend to a proof of quasipolynomiality for all
n, which turns out to be a far delicate problem. It is crucial in the context
of saturated integer programming.
Theorem 1.6.2 (Finitely generated cone)
For a fixed partition µ, let Tµ be the set of pairs (π, λ) such that the
irreducible representation Vπ(H) of H = GLn(C) occurs in the irreducible
representation Vλ(G) of G = GL(Vµ(H)) with nonzero multiplicity. Then
Tµ is a finitely generated semigroup with respect to addition.
This is proved by an extension of Brion and Knop’s proof of the analogous
result for Littlewood-Richardson coefficients based on invariant theory. In
the case of Littlewood-Richardson coefficients, this again has an elementary
polyhedral proof [Z].
Theorem 1.6.3 (PSPACE)
Given partitions λ, µ, π, the plethysm constant aπ
can be computed in
poly(〈λ〉, 〈µ〉, 〈π〉) space.
The main observation in the proof of Theorem 1.6.3 is that the oldest
algorithm for computing the plethysm constant, based on the Weyl character
formula, can be efficiently parallelized so as to work in polynomial parallel
time using exponentially many processors. After this, the result follows from
the relationship between parallel and space complexity classes. It may be
remarked that the known algorithms for computing aπ
in the literature–
e.g., the one based on Klimyk’s formula [FH]–take exponential time as well
as space.
Theorems 1.6.1, 1.6.2 and 1.6.3 lead to the following conjectural sat-
uration and positivity hypotheses for the plethysm constant. These are
analogues of PH1,PH2,PH3, SH in Section 1.2 for Littlewood-Richardson
coefficients.
Hypothesis 1.6.4 (PH1)
For every (λ, µ, π) there exists a polytope P = P π
⊆ Rm such that:
(1) The Ehrhart quasi-polynomial of P coincides with the stretching quasi-
polynomial ãπ
(n) in Theorem 1.6.1. (This means P is given by a linear
system of the form
Ax ≤ b, (1.7)
where A does not depend on λ and π and b depends only on λ and π in a
homogeneous, linear fashion.) In particular,
aπλ,µ = φ(P ), (1.8)
where φ(P ) is equal to the number of integer points in P .
(2) The dimension m of the ambient space, and hence the dimension of P
as well, and the bitlength of every entry in A are polynomial in the bitlength
of µ and the heights of λ and π.
(3) Whether a point x ∈ Rm lies in P can be decided in poly(〈λ〉, 〈µ〉, 〈π〉, 〈x〉)
time. That is, the membership problem belongs to the complexity class P .
If x does not lie in P , then this membership algorithm also outputs, in the
spirit of [GLS], the specification of a hyperplane separating x from P .
The first statement here, in particular, would imply a positive, polyhedral
formula for a
, in the spirit of the known positive polyhedral formulae for
the Littlewood-Richardson coefficients in terms of the BZ- [BZ], hive [KT1]
or other types of polytopes [Dh]. It would also imply polyhedral proofs for
Theorem 1.6.1 (a), (b), (d), and Theorem 1.6.2. Conversely, Theorem 1.6.1
(a), (b), (d), and Theorem 1.6.2 constitute a theoretical evidence for exis-
tence of such a positive polyhedral formula.
The second statement in PH1 is justified by Theorem 1.6.3. Specifi-
cally, it should be possible to compute the number of integer points in P
in PSPACE in view of Theorem 1.6.3. If dim(P ) and m were exponential,
then the usual algorithms for this problem, e.g. Barvinok [Bar], cannot be
made to work in PSPACE. Indeed, it may be conjectured that the number
of integer points in a general polytope P ⊆ Rm can not be computed in
o(m) space.
The number of constraints in the hive [KT1] or the BZ-polytope [BZ]
for the Littlewood-Richardson coefficient cλ
is polynomial in the number
of parts of α, β, λ. In contrast, the number of constraints defining P π
be exponential in the 〈µ〉 and the number of parts of λ and π. But this is
not a serious problem. As long as the faces of the polytope P have a nice
description, the third statement in PH1 is a reasonable assumption. This
has been demonstrated in [GLS] for the well-behaved polytopes in combina-
torial optimization with exponentially many constraints. The situation in
representation theory should be similar, or even better. For example, the
facets of the hive polytope [KT1] are far nicer than the facets of a typical
polytope in combinatorial optimization.
It is known that membership in a polytope is a “very easy” problem.
Formally, if a polytope has polynomially many constraints, this problem
belongs to the complexity class NC ⊆ P [KR], the subclass of problems
with efficient parallel algorithms, which is very low in the usual complexity
hierarchy. Even if the number of constraints of P πλ,µ in PH1 is exponen-
tial, the membership problem may still be conjectured to be in NC (cf.
Remarkrnc)–which would be “very easy” compared to the decision problem
we began with (Problem 1.1.2). For this reason, PH1 is primarily a mathe-
matical positivity hypothesis as against PHflip (Hypothesis 1.1.6), and the
positive, polyhedral formula for aπ
in (1.8) is its main content.
The remaining positivity hypotheses are purely mathematical. They
generalize SH,PH2 and PH3 for the Littlewood-Richardson coefficients to
the plethysm constants. We turn their specification next. We can begin
by asking if the stretching quasipolynomial ãπ
(n) is strictly saturated or
positive. This need not be so. The recent article [Ro] shows that strict
saturation need not hold for the Kronecker coefficients, as was conjectured
in the earlier version of this paper. A similar phenomenon was also reported
in [GCT7, GCT8], where it was observed that the structural constants of
the nonstandard quantum groups associated with the plethysm problem (of
which the Kronecker problem is a special case) need not satisfy an analogue
of PH2. But it was observed there that the positivity (and hence saturation)
indices of these structural constants are small, though not always zero; eg.
see Figures 30,33,35 in [GCT8]. The same can be expected here. This is
also supported by the experimental evidence in [BOR] where too it may be
observed that the positivity index is small. Furthermore, in the special case
(n = 2) of the Kronecker problem analysed in [BOR], the saturation index
is zero for almost all Kronecker coefficients.
These considerations suggest:
Hypothesis 1.6.5 (SH)
(a): The saturation index (Definition 1.2.4) of ãπ
(n) is bounded by a poly-
nomial in the dimension of G in Problem 1.1.2 and the heights of λ and π.
This means there exist absolute nonnegative constants c and c′, independent
of n, λ, µ and π, such that the saturation index is bounded above by chc
where h = dimG+ htλ+ htπ.
(b): The quasi-polynomial ãπ
(n) is strictly saturated, i.e. the saturation
index is zero, for almost all λ, µ, π. Specifically, the density of the triples
(λ, µ, π) of total bit length N with nonzero aπ
for which the saturation index
is not zero is less than 1/N c
, for any positive constant c′′, as N → ∞.
A stronger form of (a) is:
Hypothesis 1.6.6 (PH2) The positivity index (Definition 1.2.2) of the
stretching quasi-polynomial ãπλ,µ(n) is bounded by a polynomial in the di-
mension of G and the heights of λ and π.
The following is another stronger form of SH (a). For this, we observe
that the positive rational form in Theorem 1.6.1 (d) is not unique. Indeed,
there is one such form for every h.s.o.p. (homogeneous sequence of param-
eters) of the homogenenous coordinate ring S; the a(j)’s in eq.(1.6) are the
degrees of these parameters.
Kirillov asked if the only possible pole of Aπ
is at t = 1–i.e. if a
(n) is
a polynomial. This is not so (cf. Section 6.2). But it may be conjectured that
the structural constants a(j)’s are small. Specifcally, consider an h.s.o.p. of
S with a (lexicographically) minimum degree sequence, and call the (unique)
positive rational form in Theorem 1.6.1 (d) associated with such an h.s.o.p.
minimal. The modular index χ(aπ
) of the plethysm constant is defined to
be the modular index (Definition 1.2.7) of this minimal positive form. Then:
Hypothesis 1.6.7 (PH3)
The function Aπ
(t) associated with aπ
has a positive rational form
with modular index bounded by a polynomial in the dimension of G and the
heights of λ and π.
More specifically, this is so for the minimial positive rational form of
(t) as above; i.e., the modular index χ(aπ
) is itself bounded by a poly-
nomial in the dimension of G and the heights of λ and π.
This is a conjectural analogue of a stronger form of PH3 for Littlewood-
Richardson coefficients (Hypothesis 1.2.6), which says that the modular in-
dex of a Littlewood-Richardson coefficient, defined similarly, is one. PH3
here would imply that the period of Aπ
(t) is smooth–i.e. has small prime
factors–though it may be exponential in the heights of λ, µ, π. It can be
shown that PH3 implies SH (a) (Section 3.3).
The following result addresses the second arrow in Figure 1.1 in the
context of the relaxed decision problem for the plethysm constant:
Theorem 1.6.8 The complexity theoretic positivity hypothesis PHflip (Hy-
pothesis 1.1.6) for the plethysm constant is implied by the mathematical
positivity hypotheses PH1 and SH above. Specifically, assuming PH1 and
(a) Nonvanishing of abπbλ,bµ for any b > ch
c′ , with c, c′, h as in SH, can be
decided in O(poly(〈λ〉, 〈µ〉, 〈π〉, 〈b〉)) time.
(b) There is an O(poly(〈λ〉, 〈µ〉, 〈π〉)) time algorithm for deciding if aπ
nonvanishing, which works correctly on almost all λ, µ and π; almost all
means the same as in SH.
Here (a) follows by applying Theorem 1.4.1 (2) to the polytope P π
PH1, and letting the positivity index estimate for this polytope be chc
; (b)
follows from Theorem 1.4.1 (3).
Evidence for the positivity hypotheses in special cases
Littlewood-Richardson coefficients are special cases of (generalized) plethym
constants. We have already seen that PH1 holds in this case, and that there
is considerable experimental evidence for PH2 and PH3 (Section 1.2). An-
other crucial special case of the plethym problem is the Kronecker prob-
lem (Problem 1.1.1)–in fact, this may be considered to be the crux of the
plethysm problem. It follows from the results in [GCT9] that PH1 holds for
the Kronecker problem when n = 2; the earlier known formulae [RW, Ro]
for the Kronecker coefficient in this case are not positive. It can also be seen
from the experimental evidence in [BOR] that the saturation and positivity
indices of the Kronecker coefficient, for n = 2, are very small, and almost
always zero. We also give in Chapter 6 additional experimental evidence for
PH2 for another basic special case of Problem 1.1.3, with H therein being
the symmetric group.
1.7 Towards PH1, SH, PH2,PH3 via canonial bases
and canonical models
In this section, we suggest an approach to prove PH1, SH, PH2 and PH3 for
the plethysm constant and the analogous hypotheses for the other structural
constants in Problems 1.1.3, and 1.1.4, with X = G/P or a class variety.
In the case of Littlewood-Richardson coefficients of type A, PH1 and SH
have purely combinatorial proofs. But it seems unrealistic to expect such
proofs of the saturation and positivity hypotheses for the plethysm and
other structural constants under consideration here given their substantially
higher complexity.
The approach that we suggest is motivated by the proof of PH1 for
Littlewood-Richardson coefficients of arbitrary types based on the canonical
(local/global crystal) bases of Kashiwara and Lusztig for representations of
Drinfeld-Jimbo quantum groups [Dh, Kas2, Li, Lu2, Lu4]. By a Drinfeld-
Jimbo quantum group we shall mean in this paper quantization Gq of a
complex, semisimple group G as in [RTF] that is dual to the Drinfeld-Jimbo
quantized enveloping algebra [Dri]. Canonical bases for representions of a
Drinfeld-Jimbo quantum group in type A are intimately linked [GrL] to the
Kazhdan-Lusztig basis for Hecke algebras [KL1, KL2]. A starting point for
the approach suggested here is:
Observation 1.7.1 (PH0) The homogeneous coordinate rings of the canon-
ical models associated by Brion with the Littlewood-Richardson coefficients
have quantizations endowed with canonical bases as per Kashiwara and Lusztig.
This is a consequence of the work of Kashiwara [Kas3] and Lusztig [Lu3,
Lu4]; see Proposition 4.2.1 for its precise statment. This is why we call the
models here canonical models.
We shall refer to the property above as the zeroeth positivity hypothesis
PH0. Positivity here refers to the deep characteristic positivity property of
the canonical basis proved by Lusztig: namely its multiplicative and comul-
tiplicative structure constants are nonnegative. For this reason, we say that
a canonical basis is positive. Similar positivity property is also known for
the Kazhdan-Lusztig basis [KL2]. The proofs of these positivity properties
are based on the Riemann hypothesis over finite fields (Weil conjectures)
[Dl] and the related work of Beilinson, Bernstein, Deligne [BBD].
The property above is called PH0 because it implies PH1 for Littlewood-
Richardson coefficients of arbitrary types. Specifically, the latter is a formal
consequence of the abstract properties of these canonical bases and is inti-
mately related to their positivity; cf. Section 4.2.1, and [Dh, Kas2, Li, Lu4].
The saturation hypothesis SH in type A [KT1] is a refined property of the
polyhedral formulae in PH1. In Section 4.2 we suggest an approach to prove
SH, PH2 and PH3 for arbitrary types based on the properties of these canon-
ical bases. All this indicates that for the Littlewood-Richardson problem
PH1, SH, PH2 and PH3 are intimately linked to PH0.
This suggests the following approach for proving PH1, SH, PH2 and PH3
for the plethysm and other structural constants under consideration in this
paper (cf. Section 4.2.2):
1. Construct quantizations of the homogeneous coordinate rings of the
canonical models associated with these structural constants,
2. Show that they have canonical bases in some appropriate sense thereby
extending PH0 to this general setting.
3. Prove PH1, SH, PH2, and PH3 by a detailed analysis and study of
these canonical bases as per this extended PH0, just as in the case of
Littlewood-Richardson coefficients.
Pictorially, this is depicted in Figure 1.2.
Quantizations of the homogeneous coordinate rings of the canonical
models associated with Littlewood-Richardson coefficients and their posi-
tive canonical bases are constructed using the theory Drinfeld-Jimbo quan-
tum group. In type A, it is intimately related to the theory of Hecke al-
gebras. But, as expected, the theories of Drinfeld-Jimbo quantum groups
and Hecke algebras do not work for the plethysm problem. What is needed
is a quantum group and a quantized algebra that can play the same role
in the plethysm problem that the Drinfeld-Jimbo quantum group and the
Hecke algebra play in the Littlewood-Richardson problem. These have been
constructed in [GCT4] for the Kronecker problem (Problem 1.1.1) and in
[GCT7] for the generalized plethysm problem (Problem 1.1.2). We shall call
them nonstandard quantum groups and nonstandard quantized algebras; cf.
Section 4.3 for their brief overview. In the special case of the Littlewood-
Richardson problem, these specialize to the Drinfeld-Jimbo quantum group
and the Hecke algebra, respectively. The article [GCT8] gives conjecturally
correct algorithms to construct canonical bases of the matrix coordinate
rings of the nonstandard quantum groups and of nonstandard algebras that
have conjectural positivity properties analogous to those of the canonical
Construction of quantizations of the coordinate rings of canonical models
Construction of canonical bases for these quantizations (PH0)
Positivity and saturation hypotheses PH1, SH |
Polynomial-time algorithms for the relaxed decision problems
Figure 1.2: Pictorial depiction of the approach
(global crystal) bases, as per Kashiwara and Lusztig, of the coordinate ring
of the Drinfeld-Jimbo quantum group, and the Kazhdan-Lusztig basis of the
Hecke algebra. These conjectures lie at the heart of the approach suggested
here, since they are crucial for the extension of PH0 (cf. Figure 1.2) to the
general setting here. Their verification seems to need substantial extension
of the work surrounding the Riemann hypothesis over finite fields mentioned
above.
1.8 Basic plan for implementing the flip
The main application of the results and hypotheses in this paper in the
context of the flip is the following result. As mentioned in Section 1.1, and
described in more detail in Sections 7.6-7.7, each lower bound problem, such
as the P vs. NP problem over C, is reduced in [GCT1, GCT2] to the prob-
lem of proving obstructions to embeddings among the class varieties that
arise in the problem. In Chapter 7 we define a robust obstruction, which is
an obstruction that is well behaved with respect to relaxation, and whose
validity (correctness) depends only on an appropriate PH1 but not SH. It is
conjectured that in each of the lower bound problems under consideration,
robust obstructions exist (Section 7.6.6). In the lower bound problems un-
der consideration, ultimately one is only interested in proving existence of
obstructions. So one may as well search for only robust obstructions.
Theorem 1.8.1 (cf. Chapter 7) Consider the P vs. NP or the NC vs.
P#P problem over C [GCT1]. Assume that the homogeneous coordinate
rings of the relevant class varieties [GCT1, GCT2] in this context have ra-
tional singularities. Also assume that the structural constants associated
with these class varieties satisfy analogous PH1 as specified in Chapter 7.
Then:
(a) The problem of verifying a robust obstruction in each of these problems
belongs to P , so also the relaxed form of the problem of verifying any ob-
struction (not necessarily robust).
(b) There exists an explicit family of robust obstructions in each of these
problems assuming an additional hypothesis OH specified in Chapter 7; the
meaning of the term explicit is also given there.
(b) The problem of deciding existence of a geometric obstruction also belongs
to P , assuming a stronger form of PH1 specified in Chapter 7. Here geomet-
ric obstruction is a simpler type of robust obstruction, defined in Chapter 7,
which is conjectured to exist in the lower bound problems under considera-
tion.
For a precise statement of this theorem, see Chapter 7.
This theorem needs only PH1, but not SH, which is only needed to
argue why robust obstructions should exist (Section 7.6.6), and furthermore,
it is only needed for Problems 1.1.1-1.1.3 and not for the GIT Problem
1.1.4. Thus PH1 is the main positivity hypothesis of GCT in the context
proving existence of (robust) obstructions for the lower bound problems
under consideration.
A basic plan for implementing the flip suggested by the considerations
above is summarized in Figure 1.3. It is an elaboration of Figure 1.1. Ques-
tion marks in the figure indicate open problems.
1.9 Organization of the paper
The rest of this paper is organized as follows.
Negative hypotheses in complexity theory (Lower bound problems)
The flip
Positive hypotheses in complexity theory (Upper bound problems)
Saturated and positive integer programming, and
the quasi-polynomiality results in this paper
Mathematical saturation and positivity hypotheses: PH1,SH (PH2,3)
Construction of the canonical models in this paper, and
construction of the quantum groups in GCT4,7
(PH0): Construction of quantizations of the coordinate
rings of the canonical models and their canonical bases
(?): Problems related to the Riemann Hypothesis over finite
fields, and their generalizations
Figure 1.3: A basic plan for implementing the flip
In Chapter 2 we describe the basic complexity theoretic notions that we
need in this paper and describe their significance in the context of represen-
tation theory.
In Chapter 3, we give a polynomial time algorithm for saturated integer
programming (Theorem 1.4.1), and give precise statements of the results
and positivity hypotheses for Problems 1.1.3 and 1.1.4 (with X = G/P or
a class variety) mentioned in Section 1.5. These generalize the ones given
in Section 1.6 for the plethysm constant. The framework of saturated in-
teger programming in this paper may be applicable to many other struc-
tural constants in representation theory and algebraic geometry, such as the
Kazhdan-Lusztig polynomials (cf. Sections 3.7).
In Chapter 4, we prove the basic quasi-polynomiality results–Theorem 1.6.1
and its generalizations for Problems 1.1.3 and 1.1.4. We also define canonical
models for the structural constants under consideration, and briefly describe
the relevance of the nonstandard quantum groups and the related results in
[GCT4, GCT7, GCT8] in the context of quantizing the coordinate rings of
these canonical models and extending PH0 to them (Figure 1.2).
In Chapter 5, we prove the basic PSPACE results–Theorem 1.6.3 and
its extensions for the various cases of Problem 1.1.3.
In Chapter 6, we give experimental evidence for the positivity hypotheses
PH2 and PH3 in some special cases of the Problems 1.1.1-1.1.4.
In Chapter 7, we describe an application (Theorem 1.8.1) of the re-
sults and positivity hypotheses in this paper to the problem of verifying or
discovering a robust obstruction, i.e., a “proof of hardness” [GCT2] in the
context of the P vs. NP and the permanent vs. determinant problems in
characteristic zero.
1.10 Notation
We let 〈X〉 denote the total bitlength of the specification of X. Here X can
be an integer, a partition, a classifying label of an irreducible representation
of a reductive group, a polytope, and so on. The exact meaning of 〈X〉 will
be clear from the context. The notation poly(n) means O(na), for some
constant a. The notation poly(n1, n2, . . .) similarly means bounded by a
polynomial of a constant degree in n1, n2, . . .. Given a reductive group H,
Vλ(H) denotes the irreducible representation of H with the classifying label
λ. The meaning depends on H. Thus if H = GLn(C), λ is a partition and
Vλ(H) the Weyl module indexed by λ, if H = Sm, then λ is a partition of
size |λ| = m, and Vλ(H) the Specht module indexed by λ, and so on.
Chapter 2
Preliminaries in complexity
theory
In this chapter, we recall basic definitions in complexity theory, introduce
additional ones, and illustrate their significance in the context of represen-
tation theory.
2.1 Standard complexity classes
As usual, P , NP and PSPACE are the classes of problems that can be
solved in polnomial time, nondeterministic polynomial time, and polyno-
mial space, respectively. The class of functions that can be computed in
polynomial time (space) is sometimes denoted by FP (resp. FPSPACE).
But, to keep the notation simple, we shall denote these classes by P and
PSPACE again.
Let SPACE(s(N)) denote the class of problems that can be solved in
O(s(N)) space on inputs of bit length N ; by convention s(N) counts only
the size of the work space. In other words, the size of the input, which is on
the read-only input tape, and the output, which is on the write-only output
tape is not counted. Hence s(N) can be less than the size of the input or the
output, even logarithmic compared to these sizes. The class space(log(N))
is denoted by LOGSPACE.
An algorithm is called strongly polynomial [GLS], if given an input x =
(x1, . . . , xk),
1. the total number of arithmetic steps (+, ∗,− and comparisones) in the
algorithm is polynomial in k, the total number of input parameters,
but does not depend 〈x〉, where 〈x〉 =
i〈xi〉 denotes the bitlength of
2. the bit length of every intermediate operand in the computation is
polynomial in 〈x〉.
Clearly, a strongly polynomial algorithm is also polynomial. let strong P ⊆
P denote the subclass of problems with strongly polynomial time algorithms.
The counting class associated with NP is denoted by #P . Specifically,
a function f : Nk → N, where N is the set of nonnegative integers, is in #P
if it has a formula of the form:
f(x) = f(x1, · · · , xk) =
χ(x, y), (2.1)
where χ is a polynomial-time computable function that takes values 0 or 1,
and y runs over all tuples such that 〈y〉 = poly(〈x〉). The formula (2.1) is
called a #P -formula. An important feature of a #P -formula in the context
of representation theory is that it is positive; i.e., it does not contain any
alternating signs.
The formula (2.1) is called a strong #P -formula, if, in addition, l is
polynomial in k and χ is a strongly polynomial-time computable function.
Let strong #P be the class of functions with strong #P -fomulae.
It is known and easy to see that
#P ⊆ PSPACE. (2.2)
2.1.1 Example: Littlewood-Richardson coefficients
By the Littlewood-Richardson rule [FH], the coefficient cλ
(cf. Prob-
lem 1.2.1) in type A is given by:
cλα,β =
χ(T ), (2.3)
where T runs over all numbering of the skew shape λ/α, and χ(T ) is 1 if
T is a Littlewood-Richardson skew tableau of content β, and zero, other-
wise. The total number of entries in T is quadratic in the total number of
nonzero parts in α, β, λ, and the number of arithmetic steps needed to com-
pute χ(T ) is linear in this total number. Hence (2.3) is a strong #P -formula,
and Littlewood-Richardson function c(α, β, λ) = cλα,β belongs to strong #P .
It may be remarked that the character-based formulae for the Littlewood-
Richardson coefficients are not #P -formulae, since they involve alternat-
ing signs. But the algorithms based on the these formulae for computing
Littlewood-Richardson coefficients run in polynomial space. Thus, from the
perspective of complexity theory, the main significance of the Littlewood-
Richardson rule is that it puts the problem, which at the surface is only in
PSPACE, in its smaller subclass (strong) #P .
Though the Littlewood-Richardson rule is often called efficient in the
representation theory literature, it is not really so from the perspective of
complexity theory. Because computation of cλ
using this formula takes
time that is exponential in both the total number of parts of α, β and λ, and
their bit lengths. This is inevitable, since this problem is #P -complete [N].
Specifically, this means there is no polynomial time algorithm to compute
, assuming P 6= NP .
As remarked in earlier, nonzeroness (nonvanishing) of cλ
can be decided
in poly(〈α〉, 〈β〉, 〈λ〉) time; [DM2, GCT3, KT1]. Furthermore, the algorithm
in [GCT3] is strongly polynomial; i.e., the number of arithemtic steps in
this algorithm is a polynomial in the total number of parts of α, β, λ, and
does not depend on the bit lengths of α, β, λ. Hence the problem of deciding
nonvanishing of cλ
(type A) belongs to strong P .
The discussion above shows that the Littlewood-Richardson problem is
akin to the problem of computing the permanent of an integer matrix with
nonnegative coefficients. The latter is known to be #P -complete [V], but
its nonvanishing can be decided in polynomial time, using the polynomial-
time algorithm for finding a perfect matching in bipartite graphs [Sc]. If
the positivity hypotheses in this paper hold, the situation would be similar
for many fundamental structural constants in representation theory and
algebraic geometry in a relaxed sense.
2.2 Convex #P
Next we want to introduce a subclass of #P called convex #P .
Given a polytope P ⊆ Rl, let χP denote the characteristic (membership)
function of P : i.e., χP (y) = 1, if y ∈ P , and zero otherwise. We say that
f = f(x) = f(x1, . . . , xk) has a convex #P -formula if, for every x ∈ Z
there exists a convex polytope (or, more generally, a convex body) Px ⊆ R
such that
1. The membership function χPx(y) can be computed in poly(〈x〉, 〈y〉)
time, each integer point in Px has O(poly(〈x〉)) bitlength, and
f(x) = φ(Px), (2.4)
where φ(Px) denotes the number of integer points in Px. Equivalently,
f(x) =
χPx(y), (2.5)
where y runs over tuples in Zl of poly(〈x〉) bitlength, and χPx denotes
the membership function of the polytope Px.
Equation (2.5) is similar to eq.(2.1). The main difference is that χ is now
the membership function of a convex polytope. Clearly, eq.(2.5), and hence,
eq.(2.4) is a #P -formula, when χPx can be computed in polynomial time.
Let convex #P be the subclass of #P consisting of functions with convex
#P -formulae.
We say that eq.(2.4) is a strongly convex #P -formula, if the character-
istic function of Px is computable in strongly polynomial time. Let strongly
convex #P be the subclass of #P consisting of functions with strongly con-
vex #P -formulae.
We do not assume in eq.(2.4) that the polytope Px is explicitly specified
by its defining constraints. Rather, we only assume, following [GLS], that
we are given a computer program, called a membership oracle, which, given
input parameters x and y, tells whether y ∈ Px in poly(〈x〉, 〈y〉) time.
If the number of constraints defining Px is polynomial in 〈x〉, then it
is possible to specify Px by simply writing down these constraints. In this
case the membership question can be trivially decided in polynomial time–in
fact, even in LOGSPACE–by verifying each constraint one at a time. This
would not work if Px has exponentially many constraints. In good cases,
it is possible to answer the membership question in polynomial time even
if Px has exponentially many facets. Many such examples in combinatorial
optimization are given in [GLS]. One such illustrative example in repre-
sentation theory is given in Section 2.2.2. The polytopes that would arise
in the plethysm and other problems of main interest in this paper are also
expected to be of this kind.
We now illustrate the notion of convex #P with a few examples in rep-
resentation theory.
2.2.1 Littlewood-Richardson coefficients
A geeneralized Littlewood-Richardson coefficient cλα,β for arbitrary semisim-
ple Lie algebra (Problem 1.2.1) has a strong, convex #P -formula, because
cλα,β = φ(P
α,β),
where P λα,β is the BZ-polytope [BZ] associated with the triple (α, β, λ).
It is easy to see from the description in [BZ] that the number of defin-
ing constraints of P λα,β is polynomial in the total number of parts (coor-
dinates) of α, β, λ. Given α, β, λ, these constraints can be computed in
strongly polynomial time. Hence, the membership problem for P λ
belongs
to LOGSPACE ⊆ P . It follows that the Littlewood-Richardson function
c(α, β, λ) = cλ
belongs to strongly convex #P .
2.2.2 Littlewood-Richardson cone
We now give a natural example of a polytope in representation theory, the
number of whose defining constraints is exponential, but whose membership
function can still be computed in polynomial time.
Given a complex, semisimple, simply connected groupG, let the Littlewood-
Richardson semigroup LR(G) be the set of all triples (α, β, λ) of dominant
weights of G such that the irreducible module Vλ(G) appears in the tensor
product Vα(G) ⊗ Vβ(G) with nonzero multiplicity [Z]. Brion and Knop [El]
have shown that LR(G) is a finitely generated semigroup with respect to
addition. This also follows from the polyhedral expression for Littlewood-
Richardson coefficients in terms of BZ-polytopes [Z]. Let LRR(G) be the
polyhedral cone generated by LR(G).
When G = GLn(C), the facets of LRR(G) have an explicit description by
the affirmative solution to Horn’s conjecture in [Kl, KT1]. But their number
can be quite large (possibly exponential). Nevertheless, membership of any
rational (α, β, λ) (not necessarily integral) in LRR(G) can be decided in
strongly polynomial time.
This is because LRR(G) is the projection of a polytope P (G), the num-
ber of whose constraints is polynomial in the heights of α, β, λ [Z]. If
φ : P (G) → LR(G) is this projection, we can choose P (G) so that for
any integral (α, β, λ), φ−1(α, β, λ) is the BZ-polytope associated with the
triple (α, β, λ). To decide if (α, β, λ) ∈ LR(G), we only have to decide if the
polytope φ−1(α, β, λ) is nonempty. This can be done in strongly polynomial
time using Tardos’ linear programming algorithm [Ta].
2.2.3 Eigenvalues of Hermitian matrices
Here is another example of a polytope in representation theory with expo-
nentially many facets, whose membership problem can still belong to P .
For a Hermitian matrix A, let λ(A) denote the sequence of eigenvalues
of A arranged in a weakly decreasing order. Let HEr be the set of triple
(α, β, λ) ∈ Rr such that α = λ(A + B), β = λ(A), λ = λ(B) for some
Hermitian matrices A and B of dimension r. It is closely related to the
Littlewood-Richardson semisgroup LRr = LR(GLr(C)): HEr ∩ P
r = LRr,
where Pr is the semigroup of partitions of length ≤ r. I. M. Gelfand asked
for an explicit description of HEr. Klyachko [Kl] showed that HEr is a
convex polyhedral cone. An explicit description of its facets is now known
by the affirmative answer to Horn’s conjecture. But their number may be
exponential. Hence, membership in HEr is still not easy to check using this
explicit description. This leads to the following complexity theoretic variant
of Gelfand’s question:
Question 2.2.1 Does the memembership problem for HEr belong to P?
Given that the answer is yes for the closely related LRr = LR(GLr(C))
(Section 2.2.2), this may be so. If HEr were a projection of some polytope
with polynomially many facets, this would follow as in Section 2.2.2. But
this is not necessary. For example, Edmond’s perfect matching polytope for
non-bipartite graphs is not known to be a projection of any polytope with
polynomially many constraints. Still the associated membership problem
belongs to P [Sc].
2.3 Separation oracle
Suppose P ⊆ Rl is a convex polytope whose membership function χP is
polynomial time computable. If χP (y) = 0 for some y ∈ R
r, it is natural to
ask, in the spirit of [GLS], for a “proof” of nonmembership in the form of a
hyperplane that separates y from P .
In this paper, we assume that all polytopes are specified by the separation
oracle. This is a computer program, which given y, tells if y ∈ P , and if
y 6∈ P , returns such a separting hyperplane as a proof of nonmembership. We
assume that the hyperplane is given in the form l = 0, where a linear function
l such that P is contained in the half space l ≥ 0, but l(y) < 0. Furthermore.
we assume that P is a well-described polyhedron in the sense of [GLS]. This
means P is specified in the form of a triple (χP , n, φ), where P ⊆ R
n, χP
is a program for computing the membership function given y ∈ Rn, and
there exists a system of inequalities with rational coefficients having P as
its solution set such that the encoding bit length of each inequality is at
most φ. We define the encoding length 〈P 〉 of P as n+ φ. We also assume
that the separation oracle works in O(poly(〈P 〉, 〈y〉) time.
For example, the polynomial time algorithm for the membership function
of the Littlewood-Richardson cone (cf. Section 2.2.2) can be easily modified
to return a separating hyperplane as a proof of nonmembership.
In what follows, we shall assume, as a part of the definition of a convex
#P -formula, that Px in (2.4) is a well-described polyhedron specified by
a separation oracle that works in polynomial time with 〈Px〉 = poly(〈x〉).
These additional requirements are needed for the saturated integer program-
ming algorithm in Chapter 3.
Chapter 3
Saturation and positivity
In this chapter we describe (Section 3.1) a polynomial time algorithm for
saturated and positive integer programming (Theorem 1.4.1). In Section 3.3
we state the main results and positivity hypotheses for the relaxed forms of
Problem 1.1.3 and Problem 1.1.4, with X = G/P or a class variety therein.
Together they say that these relaxed decision problems can be efficiently
transformed into saturated (more strongly, positive) integer programming
problems, and hence can be solved in polynomial time.
3.1 Saturated and positive integer programming
We begin by proving Theorem 1.4.1.
Let P ⊆ Rn be a polytope given by a separation oracle (Section 2.3).
Let 〈P 〉 be the encoding length of P as defined in Section 2.3. An oracle-
polynomial time algorithm [GLS] is an algorithm whose running time is
O(poly(〈P 〉)), where each call to the separation oracle is computed as one
step. Thus if the separation oracle works in polynomial time, then such
an algorithm works in polynomial time in the usual sense. Let φ(P ) be
the number of integer points in P . Let fP (n) = φ(nP ) be the Ehrhart
quasi-polynomial [St1] of P . Let l(P ) be the least period of fP (n), if P
is nonempty. Let fi,P (n), 1 ≤ i ≤ l(P ), be the polynomials such that
fP (n) = fi,P (n) if n = i modulo l(P ). Let FP (t) =
n≥0 fP (n)t
n denote
the Ehrhart series of P . It is a rational function.
Theorem 3.1.1 (a) The index of fP (n), index(fP ), can be computed in
oracle-polynomial time, and hence, in polynomial time, assuming that the
oracle works in polynomial time. Furthermore, if index(fP ) 6= 0 (i.e. if P
is nonempty), then fi,P (n) is not an identically zero polynomial for every i
divisible by index(fP ).
(b) The saturated, and hence, positive integer programming problem, as de-
fined in Section 1.4, can be solved in oracle-polynomial time. Here it is
assumed that the specification of P also contains the saturation index esti-
mate sie(P ), or the positivity index estimate pie(P ), and that the bitlength
of this estimate is O(poly(〈P 〉)). Given a relaxation parameter c > sie(P )
(or pie(P )), the problem is to determine if cP contains an integer point in
O(poly(〈P 〉, 〈c〉)) time.
(c) Suppose {Px} is a family of polytopes, indexed by some parameter x,
with the following property: wherenver Px is nonempty, the Ehrhart quasi-
polynomial fPx(n) is “almost always” strictly saturated. Almost always
means, the density of x’s of bitlength ≤ N , with nonempty Px for which
fPx(n) is not strictly saturated is less than 1/N
c′′ , for any positive c′′, as
N → 0. We also assume that Px is given by a separation oracle that works in
O(poly(〈x〉)) time, where 〈x〉 is the bitlength of x, and 〈Px〉 = O(poly(〈x〉)).
Then there exists a O(poly(〈x〉)) time algorithm for deciding if Px con-
tains an integer point that works correctly “almost always”; i.e., on almost
all x.
Proof:
Nonemptyness of P can be decided in oracle-polynomial time using the
algorithm of Grötschel, Lovász and Schrijver [GLS] (cf. Theorem 6.4.1
therein). An extension of this algorithm, furthermore, yields a specifica-
tion of the affine space span(P ) containing P if P is nonempty (cf. Theo-
rems 6.4.9, and 6.5.5 in [GLS]). Specifically, it outputs an integral matrix
C and an integral vector d such that span(P ) is defined by Cx = d. This
final specification is exact, even though the first part of the algorithm in
[GLS] uses the ellipsoid method. Indeed, the use of simultneous diophan-
tine approximation based on basis reduction in lattices is precisely to ensure
this exactness in the final answer. This is crucial for the next step of our
algorithm.
If P is empty, index(fP ) = 0. So assume that it is nonempty. Let C̄ be
the Smith normal form of C; i.e., C̄ = ACB for some unimodular matrices
A and B, where the leftmost principal submatrix of C̄ is a diagonal, integral
matrix, and all other columns are zero.
The matrices C̄, A and B can be computed in polynomial time using
the algorithm in [KB]. After a unimodular change of coordinates, by letting
z = B−1x, span(P ) is specified by the linear system C̄z = d̄ = Ad. The
equations in this system are of the form:
c̄izi = d̄i, (3.1)
i ≤ codim(P ), for some integers c̄i and d̄i. By removing common factors if
necessary, we can assume that c̄i and d̄i are relatively prime for each i. Let
c̃ be the l.c.m. of c̄i’s.
The statement (a) follows from:
Claim 3.1.2 index(fP ) = c̃ and fi,P (n) is not an identically zero polyno-
mial for every i divisible by c̃.
Proof of the claim: Indeed, nP = {nz | z ∈ P} contains no integer point
unless c̃ divides n. Hence, it is easy to see that FP (t) = FP̄ (t
c̃), where
FP̄ (x) is the Ehrhart series of the dilated polytope P̄ = c̃P . By eq.(3.1),
the equations defining P̄ are:
zi = d̄i(c̃/c̄i), (3.2)
Clearly, c̃ divides the least period l(P ) of fP , and l(P̄ ) = l(P )/c̃ is the period
of the Ehrhart quasipolynomial fP̄ (n). It suffices to show that the index of
fP̄ (n) is one and that fj,P̄ (n) is not an identically zero polynomial for every
1 ≤ j ≤ l(P̄ ). This is equivalent to showing that P̄ contains a point z with
with zi = ai/b, for some integers ai’s and b such that b = j modulo l(P̄ ).
Let us call such a point j-admissible. Because of the form of the equations
(3.2) defining span(P̄ ), we can assume, without loss of generality, that P̄ is
full dimensional. This means the system (3.2) is empty. Then this follows
from denseness of the set of j-admissible points. This proves the claim, and
hence (a).
(b): Let s = sie(P ) be the given saturation index estimate. This means
fP (n + s) is strictly saturated. This in conjunction with (a) implies that,
given a relaxation parameter c > s, cP contains an integer point, iff c
is divisible by index(fP ) (by letting n = c − s). This can be checked in
O(poly(〈P 〉, 〈c〉)) time since index(fP ) can be computed in polynomial time
by (a).
(c) The algorithm computes index(fPx) and says “Probably Yes”if the index
is one, and “No” otherwise. Since the saturation index of fPx(n) is zero
almost always, by the argument in (b) with s = 0 and c = 1, “Probably
Yes” really means “Yes” almost always. Q.E.D.
The algorithm in (c) has one drawback. If the answer is “Probably
Yes”, we have no easy way of checking if Px really contains an integer point.
Ideally, we would like an algorithm that says “Yes”, with an integer point
in Px as a proof certificate, or “No”, or “Unsure”, and the density of x’s on
which it says “Unsure” should be very small. This problem can be overcome
if the family {Px} has the following stronger property, akin to the family of
hive polytopes [KT1]: there is a linear function lx such that, for almost all x,
if {Px} is nonempty, then the lx-optimum of Px is integral (this is stronger
than saying that fPx(n) is strictly saturated). In this case, the algorithm in
(c) can be extended to yield the integral lx-optimum as a proof certificate. If
the lx-optimum is not integeral, the algorithm says “Unsure”. PH1 and SH
(Section 1.6) for the plethysm (and more generally, the subgroup restriction)
problem may be strengthened by stipulating that the polytopes therein have
this property. But this is not needed in this paper.
We note down one corollary of the proof of Theorem 3.1.1 (this should
be well known):
Proposition 3.1.3 The rational function FP (t) = FP̄ (t
c̃), where FP̄ (x) is
the Ehrhart series of the dilated polytope P̄ = c̃P , and c̃ is the index of
fP (n).
If P is explicitly specified in the form a linear system
Ax ≤ b, (3.3)
where A is an m × n matrix, b an m vector and m = poly(n), then the
following stronger version of Theorem 3.1.1 holds. Let 〈A〉 and 〈A, b〉 denote
the bitlength of the specification of A and of the linear system (3.3).
Theorem 3.1.4 Suppose P is specified in terms of an explicit linear system
(3.3). Then the index of the Erhart quasi-polynomial fP (n) can be computed
in poly(〈A, b〉) time, using poly(〈A〉) arithmetic operations.
Thus, saturated, and hence, positive integer programming problem spec-
ified in the form (3.3) can be solved in in poly(〈A, b, c〉) time, where c is the
relaxation parameter, using poly(〈A〉) arithmetic operations.
Proof: This is proved exactly as Theorem 3.1.1, but with Tardos’ strongly
polynomial time algorithm for combinatorial linear programming [Ta] used
in place of the algorithm in [GLS]. Q.E.D.
3.1.1 A general estimate for the saturation index
Now we give a general estimate for the saturation index of any polytope P
with a specification of the form
Ax ≤ b, (3.4)
where A is an m × n matrix, m possibly exponential. Let ‖P‖ = n + ψ,
where ψ is the maximum bitlength of any entry of A. Trivially, ‖P‖ ≤ 〈P 〉.
We do not assume that we know the specification (3.4) of P explicitly. We
only assume that it exists, and that we are told ‖P‖. Then:
Theorem 3.1.5 The saturation index of P is O(2poly(‖P‖)). Thus the
bitlength of the saturation index is O(poly(‖P‖)).
Conjecturally, this also holds for the positivity index. This estimate is
very conservative, but useful when no better estimate is available.
Proof: There exists a triangulation of P into simplices such that every vertex
of any simplex is also a vertex of P . Then
fP (n) =
f∆(n),
where ∆ ranges over all open simplices in this triangulation; a zero-dimensional
open simplex is a vertex. The saturation index of fP (n) is clearly bounded
by the maximum of the saturation indices of f∆(n).
Hence, we can assume, without loss of generality, that P is an open sim-
plex. Let v0, . . . , vn be its vertices. Then, by Ehrhart’s result (cf. Theorem
1.3 in [st5]),
FP (t) =
i hit
j=0(1− t
, (3.5)
where h0 = 1, hi’s are nonnegative, and aj is the least positive integer
such that ajvj is integral. By Cramer’s rule, the bit length of each aj is
poly(‖P‖). Without loss of generality, we can also assume that aj’s are
relatively prime. Otherwise, the estimate on the saturation index below has
to be multiplied by the g.c.d. of aj ’s. Then the result follows by applying
the following lemma to FP (t), since 〈aj〉 = O(poly(‖P‖)). Q.E.D.
Lemma 3.1.6 Let f(n) be a quasipolynomial whose generating function
F (t) has a positive form
F (t) =
i hit
j=0(1− t
, (3.6)
where h0 = 1, hi’s are nonnegative, and aj’s are positive and relatively
prime. Let a = max{aj}. Then the saturation index s(f) of f(n) is
O(poly(a, n)).
Proof: Let g(n) be the quasi-polynomial whose generating function G(t) =
g(n)tn is 1/
j=0(1− t
aj ). It is known that this is the Ehrhart quasipoly-
nomial of the polytope N(a0, . . . , an) defined by the linear system
ajxj = 1, xj > 0.
The saturation index s(g) of g(n) is bounded by the Frobenius number
associated with the set of integers {aj}–this is the largest positive integer m
such that the diophantine equation
ajxj = m
has no positive integeral solution (x0, . . . , xn). It is known (e.g. [BDR]) that
the Frobenius number is bounded by
a0a1a2(a0 + a1 + a2) = O(poly(a)),
assumming that a0 ≤ a1 . . .. Hence, s(g) = O(poly(a)).
Since f(n) is a quasi-polynomial, the degree of the numerator of F (t) is
less than the degree of the denominator. Thus the maximum value of i that
occurs in (3.6) is an.
Let gi(n), i ≤ an, be the quasi-polynomial whose generating function is
j=0(1− t
aj ). Then
s(gi) ≤ i+ s(g) = O(poly(a, n)).
Since, hi’s in (3.6) are nonnegative, s(f) = max s(gi). The result follows.
Q.E.D.
3.1.2 Extensions
We now mention a few straightforard extensions of Theorem 3.1.1.
First, it is not necessary that P be a closed polytope. We can allow
it to be half-closed. Specifically, it can be a solution set of a system of
inequalitites of the form:
A1x ≤ b1 and A2x < b2, (3.7)
where we have allowed strict inequalities. The function FP (n) = φ(nP ), the
number of integer points in nP , is again a quasi-polynomial. Hence, the
notions of saturation and positivity can be generalized to this setting in a
natural way.
Second, the algorithm in Theorem 3.1.1 (b) only needs a nonnegative
number s(P ) such that, for any positive integer c > s(P ):
Saturation guarantee: If the affine span of cP , contains an integer point,
then cP is guaranteed to contain an integer point.
If s(P ) = sie(P ), then this guarantee holds, as can be seen from the
proof of Theorem 3.1.1.
3.1.3 Is there a simpler algorithm?
Though the algorithm for saturated integer programming in Theorem 3.1.1
is conceptually very simple, in reality it is quite intricate, because the work
of Grötschel, Lovász and Schrijver [GLS] needs a delicate extension of the el-
lipsoid algorithm [Kh] and the polynomial-time algorithm for basis reduction
in lattices due to Lenstra, Lenstra and Lovász [LLL]. As has been empha-
sized in [GLS], such a polynomial-time algorithm should only be taken as a
proof of existence of an efficient algorithm for the problem under consider-
ation. It may be conjectured that for the problems under consideration in
this paper such simple, combinatorial algorithms exist. But for the design
of such algorithms, saturation alone does not suffice. The stronger property
(PH3), and more, is necessary. We shall address this issue in Section 3.6.
3.2 Littlewood-Richardson coefficients again
Theorem 3.1.4 applied to the BZ-polytope [BZ], with saturation index esti-
mate equal to zero, specializes to the following in the setting of the Littlewood-
Richardson problem (Problem 1.2.1):
Theorem 3.2.1 [GCT5] Assuming SH (Hypothesis 1.2.5), nonvanishing of
, given α, β, λ, can be decided in strongly polynomial time (Section 2.1)
for any semisimple classical Lie algebra G.
It is assumed here that α, β, λ are specified by their coordinates in the
basis of fundamental weights. For type A, this reduces to the result in
[GCT3], which holds unconditionly.
The saturation conjecture for type A arose [Z] in the context of Horn’s
conjecture and the related result of Klyachko [Kl]. We now turn to implica-
tions of Theorem 3.2.1 in this context.
Given a complex, semisimple, simply connected, classical group G, let
LR(G) be the Littlewood-Richardson semigroup as in Section 2.2.2. The
following is a natural generalization of the problem raised by Zelevinsky [Z]
to this general setting:
Problem 3.2.2 Give an efficient description of LR(G).
Zelevinsky asks for a mathematically explicit description. This is a com-
puter scientist’s variant of his problem.
Let LRR(G) be the polyhedral convex cone generated by LR(G). For
G = GLn(C), by the saturation theorem, a triple (α, β, λ) of dominant
weights belongs to LR(G) iff it belongs to LRR(G). Assuming SH (Hy-
pothesis 1.2.5), Theorem 3.2.1 provides the following efficient description
for LR(G) in general. Recall that the period of the Littlewood-Richardson
stretching polynomial c̃λ
(n) divides a fixed constant d(G), which only de-
pends on the types of simple factors of G [DM2, GCT5]. Let αi’s denote
the coordinates of α in the basis of fundamental weights.
Corollary 3.2.3 (a) Assuming SH, whether a given (α, β, λ) belongs to
LR(G) can be determined in strongly polynomial time.
(b) There exists a decomposition of LRR(G) into a set of polyhedral cones,
which form a cell complex C(G), and, for each chamber C in this complex,
a set M(C) of O(rank(G)2) modular equations, each of the form
aiαi +
biβi +
ciλi = 0 (mod d),
for some d dividing d(G), such that
1. SH (Hypothesis 1.2.5) is equivalent to saying that: (α, β, λ) ∈ LR(G)
iff (α, β, λ) ∈ LRR(G) and (α, β, λ) satisfies the modular equations in
the set M(Cα,β,λ) associated with the cone Cα,β,λ containing α, β, λ.
2. Given (α, β, λ), whether (α, β, λ) ∈ LRR(G) can be determined in
strongly polynomial time (cf. Section 1.2.5).
3. If so, the cone Cα,β,λ and the associated set M(C
) of modular equa-
tions can also be determined in strongly polynomial time. After this,
whether (α, β, λ) satisfies the equations in M(Cλα,β) can be trivially
determined in strongly polynomial time.
Proof: (a) is a consequence of Theorem 3.2.1. (b) follows from a careful
analysis of the algorithm therein; see the proof of a more general result
(Theorem 4.4.2) later. Q.E.D.
We call the labelled cell complex C(G), in which each cell C ∈ C(G)
is labelled with the set of modular equations M(C), the modular complex,
associated with LRR(G). When G = SLn(C), the modular complex is
trivial: it just consists of the whole cone LRR(G) with only one obvious
modular equation attached to it. But, for general G, the modular complex
and the map C → M(C) are nontrivial. We do not know their explicit
description. Corollary 3.2.3 says that, given x = (α, β, λ), whether x ∈
LRR(G), and whether the relevant modular equations are satisfied can be
quickely verified on a computer, though the modular equations cannot be
easily determined and verified by hand, as in type A. This is the main
difference between type A and general types.
This naturally leads to:
Question 3.2.4 Is there a mathematically explicit description of the mod-
ular complex C(G) for a general G?
3.3 The saturation and positivity hypotheses
Now let f(x), x ∈ Nk, be a counting function associated with a structural
constant in representation theory or algebraic geometry. Here x denotes
the sequence of parameters associated with the constant. Let 〈x〉 denote
the bitlength of x. Let ‖x‖ and rank(x) denote its combinatorial size and
combinatorial rank–these measure complexity of the nonstretchable part in
the specification of x and will be specified later for the f ’s of interest in this
paper.
For example, in the Littlewood-Richardson problem, x is the triple (α, β, λ),
f(x) = f(α, β, λ) = cλ
, 〈x〉 is the total bitlength of the coordinates of
α, β, λ, ‖x‖ is the total number of coordinates of α, β and λ, and rank(x) =
‖x‖. The number of coordinates does not change during stretching, and
hence, constitute the nonstretchable part of the input specification here.
Assume that f(x) is nonnegative for all x ∈ Nk, Then we can successively
ask the following questions:
1. Does f ∈ PSPACE? That is, can f(x) be computed in poly(〈x〉)
space?
2. Does f ∈ #P? (cf. Section 2.1)
3. Does f ∈ convex#P? (cf. Section 2.2)
4. Can a stretching function f̃(x, n) be associated with f(x) intrinsically
so that f̃(x, n) is quasi-polynomial?
5. (PH1?): Is there a polytope Px, for every x, with 〈Px〉 = O(poly(〈x〉))
and ‖Px‖ = O(poly(‖x‖)), such that f̃(x, n) = fPx(n)?
6. Are there good analogues of SH and/or PH2, PH3 for f̃(x, n)? If
so, nonvanishing of f(x), modulo small relaxation, can be decided in
O(poly(〈Px〉)) time by Theorem 3.1.1.
In the rest of this paper, we study these questions when f = f(x) is a
nonnegative function associated with a structural constant in any of the deci-
sion problems in Section 1.1. Exact specifications of x, 〈x〉, ‖x‖, rank(x), f(x),
and f̃(x, n) for these decision problems are given in Sections 3.4-3.5. It is
shown in Chapter 5 that f(x) ∈ PSPACE for Problem 1.1.2 and the special
cases of Problem 1.1.3 that arise in the flip. This may be conjectured to be
so for the f ’s in Problem 1.1.4, with X therein a class variety; cf [GCT10]
for its justification. Quasipolynomiality of f̃(x, n) is addressed in Chapter 4.
The hypotheses PH1, SH, PH2, and PH3 in these cases have the following
unified form.
Hypothesis 3.3.1 (PH1) Let f = f(x) be the function associated with a
structural constant in
1. Problem 1.1.1, or
2. 1.1.2, or
3. Problem 1.1.3, or
4. Problem 1.1.4, with X being a class variety therein.
Then the function f(x) has a convex #P -formula (cf. (2.4))
f(x) = φ(Px),
such that:
1. for every fixed x, the Ehrhart quasi-polynomial fPx(n) of Px coincides
with f̃(x, n).
2. 〈P 〉 = O(poly(P )) and ‖P‖ = O(poly(‖x‖)).
Hypothesis 3.3.2 (SH)
(a) Suppose f(x) is a structural constant as in PH1 above. Then for every x,
the saturation index s(f̃) of f̃(x, n) is O(poly(rank(x))). This means there
exist absolute nonnegative constants c, c′ such that s(f̃) ≤ c(rank(x))c
(b) For f(x) in Problems 1.1.1-1.1.3, the saturation index of f̃(x, n) is zero–
i.e., f̃(x, n) is strictly saturated–for almost all x. This means the density of
x, with 〈x〉 ≤ N and f(x) nonzero, for which the saturation index s(f̃) is
nonzero is ≤ 1/N c
, for any positive costant c′′, as N → ∞.
More strongly than (a),
Hypothesis 3.3.3 (PH2) For f(x) as in PH1, the positivity index of f̃(x, n)
is O(poly(rank(x))).
Hypothesis 3.3.4 (PH3) For f(x) as in PH1, the generating function
F (x, t) =
n f̃(x, n)t
n has a positive rational form of modular index O(poly(rank(x))).
More specifically, the modular index of f̃(x, n), as defined in Section 4.1.1
for f ’s that arise in this paper, is O(poly(rank(x))).
PH3 implies SH (a); this follows from Lemma 3.1.6.
The following conservative bound follows from Theorem 3.1.5.
Theorem 3.3.5 (Weak SH)
Assuming PH1 (Hypothesis 3.3.1), the saturation index of f̃(x, n) is
bounded by 2O(poly(‖x‖)); hence its bitlength is bounded by O(poly(‖x‖)).
The following result addresses the relaxed forms of the decision problems
for the structural constants under consideration (cf. Section 1.1).
Theorem 3.3.6 Suppose f(x) is a structural constant as in PH1 above.
Then PH1 (Hypothesis 3.3.1) and SH (Hypothesis 3.3.2) imply Hypothe-
sis 1.1.6 (PHflip) in this case. Specifically:
(a) For f(x) in Problems 1.1.1-1.1.4, nonvanishing of f̃(x, a), for a given x
and a relaxation parameter a > c(rank(x))c
, with c, c′ as in Hypothesis 3.3.2,
can be decided in poly(〈x〉, 〈a〉) time.
(b) For f(x) as in Problems 1.1.1-1.1.3, there is a poly(〈x〉) time algorithm
for deciding nonvanishing of f(x) that works correctly on almost all x.
This follows from Theorem 3.1.1.
The following sections give precise descriptions of x, 〈x〉, ‖x‖, rank(x)
and f̃(x, n) for the structural constants under consideration.
3.4 The subgroup restriction problem
In this section we consider the subgroup restriction problem (Problem 1.1.3).
The Kronecker and the plethysm problems (Problems 1.1.1, 1.1.2) are its
special cases.
Let G,H, ρ, λ, π,mπλ be as in Problem 1.1.3. We shall define below an ex-
plicit polynomial homomorphism ρ : H → G, as needed in the statement of
Problem 1.1.3, and also the precise specifications [H], [ρ], [λ], [π] of H, ρ, λ, π,
respectively. We shall also define the bitlengths 〈H〉, 〈ρ〉, 〈λ〉, 〈π〉 and the
combinatorial bit lengths ‖λ‖, ‖π‖. We let ‖H‖ = 〈H〉 and ‖ρ‖ = 〈ρ〉, since
H and ρ belong to the nonstretchable part of the input. On the other hand,
λ and π will be stretched in the definition of f̃(x, n), and hence their com-
binatorial bit lengths will differ from the usual bit lengths. The input x in
the subgroup restriction problem is the tuple ([H], [ρ], [λ], [π]). Its bitlength
〈x〉 is defined to be the sum of the bitlengths 〈H〉, 〈ρ〉, 〈λ〉, 〈π〉, and ‖x‖ is
defined to be the sum of ‖H‖, ‖ρ‖, ‖λ‖ and ‖π‖. Finally rank(x) is defined
to the sum of the ranks of H and G and ‖λ‖ and ‖π‖. Here that rank of
a (reductive) group is defined in a standard way. For example, the rank of
the symmentric group Sn is n, that of GLn(C) is n. The rank of a general
finite or connected simple group can be defined similarly, and the rank of a
more complex reductive group is defined to be the sum of the ranks of its
simple components. With this terminology, we let f(x) = mπ
, with x as
defined here in Hypotheses 3.3.1-3.3.4 and Theorem 3.3.6 for the subgroup
restriction problem. Here H and ρ are implicit in the definition of mπ
For example, in the plethym problem (Problem 1.1.2), these specifi-
cations are as follows. The specification [H] is just the root system for
H = GLn(C). Its bitlength 〈H〉 is n. The specification [ρ] of the repre-
sentation map ρ : H → G = GL(Vµ(H)) consists of just the partition µ
specified in terms of its nonzero parts. Its bitlength 〈ρ〉 = 〈µ〉. The ranks
of H and G are as usual. The partitions λ and µ are specified in terms of
their nonzero parts. Their bitlength is the total bitlength of the parts, and
the combinatorial bit length is the total number of parts (the height). It
is crucial here that only nonzero parts of λ are specified, because the rank
of G can be exponential in the rank of H and the bitlength of µ. Hence,
the bitlength of this compact representation of λ can be polynomial in the
rank of H and the bitlength of µ, even if the dimension of G is exponential.
The main difference between 〈x〉 and ‖x‖ is that the stretchable data λ and
π contribute their bitlengths to the former, and their heights to the latter.
The plethysm problem is the main prototype of the subgroup restriction
problem. If the reader wishes, (s)he can skip the rest of this subsection and
jump to Section 3.4.3 in the first reading.
In general, we assume that H in Problem 1.1.3 is a finite simple group, or
a complex simple, simply connected Lie group, or an algebraic torus (C∗)k,
or a direct product of such groups. The results and hypotheses in this paper
are also applicable if we allow simple types of semidirect products, such as
wreath products, which is all that we need for the sake of the flip. But these
extensions are routine, and hence, for the sake of simplicity, we shall confine
ourselves to direct products.
3.4.1 Explicit polynomial homomorphism
Now let us define an explicit polynomial homomorphism. This will be done
by defining basic explicit homomorphisms, and composing them functorially.
Basic explicit homomorphisms:
Let V be an irreducible polynomial representation of H (character-
istic zero), or more generally, an explicit polynomial representation that
is constructed functorially from the irreducible polynomial representations
using the operations ⊕ and ⊗. Then the corresponding homomorphism
ρ : H → G = GL(V ) is an explicit polynomial homomorphism. The iden-
tity map H → H is also an explicit polynomial homomorphism.
The polynomiality restriction here only applies to the torus component
of H. If H is a finite simple group, or a complex semisimple group, then
any irreducible representation of H is, by definition, polynomial. In general,
a representation is polynomial if its restriction to the torus component is
polynomial; i.e., a sum of polynomial (one dimensional) characters.
To see why the polynomiality restriction is essential, let H be a torus,
V its rational representation, and G = GL(V ). Let Vλ(G) = Sym
d(V ),
the symmetric representation of G, and let π be the label of the trivial
character of H. Then the multiplicity mπ
is the number of H-invariants in
Symd(V ). This is easily seen to be the number of nonnegative solutions of a
system of linear diophontine equations. But the problem of deciding whether
a given system of linear diophontine equations has a nonnegative solution
is, in general, NP -complete. Though the system that arises above is of a
special form, it is not expected to be in P if V is allowed to be any rational
representation; the associated decision problem may be NP -complete even
in this special case. If V is a polynomial representation of a torus H, then
all coefficients of the system are nonnegative, and the decision problem is
trivially in P .
Composition:
We can now compose the basic explicit (polynomial) homomorphisms
above functorially:
1. If ρi : H → Gi are explicit, the product map ρ : H →
iGi is also
explicit.
2. If ρi : Hi → Gi are explicit, the product map ρ :
Gi is also
explicit.
Instead of products, we can also allow simple semi-direct products such
as wreath products here. We may also allow other functorial constructions
such as induced representations and restrictions. For example, if ρ : H → G
is an explicit polynomial homomorphism, and G′ ⊆ G is an explicit subgroup
of G such that ρ(H) ⊆ G′, then the restricted homomorphism ρ′ : H → G′
can also be considered to be an explicit polynomial homomorphism. But
for the sake of simplicity, we shall confine ourselves to the simple functorial
constructions above.
3.4.2 Input specification and bitlengths
Now we describe the specifications [H], [ρ], [λ], [µ], their bitlengths. These
are very similar to the ones in the plethysm problem.
The specification [H]:
We assume that H is specified as follows.
(1) IfH is a complex, simple, simply connected Lie group, then the specifica-
tion [H] consists of the root system of H or the Dynkin diagram. Let 〈H〉 be
the bitlength of this specification. Thus, if H = SLn(C), then 〈H〉 = O(n).
(2) If H is a simple group of Lie type (Chevalley group) then it has a similar
specification [Ca]. The only finite groups of Lie type that arise in GCT are
SLn(Fpk) and GLn(Fpk). In this case the specification [H] is easy: we only
have to specify n, p, k. We define 〈H〉 in this case to be n + k + log2 p; not
log2 n+ log2 k + log2 n. As a rule, 〈H〉 is defined to be the sum of the rank
parameters (such as n and k here) and bit lengths of the weight parameters
(such as p here) in the specification. This is equivalent to assuming that the
rank parameters are specified in unary.
(3) If H is the alternating group An, we only specify n. Let 〈H〉 = n.
(4) The torus is specified by its dimension. We define 〈H〉 to be the dimen-
sion.
(5) If H is a product of such groups, its specification is composed from the
specifications of its factors, and the bitlength 〈H〉 is defined to be the sum
of the bitlengths of the constituent specifcations.
The specification [ρ]:
Let us first assume that ρ is a basic explicit polynomial homomorphism.
In this case the specification of ρ : H → G = GL(V ) is a pair [ρ] = ([H], [V ])
consisting of the specification [H] of H as above, and the combinatorial
specification [V ] of the representation V as defined below:
(1) If H is a semisimple, simply connected Lie group, and V = Vµ(H) its
irrreducible representation for a dominant weight µ of H, then V is specified
by simply giving the coordinates of µ in terms of the fundamental weights
of H. Thus [V ] = µ, and its bitlength 〈V 〉 is the total bitlength of all
coordinates of µ, and the combinatorial bit length ‖V ‖ is the total number
of coordinates of µ.
(2) If H = Sn, and V = Sγ its irreducible representation (Specht module),
then [V ] is the partition γ labelling this Specht module. We define 〈V 〉 to
be the bitlength of this partition, and ‖V ‖ = 〈V 〉.
(3) If H is a finite general linear group GLn(Fpk), and V its irreducible rep-
resentation, as classified by Green [Mc], then [V ] is the combinatorial clas-
sifying label of V as given in [Mc]. It is a certain partition-valued function,
which can be specified by listing the places where the function is nonzero
and the nonzero partition values at these places. Let 〈V 〉 be the bitlength
of this specification; it is O(poly(n, k, 〈p〉)). We let ‖V ‖ = 〈V 〉. More gener-
ally, if H is a finite group of Lie type, and V its irreducible representation,
then [V ] is the combinatorial classifying label of V as given by Lusztig [Lu1].
(4) If H is a torus and V is a polynomial character, then [V ] is the speci-
fication of the character. Its bitlength is the bitlength of the specification,
and combinatorial bit length is the dimension of H.
(5) If V is composed from irreducible representations, then [V ] is composed
from the specifications of the irreducible representations in an obvious way.
Bitlengths and combinatorial bitlengths are defined additively.
The bitlength 〈ρ〉 is defined to be 〈H〉+ 〈V 〉, where 〈V 〉 is the bitlength
of [V ].
If ρ is a composite homomorphism, its specification [ρ] is composed from
the specifications of its basic constituents in an obvious way. The bitlength
〈ρ〉 is defined to be the sum of the bitlengths of these basic specifications.
The specifications [λ] and [π]:
Vπ(H) is the tensor product of the irreducible representations of the
factors of H. We let [π] be the tuple of the combinatorial classifying labels
of each of these irreducible representations, as specified above. Let 〈π〉 be
their total bit length, and ‖π‖ the total combinatorial bit length. Similarly,
Vλ(G) is the tensor product of the irreducible representations of the factors
of G. When G = GLm(C), λ is a partition, which we specify by only giving
its nonzero parts, whose number is equal to the height of λ. This is crucial
since the height of λ can be much less than than the rank m of G, as in
the plethysm problem (Problem 1.1.2). We shall leave a similar compact
specification [λ] for a general connected, reductive G to the reader. Let 〈λ〉
be its bitlength and ‖λ‖ its combinatorial bit length.
3.4.3 Stretching function and quasipolynomiality
Let f(x) = mπ
as above, with x = ([H], [ρ], [λ], [π]). Here λ is the dominant
weight of G. First, assume that H is connected, reductive. Then π is the
dominant weight of H. For a given x, let us define the stretching function
f̃(x, n) = m̃πλ(n) = m
nλ, (3.8)
which is the multiplicity of Vnπ(H) in Vnλ(G), considered as an H-module
via ρ : H → G. Let Mπ
(t) =
n≥0 m̃
(n)tn be the generating function of
this stretching quasi-polynomial.
The following is the generalization of Theorem 1.6.1 in this setting.
Theorem 3.4.1 (a) (Rationality) The generating function Mπ
(t) is ratio-
(b) (Quasi-polynomiality) The stretching function m̃π
(n) is a quasi-polynomial
function of n.
(c) There exist graded, normal C-algebras S = S(mπ
) = ⊕nSn and T =
T (mπ
) = ⊕nTn such that:
1. The schemes spec(S) and spec(T ) are normal and have rational singu-
larities.
2. T = SH , the subring of H-invariants in S.
3. The quasi-polynomial m̃π
(n) is the Hilbert function of T .
(d) (Positivity) The rational function Mπ
(t) can be expressed in a positive
form:
Mπλ (t) =
h0 + h1t+ · · ·+ hdt
j(1− t
a(j))d(j)
, (3.9)
where a(j)’s and d(j)’s are positive integers,
j d(j) = d+1, where d is the
degree of the quasi-polynomial, h0 = 1, and hi’s are nonnegative integers.
The specific rings S(mπ
) and T (mπ
) constructed in the proof of this
result are called the canonical rings associated with the structrural con-
stant mπ
. The projective schemes Y (mπ
) = Proj(S(mπ
)), and Z(mπ
Proj(T (mπλ)) are called the canonical models associated with m
Theorem 3.4.1 and its generalization, when H can be disconnected, is
proved in Chapter 4; cf. Theorem 4.1.1.
Finitely generated semigroup
The following is an analogue of Theorem 1.6.2.
Theorem 3.4.2 Assume that H is connected. For a fixed ρ : H → G, let
T (H,G) be the set of pairs (µ, λ) of dominant weights of H and G such that
the irreducible representation Vπ(H) of H occurs in the irreducible repre-
sentation Vλ(G) of G with nonzero multiplicity. Then T (H,G) is a finitely
generated semigroup with respect to addition.
This is proved in Section 4.4.
PSPACE
The following is a generalization of Theorem 1.6.3.
Theorem 3.4.3 Assume that H in Problem 1.1.3 is a direct product, whose
each factor is a complex simple, simply connected Lie group, or an alternat-
ing (or symmetric) group, or SLn(Fpk) (or GLn(Fpk)), or a torus. Then
f(x) = mπ
can be computed in poly(〈x〉) space, with x as specified above.
This is proved in Chapter 5. It may be conjectured that Theorem 3.4.3
holds even when the composition factors of H are allowed to be general
finite simple groups of Lie type. This will be so if Lusztig’s algorithm [Lu5]
for computing the characters of finite simple groups of Lie type can be
parallelized; cf. Section 5.4.
Positivity hypotheses
Theorem 3.4.1-3.4.3, along with the experimental results in special cases
(cf. Chapter 6), constitute the main evidence in support of the positivity
Hypotheses 3.3.1-3.3.4 for the subgroup restrition problem.
3.5 The decision problem in geometric invariant
theory
Finally, let us turn to the most general Problem 1.1.4.
3.5.1 Reduction from Problem 1.1.3 to Problem 1.1.4
First, let us note that the subgroup restriction problem (Problem 1.1.3)
is a special case of Problem 1.1.4. To see this, let H, ρ and G be as in
Problem 1.1.3, and let X be the closed G-orbit of the point vλ corresponding
to the highest weight vector of Vλ(G) in the projective space P (Vλ(G)). Then
X = Gvλ ∼= G/Pλ, (3.10)
where the P = Pλ = Gvλ is the parabolic stabilizer of vλ. We have a natural
action of H on X via ρ. Let R be the homogeneous coordinate ring of X. By
[Ha, MR, Rm, Sm], the singularities of spec(R) are rational. By Borel-Weil
[FH], the degree one component R1 of the homogeneous coordinate ring R
of X is Vλ(G). Hence, s
1 in this special case of Problem 1.1.4 is precisely m
in Problem 1.1.3. The results in Section 3.4 for sπ1 generalize in a natural
way for sπd .
3.5.2 Input specification
The variety X in the above example is completely specified by H, ρ and λ.
Hence its specification [X] can be given in the form a tuple ([H], [ρ], [λ]),
where [H], [ρ] and [λ] are the specifications of H, ρ and λ as in Section 3.4,
The input specification x for Problem 1.1.4 in the special case above is the
tuple ([X], d, [π]) = ([H], [ρ], [λ], d, [π]), where [π] is the specification of π as
in Section 3.4.
We now describe a class of varieties X which have similar compact spec-
ifications.
Let G be a connected, reductive group, H a reductive, possibly discon-
nected, reductive group, and ρ : H → G an explicit polynomial homomor-
phism as in Section 3.4. Let V = Vλ(G) be an irreducible representation of G
for a dominant weight λ. Let P (V ) be the projective space associated with
V . It has a natural action of H via ρ. Let v ∈ P (V ) be a point that is char-
acterized by its stabilizer Gv ⊆ G. This means it is the only point in P (V )
that is stabilized by Gv. For example, the point vλ above is characterized by
its parabolic stabilier. We assume that we know the Levi decompositioon of
Gv explicity, and its compact specification [Gv ], like that of H, and also an
explicit compact specification of the embedding ρ′ : Gv → G, aking to that
of the explicit homomorphism ρ : H → G. Let X ⊆ P (V ) be the projective
closure of the G-orbit of v in P (V ). Then X as well as the action of H on
X are completely specified by λ,H, ρ,Gv and ρ
′. Hence, we can let [X] be
the tuple (λ, [H], [ρ], [Gv ], [ρ
′]). The input specification x for Problem 1.1.4
with the X of this form is the tuple ([X], d, [π]). The bitlengths 〈x〉 and ‖x‖
are defined additively. The rank(x) is defined to be the sum of the ranks of
H and G, dim(V ) and ‖π‖. Since the point vλ above is characterized by its
stabilizer, G/P is a variety of this form.
The class varieties [GCT1, GCT2] are either of this form, or a slight ex-
tension of this form, and admit such compact specifications. The algebraic
geometry of an X of the above form is completely determined by the repre-
sentation theories of the two homomorphisms ρ : H → G and ρ′ : Gv → G.
Furthermore, the results in [GCT2] say that Problem 1.1.4 for a class variety
is intimately linked with the subgroup restriction problem and its variants
for the homomomorphisms ρ and ρ′. Hence it is qualitatively similar to the
subgroup restriction problem in this case; cf. [GCT10] for further elabora-
tion of the connection between these two problems.
3.5.3 Stretching function and quasi-polynomiality
Now let H,X,R and sπ
be as in Problem 1.1.4, with H therein assumed to
be connected. We associate with f(x) = sπd the following stretching fucntion:
f̃(x, n) = s̃πd (n) = s
nd , (3.11)
where snπ
is the multiplicity of the irrreducible representation Vnπ(H) of H
in Rnd, the componenent of the homogeneous coordinate ring R of X with
degree nd. Let S(t) =
n≥0 s̃
(n)tn.
Theorem 3.5.1 Assume that the singularities of spec(R) are rational.
(a) (Rationality) The generating function Sπ
(t) is rational.
(b) (Quasi-polynomiality) The stretching function s̃πd(n) is a quasi-polynomial
function of n.
(c) There exist graded, normal C-algebras S = S(sπ
) = ⊕nSn and T =
T (sπ
) = ⊕nTn such that:
1. The schemes spec(S) and spec(T ) are normal and have rational singu-
larities.
2. T = SH , the subring of H-invariants in S.
3. The quasi-polynomial s̃πd(n) is the Hilbert function of T .
(d) (Positivity) The rational function Sπ
(t) can be expressed in a positive
form:
Sπd (t) =
h0 + h1t+ · · ·+ hkt
j(1− t
a(j))k(j)
, (3.12)
where a(j)’s and k(j)’s are positive integers,
j k(j) = k + 1, where k is
the degree of the quasi-polynomial s̃πd(n), h0 = 1, and hi’s are nonnegative
integers.
This is proved in Chapter 4. Theorem 3.4.1 is a special case of this theorem,
in view of the reduction in Section 3.5.1. Theorem 3.5.1 is applicable when
X is a class variety, assuming that its singularities are rational.
3.5.4 Positivity hypotheses
Even though Theorem 3.5.1 holds for any X, with spec(R) having ratio-
nal singularities, the positivity hypotheses PH1, SH, PH2, and PH3 can
be expected to hold for only very special X’s. In general, characterizing
the X’s with compact specification for which these hypotheses hold is a
delicate problem. Hypotheses 3.3.1-3.3.4 say that these hold when X in
Problem 1.1.4 is G/P (as in Section 3.5.1) or a class variety, with the input
specification x as described above. For future reference, we shall reformulate
these hypotheses purely in geometric terms.
For this we need a definition.
Let T =
n Tn be a graded complex C-algebra so that the singularities
of spec(T ) rational. Let Z = Proj(T ). Assume that Z has a compact
specification [Z]; we shall specify it below for the Z’s of interest to us.
We let [T ], the specification of T , to be [Z]. This will play the role of
the input in the definition below. Let 〈T 〉 denote its bitlength, and ‖T‖
combinatorial bit length. Let hT (n) = dim(Tn) be its Hilbert function,
which is a quasipolynomial, since the singularities of spec(T ) are rational;
cf. Lemma 4.1.3.
Definition 3.5.2 We say that PH1 holds for T (or Z) if the Hilbert quasi-
polynomial hT (n) is convex. This means there exists a polytope P = PT
depending on the input [T ], whose Ehrhart quasipolynomial fP (n) coincides
with the Hilbert function hT (n), and whose membership function χP (y) can
be computed in poly(〈T 〉, y) time. We assume that a separating hyperplane
can also be computed in polynomial time if y 6∈ P (Section 2.3).
If PH1 holds we can also ask if analogues of SH, PH2, and PH3–whose
formulation is similar and hence omitted–hold.
3.5.5 G/P and Schubert varieties
Let us illustrate this definition with an example. Let X ∼= G/Pλ be as in
Section 3.5.1 and R its homogeneous coordinate ring. We have already seen
that it has a compact specification: namely [X] = λ. Since singularities
of spec(R) are rational, PH1 makes sense. For G/P it follows from the
Borel-Weil theorem. The Hilbert series of R is of the form
h0 + · · ·+ hdt
(1− t)d+1
with h0 = 1 and hi’s nonnegative. This is so because R is Cohen-Macauley
[Rm] and is generated by its degree one component. Hence, the modular
index of the Hilbert function is one (PH3). PH2 turns out to be nontriv-
ial. Experimental evidence in its support for the classical G/P is given in
Section 6.3. Considerations for the Schubert subvarieties are similar. Ex-
perimental evidence for PH2 for the classical Schubert varieties is also given
in Section 6.3.
Now let s = sπ
be the multiplicity as Problem 1.1.4, with X having a
compact specification [X] as above. Let T = T (s) be the ring associated
with s as in Theorem 3.5.1 (c). Let Z = Z(s) = Proj(T ). We let the
specification [Z] = ([X], d, π). Let 〈Z〉 be its bitlength.
So Theorem 3.1.1 in this context implies:
Theorem 3.5.3 If PH1 and SH holds for Z(s) then nonvanishing of s,
modulo small relaxation, can be decided in poly(〈Z〉) time.
We also have the following reformulation:
Proposition 3.5.4 Hypotheses 3.3.1-3.3.4 are equivalent to PH1,SH,PH2,PH3
for Z(s), where s is a stucture constant that corresponds the structure con-
stant f(x) in Hypotheses 3.3.1. Thus, in the case of the subgroup restriction
problem, s = sπ1 = m
λ as in Section 3.5.1.
This is just a consequence of definitions.
3.6 PH3 and existence of a simpler algorithm
As we remarked in Section 3.1.3, the use of the ellipsoid method and basis
reduction in lattices makes the the algorithm for saturated integer program-
ming (cf. Theorem 3.1.1) fairly intricate. For the flip (cf. [GCTflip] and
Chapter 7), it is desirable to have simpler algorithms for the relaxed forms
of the decision problems under consideration, akin to the the polynomial
time combinatorial algorithms in combinatorial optimization [Sc] that do
not rely on the elliposoid method or basis reduction. We briefly examine in
this section the role of PH3 in this context.
The simple combinatorial algorithms in combinatorial optimization work
only when the problem under consideration is unimodular–in which case the
vertices of the underlying polytope P are integral–or almost unimodular–
e.g. when the vertices of P are half integral. Edmond’s algorithm for finding
minimum weight perfect matching in nonbipartite graphs [Sc] is a classic
example of the second case.
In the unimodular case, Stanley’s positivity result [St1] implies that the
rational function FP (t) has a positive form
FP (t) =
h(d)td + · · ·+ h(0)
(1− t)d+1
If PH3 (Hypothesis 3.3.4) holds for a structural function f(x) under con-
sideration then the Ehrhart series FPx(t) of the polytope Px associated with
x in PH1 (Hypothesis 3.3.1) has a minimal positive form in which each root
of the denominator has O(poly(‖x‖)) order. Roughly, this says that the
situation is “close” to the unimodular case. Hence, in such a case we can
expect a purely combinatorial polynomial-time algorithm for deciding non-
vanishing of f(x), modulo small relaxation, that does not need the ellipsoid
method or basis reduction.
3.7 Other structural constants
The paradigm of saturated and positive integer programming in this paper,
along with appropriate analogues of PH1,SH,PH2,PH3, may be applicable
several other fundamental structural constants in representation theory and
algebraic geometry, in addition to the ones in Problems 1.1.1-1.1.4 treated
above, such as
1. the value of a Kazhdan-Lusztig polynomial at q = 1, [KL1];
2. the values at q = 1 of the well behaved special cases of the parabolic
Kostka polynomials and their q-analogues [Ki];
3. the structural coefficients of the multiplication of Schubert polynomi-
als, and so on.
Chapter 4
Quasi-polynomiality and
canonical models
In this chapter we prove quasipolynomiality of the stretching functions as-
sociated with the various structural constants under consideration (Sec-
tion 4.1), describe the associated canonical models (Section 4.2), describe
the role of nonstandard quantum groups in [GCT4, GCT7, GCT8] in the
deeper study of these models (Section 4.3), prove finite generation of the
semigroup of weights (Theorem 3.4.2) (Section 4.4), and give an elementary
proof of rationality in Theorem 3.4.1 (a) (Section 4.5).
4.1 Quasi-polynomiality
Here we prove Theorem 3.5.1; Theorems 1.6.1 and 3.4.1 are its special cases
in view of the reduction in Section 3.5.1. This, in turn, follows from the
following more general result.
Let R = ⊕kRd be a normal graded C-algebra with an action of a reduc-
tive group H. Assume that spec(R) has rational singularities. Let H0 be
the connected component of H containing the identity. Let HD = H/H0 be
its discrete component. Given a dominant weight π of H0, we consider the
module Vπ = Vπ(H0), an H-module with trivial action of HD. Let s
denote
the multiplicity of the H-module Vπ in Rd. Let s̃
(n) be the multiplicity of
the H-module Vnπ in Rnd. This is a stretching function associated with the
mulitplicity sπ
. Let Sπ
(t) =
n≥0 s̃
(n)tn.
Theorem 4.1.1 (a) (Rationality) The generating function Sπ
(t) is ratio-
(b) (Quasi-polynomiality) The stretching function s̃π
(n) is a quasi-polynomial
function of n.
(c) There exist graded, normal C-algebras S = S(sπ
) = ⊕nSn and T =
T (sπd ) = ⊕nTn such that:
1. The schemes spec(S) and spec(T ) are normal and have rational singu-
larities.
2. T = SH , the subring of H-invariants in S.
3. The quasi-polynomial s̃π
(n) is the Hilbert function of T .
(d) (Positivity) The rational function Sπd (t) can be expressed in a positive
form:
Sπd (t) =
h0 + h1t+ · · ·+ hkt
j(1− t
a(j))k(j)
, (4.1)
where a(j)’s and k(j)’s are positive integers,
j k(j) = k + 1, where k is
the degree of the quasi-polynomial s̃π
(n), h0 = 1, and hi’s are nonnegative
integers.
Theorem 3.5.1 follows from this by letting R be the homogeneous coordinate
ring of X.
More generally, if W is an irreducible representation of HD, we can
consider the H-module Vπ ⊗ W . Let s
be its multiplicity in Rd. Let
(n) be the multiplicity of the trivial H-representation in the H-module
Rnd ⊗ V
nπ ⊗ Sym
n(W ∗). Then
Theorem 4.1.2 Analogue of Theorem 4.1.1 holds for s̃
For the purposes of the flip, Theorem 4.1.1 suffices.
Proof: We shall only prove Theorem 4.1.1, the proof of Theorem 4.1.2 be-
ing similar. The proof is an extension of M. Brion’s proof (cf. [Dh]) of
quasi-polynomiality of the stretching function associated with a Littlewood-
Richardson coefficient of any semisimple Lie algebra.
Clearly (a) follows from (b); cf. [St1].
(b) and (c):
Let Cd be the cyclic group generated by the primitive root ζ of unity of
order d. It has a natural action on R: x ∈ Cd maps z ∈ Rk to x
kz. Let
B = RCd =
n≥0Rnd ⊆ R be the subring of Cd-invariants. By Boutot
[Bou], B is a normal C-algebra and spec(B) has rational singularities.
Assume thatH0 is semisimple; extension to the reductive case being easy.
Let π∗ be the dominant weight of H0 such that V
π = Vπ∗ . By Borel-Weil
[FH],
Cπ∗ = ⊕n≥0V
nπ = ⊕n≥0Vnπ∗ ,
is the homogeneous coordinate ring of theH0-orbit of the point vπ∗ ∈ P (Vπ∗)
corresponding to the highest weight vector. This H0-orbit is isomorphic to
H0/Pπ∗ , where Pπ∗ ⊆ H0 is the parabolic stabilizer of vπ∗ . Hence Cπ∗ is
normal and spec(Cπ∗) has rational signularities; cf. [Ha, MR, Rm, Sm].
It follows that B ⊗ Cπ∗ is also normal, and spec(B ⊗ Cπ∗) has rational
singularities. Consider the action of C∗ on B ⊗ Cπ∗ given by:
x(b⊗ c) = (x · b)⊗ (x−1 · c),
where x ∈ C∗ maps b ∈ Bn to x
nb, the action on Cπ∗ being similar. Consider
the invariant ring
S = (B ⊗ Cπ∗)
= ⊕nSn = ⊗n≥0Rnd ⊗ V
nπ. (4.2)
By Boutot [Bou], it is a normal, and spec(D) has rational singularities.
Since Vnπ is an H-module, the algebra S has an action of H. Let
T = T (sπd ) = S
H = ⊕n≥0Tn (4.3)
be its subring of H-invariants. By Boutot [Bou], it is normal, and spec(T )
has rational singularities–this is the crux of the proof. By Schur’s lemma, the
multiplicity of the trivial H-representation in Sn = Rnd⊗V
nπ is precisely the
multiplicity s̃πd (n) of the H-module Vnπ in Rnd. Hence, the Hilbert function
of T , i.e., dim(Tn), is precisely s̃
d (n), and the Hilbert series
n≥0 dim(Tn)t
is Sπ
(t). Quasipolynomiality of s̃π
(n) follows by applying the following
lemma:
Lemma 4.1.3 (cf. [Dh]) If T = ⊕∞n=0Tn is a graded C-algebra, such that
spec(T ) is normal and has rational simgularites, then dim(Tn), the Hilbert
function of T , is a quasi-polynomial function of n.
(d) Since spec(T ) has rational singularities, T is Cohen-Macaualey. Let
t1, . . . , tu be its homogeneous sequence of parameters (h.s.o.p.), where u =
k + 1 is the Krull dimension of T . By the theory of Cohen-Macauley rings
[St2], it follows that its Hilbert series Sπd (t) is of the form
h0 + h1t+ · · ·+ hkt
i=1 (1− t
, (4.4)
where (1) h0 = 1, (2) di is the degree of ti, and (3) hi’s are nonnegative
integers. This proves (d). Q.E.D.
Remark 4.1.4 A careful examination of the proof above shows that ratio-
nality of Sπd (t), and more strongly, asymptotic quasi-polynomiality of s̃
d (n)
as n → ∞, can be proved using just Hilbert’s result on finite generation of
the algebra of invariants of a reductive-group action. Boutot’s result is nec-
essary to prove quasi-polynomiality for all n. This is crucial for saturated
and positive integer programming (Chapter 3).
4.1.1 The minimal positive form and modular index
The form (4.4) of Sπd (t) is not unique because it depends on the degrees di’s
of the paramters ti’s. For future use, let us record the following consequences
of the proof. Let T be the ring constructed in the proof above.
Corollary 4.1.5 Suppose T has an h.s.o.p. t = (t1, . . . , tu) with di =
deg(ti). Then S
(T ) has a positive rational form (4.4) with di = deg(ti)
therein.
The proof above is lets us define a minimal positive form of the rational
function Sπ
(t) associated with a structural constant s. For this, let us or-
der h.s.o.p.’s of T lexicographically as per their degree sequences. Here the
degree seqeunce of an h.s.o.p. t = (t1, . . . , tu) is defined to be (d1, . . . , du),
where di = deg(ti). The form (4.4) is the same for any h.s.o.p. of lexi-
cographically minimum degree sequence. We call it the minimal positive
form of Sπ
(t). The modular index of sπ
is defined to be max{di}, where
(d1, . . . , du) is the degree sequence of a lexicographically minimal h.s.o.p.
Since Problems 1.1.1, 1.1.2,1.1.3, 1.2.1 are special cases of Problem 1.1.4,
this defines minimal positive forms of the rational generating functions of the
stretching quasi-polynomials (cf. Theorem 3.4.1) associated with the struc-
tural constants in these problems, and also the modular indices of these
structural constants.
4.1.2 The rings associated with a structural constant
The preceding proof also associates with the structural constant s a few
rings which will be important later. Specifically, let S = S(s) and T = T (s)
be the rings as in Theorem 4.1.1 (c) associated with the structural constant
s = sπ
. Let R = R(s) be the homogeneous coordinate ring of X as in
Theorem 4.1.1. We call R(s), S(s) and T (s) the rings associated with the
structure constant s.
When s = mπ
, as in the subgroup restriction problem (Problem 1.1.3),
X ∼= G/P as given in eq.(3.10. Then these rings are explicitly as follows:
) = ⊕n≥0Vnλ(G),
) = ⊕n≥0Vnλ(G)⊗ Vnπ(H)
T (mπ
) = ⊕n≥0(Vnλ(G) ⊗ Vnπ(H)
∗)H .
(4.5)
By specializing the subgroup restriction problem further to the Littlewood-
Richardson problem (Problem 1.2.1), we get the following rings associated
by Brion (cf. [Dh]) with the Littlewood-Richardson coefficient cλ
R(cλα,β) = ⊕n≥0Vnα(H)⊗ Vnβ(H),
) = ⊕n≥0Vnα(H)⊗ Vnβ(H)⊗ Vnλ(H)
T (cλ
) = ⊕n≥0(Vnα(H)⊗ Vnβ(H)⊗ Vnλ(H)
∗)H .
(4.6)
4.2 Canonical models
There are several rings other than T (cλ
) whose Hilbert function coincides
with the Littlewood-Richardson stretching quasi-polynomial c̃λ
(n). For
example, let P = P λ
be the BZ-polytope [BZ] whose Ehrhart quasi-
polynomial coincides with c̃λ
(n). We can associate with P a ring TP
as in Stanley [St3] whose Hilbert function coincides with c̃λ
(n). There
are many other choices for P . For example, in type A, we can consider a
hive polytope or a honeycomb polytope [KT1] instead of the BZ-polytope.
The rings TP ’s associated with different P ’s will, in general, be different,
and there is nothing canonical about them. In contrast, the ring T (cλα,β) is
special because:
Proposition 4.2.1 (PH0) The rings R(cλ
), S(cλ
), T (cλ
) have quan-
tizations Rq(c
), Sq(c
), Tq(c
) endowed with canonical bases in the ter-
minology of Lusztig [Lu4]. Furthermore, the canonical bases of Rq(c
), Sq(c
are compatible with the action of the Drinfeld-Jimbo quantum group associ-
ated with H = GLn(C), and the canonical basis of Sq(c
α,β) is an extension
of the canonical basis of Tq(c
α,β) in a natural way.
This follows from the work of Lusztig (cf. [Lu3], Chapter 27 in [Lu4]) and
Kashiwara (cf. Theorem2 in [Kas3]). Specializations of these canonical
bases at q = 1 will be called canonical bases of R(cλα,β), S(c
α,β), T (c
α,β).
Lusztig [Lu4] has conjectured that the structural constants associated with
the canonical bases in Proposition 4.2.1 are polynomials in q with nonnega-
tive integral coefficients as in the case of the canonical basis of the (negative
part of the) Drinfeld-Jimbo enveloping algebra. We refer to Proposition 4.2.1
as PH0 in view of this (conjectural) positivity property.
In view of this proposition, we call the rings R(cλα,β), S(c
α,β) and T (c
the canonical rings associated with the Littlewood-Richardson coefficient
, and X = Proj(R(cλ
)), Y = Proj(S(cλ
)) and Z = Proj(T (cλ
)) the
canonical models associated with cλα,β.
4.2.1 From PH0 to PH1,3
Now we study the relevance of PH0 above in the context of PH1,SH,PH2,
and PH3 for Littlewood-Richardson coefficients (Section 1.2).
As already remarked in Section 1.7, PH1 for Littlewood-Richardson coeffi-
cients is a formal consequence of the properties of Kashiwara’s crystal oper-
ators on the canonical bases in PH0 (Proposition 4.2.1); [Dh, Kas2, Li, Lu4].
Specifically, the canonical basis of the ring Rq(c
) also yields a canon-
ical basis for the tensor product Vq,α ⊗ Vq,β of the irreducible Hq modules
with highest weights α and β. The Littlwood-Richardson rule for arbitrary
types follows from the study of Kashiwara’s crystal operators on this canon-
ical basis for the tensor product; [Lu4]. This rule is equivalent to the one
in [Li] based on combinatorial interpretation of the crystal operators in the
path model therein. The article [Dh] derives a convex polyhedral formula
for Littlewood-Richardson coefficients (of arbitrary type) using this com-
binatorial interpretation. Though the complexity-theoretic issues are not
addressed in [Dh], it can be verified that the polyhedral formula therein is a
convex #P -formula. This yields PH1 for Littlewood-Richardson coefficients
of arbitrary types using PH0.
Now let us see the relevance of PH0 in the context of SH for Littlewood-
Richardson coefficients of arbitrary type.
The polytope in [Dh], mentioned above, for type A is equivalent to the
hive polytope in [KT1] in the sense that the number integer points in both
the polytopes is the same. Knutson and Tao prove SH for type A by show-
ing that the hive polytope always has in integral vertex. To extend this
proof to an arbitrary type, one has to convert the polytope in [Dh] into a
polytope that is guaranteed to contain an integral vertex if the index of the
stretching quasipolynomial c̃λ
(n) is one. The main difficulty here is that
we do not have a nice mathematical interpretation for the index. Algorithm
in Theorem 3.1.1 applied to the polytope in [Dh] computes this index in
polynomial time. But it does not give a nice interpretation that can be used
in a proof as above.
This index is simply the largest integer dividing the degrees of all ele-
ments in any basis of the canonical ring T (cλ
)–in particular, the canon-
ical basis. This follows by applying Proposition 3.1.3 to the polytope in
[Dh]. This leads us to ask: is there an interpretation for the index based on
Lusztig’s topological construction of the canonical basis in Proposition 4.2.1?
If so, this may be used to extend the known polyhedral proof for SH in type
A to arbitrary types. Alternatively, it may be possible to prove SH using
topological properties of the canonical basis in the spirit of the topological
(intersection-theoretic) proof [Bl] of SH in type A.
Now let us see the relevance of PH0 in the context of PH3 for Littlewood-
Richardson coefficients.
First, let us consider the minimal positive form (Section 4.1.1) associated
with a Littlewood-Richardson coefficient cλ
of type A. Let T = T (cλ
denote the ring that arises in this case; cf. eq.(4.6). Now we can ask:
Question 4.2.2 Are all di’s occuring in the minimal positive form (cf.
(4.4)) one in this special case? This is equivalent to asking if the ring
T = T (cλα,β) in this case is integral over T1, the degree one component of T .
If so, this would provide an explanation for the conjecture of King at al
[KTT] (cf. eq.(1.3)) in the theory of Cohen-Macauley rings:
Proposition 4.2.3 Assuming yes, the conjecture of King et al [KTT] (Hy-
pothesis 1.2.6) holds.
Remark 4.2.4 In contrast, the ring TP associated with the hive polytope
(cf. beginning of Section 4.2) need not be integral over its degree one compo-
nent, in view of the fact that the hive polytope can have nonintegral vertices
[DM1].
Remark 4.2.5 T = T (cλ
) need not be generated by its degree one compo-
nent T1. If this were always so, the h-vector (hd, · · · , h0) in eq.(1.3) would
be an M-vector (Macauley-vector) [St2]. But one can construct α, β and λ
for which this does not hold.
Proof: (of the proposition) Since T is integral over T1, it has an h.s.o.p., all of
whose elements have degree 1. By Theorem 3.4.1, the singularities of spec(T )
are rational. Hence T is Cohen-Macaulay. Now the result immediately
follows from the theory of Cohen-Macauley rings [St2]. Q.E.D.
In view of this Proposition, the conjecture of King et al will follow if all
canonical basis elements of T (cλ
) can be shown to be integral over the basis
elements of degree one. This requires a further study of the multiplicative
structure of this canonical basis. Considerations for PH3 (Hypothesis 1.2.8)
for Littlewood-Richardson coefficients of arbitrary type are similar.
Similarly, the positivity property (PH2) of the stretching quasipolynomial
associated with Littlewood-Richardson coefficients may possibly follow from
a deep study of the multiplicative structure of the canonical basis as per
PH0 (Proposition 4.2.1), just as positivity of the multiplicative structural
coefficients of the canonical basis for the (negative part of the) Drinfeld-
Jimbo enveloping algebra follows from a deep study of the multiplicative
structure of this basis [Lu4].
4.2.2 On PH0 in general
The discussion above indicates that for Littlewood-Richardson coefficients
PH1,SH,PH3, and plausibly PH2 as well are intimately related to PH0
(Proposition 4.2.1). This leads us to ask if the rings associated in Sec-
tion 4.1.2 with other structural constants under consideration in this paper
have quantizations which satisfy appropriate forms of PH0. If so, this PH0
may be used to derive PH1, SH, PH3, and PH2 (Hypotheses 3.3.1-3.3.4)
for these structural constants. Note that SH (a) follows from PH3 (see the
remark after Hypothesis 3.3.4); PH2 may also follow from PH3. Thus PH1
and PH3 are the ones to focus on.
To formalize this, let s be a structural constant which is either the Kro-
necker coefficient as in Problem 1.1.1, or the plethysm constant as in Prob-
lem 1.1.2, or the multiplicity mπ
in Problem 1.1.3, or the multiplicity sπ
in Problem 1.1.4, when X therein is a class variety. Let R(s), S(s), T (s) be
the rings associated with s (Section 4.1.2). Let X(s) = Proj(R(s)), Y (s) =
Proj(S(s)) and Z(s) = Proj(R(s)). We call R = R(s), S = S(s), T = T (s)
the canonical rings associated with s, and X(s), Y (s), Z(s) the canonical
models associated with s, because we expect these rings and models to be
special as in the case of the Littlewood-Richardson coefficients.
Let H be as in Problem 1.1.3 or Problem 1.1.4. Assume that H is
connected. Let Hq denote the Drifeld-Jimbo quantization of H. Now we
Question 4.2.6 (PH0??) Are there quatizations Rq, Sq of R,S, with Hq-
action, and a quantization Tq of T with “canonical” bases (in some appro-
priate sense) B(Rq), B(Sq), B(Tq), where B(Rq) and B(Sq) are compatible
with the Hq-action and B(Sq) is an extension of B(Tq)? Furthermore, do
these canonical bases have appropriate positivity properties?
In other words, are there quantizations of R,S and T for which PH0
(Proposition 4.2.1) can be extended in a natural way?
If so, this extended PH0 may be used to prove PH1 and SH for s just as
in the case of Littlewood-Richardson coefficients (of type A).
4.3 Nonstandard quantum group for the Kronecker
and the plethysm problems
We now consider this question when s is the kronecker or the plethysm
constant (cf. Problems 1.1.1 and 1.1.2). PH0 for Littlewood-Richardson
coefficients (Proposition 4.2.1) depends critically on the theory of Drinfeld-
Jimbo quantum groups. This is intimately related (in type A) [GrL] to the
representation theory of Hecke algebras. To extend PH0 in the context of the
kronecker and the plethysm constants, one needs extensions of these theories
in the context of Problems 1.1.1-1.1.2. In this section, we briefly review the
results in [GCT4, GCT8, GCT7] in this direction and the theoretical and
experimental evidence it provides in support of PH0–that is, affirmative
answer to Question 4.2.6–in this context.
So let us consider the generalized plethysm problem (Problem 1.1.2).
As expected, the representation theory of Drinfeld-Jimbo quantum groups
and Hecke algebras does not work in the context of this general problem.
Briefly, the problem is that if H is a connected, reductive group and V its
representation, then the homomorphism H → G = GL(V ) does not quan-
tize in the setting of Drinfeld-Jimbo quantum groups. That is, there is no
quantum group homomorphism from Hq, the Drinfeld-Jimbo quantization
of H, to Gq, the Drinfeld-Jimbo quantization of G. In [GCT4, GCT7], a new
nonstandard quantization GHq of G– called a nonstandard quantum group–is
constructed so that there is a quantum group homomorphism Hq → G
When H = G, GHq coincides with the Drinfeld-Jimbo quantum group. The
article [GCT8] gives a conjectural scheme for constructing a nonstandard
canonical basis for the matrix coordinate ring of GHq that is akin to the
canonical basis for the matrix coordinate ring of the Drinfeld-Jimbo quan-
tum group [Lu4, Kas3].
It is known that the Drinfeld-Jimbo quantum group Gq = GLq(V ) and
the Hecke algebra Hn(q) are dually paired: i.e., they have commuting ac-
tions on V ⊗nq from the left and the right that determine each other, where Vq
denotes the standard quantization of V . Furthermore, the Kazhdan-Lusztig
basis forHn(q) is intimately related to the canonical basis for Gq [GrL]. Sim-
ilarly, [GCT7] constructs a nonstandard generalization BHn (q) of the Hecke
algebra which is (conjecturally) dually paired to GHq . The article [GCT8]
gives a conjectural scheme for constructing a nonstandard canonical basis of
BHn (q) akin to the Kazhdan-Lusztig basis of the Hecke algebra Hn(q).
The nonstandard quantum groupGHq and the nonstandard algebraB
n (q)
turn out to be fundamentally different from the standard Drinfeld-Jimbo
quantum group Gq and the Hecke algebra Hn(q). For example, the non-
standard quantum group GHq is a nonflat deformation of G in general. This
means the Poincare series of the matrix coordinate ring of GHq is different
from the Poincare series of the matrix coordinate ring of G. Specifically,
the terms of the first series can be smaller than the respective terms of the
second series. Similarly, BHn (q) is a nonflat deformation of the group algebra
C[Sn] of the symmetric group Sn; i.e., its dimension can be bigger than that
of C[Sn].
Nonflatness of GHq intuitively means that it is “smaller” than G in gen-
eral. Hence, it may seem that there is a loss of information when one goes
from G to GHq . Fortunately, there is none, as per the reciprocity conjecture
in [GCT7]. This roughly says that the information which is lost in the tran-
sition from G to GHq simply gets transfered to B
n (q), which is bigger than
Hn(q). In other words, there is no information loss overall. Hence analogues
of the properties in the standard setting should also hold in the nonstandard
setting, though in a far more complex way.
That is what seems to happen to positivity. Specifically, experimental ev-
idence suggests that the conjectural nonstandard canonical bases in [GCT8]
have nonstandard positivity properties which are complex versions of the
positivity properties in the standard setting. See [GCT7, GCT8, GCT10]
for a detailed story.
4.4 The cone associated with the subgroup restric-
tion problem
In this section, we prove Theorem 3.4.2, by extending the proof of Brion
and Knop (cf. [El]) for the Littlewood-Richardson problem. The proof is in
the spirit of the proof of quasipolynomiality in Section 4.1.
Let G be a connected, reductive group, H a connected, reductive sub-
group, and ρ : H → G a homomorphism. Theorem 3.4.2 has the following
equivalent formulation. Let S(H,G) be the set of pairs (µ, λ) such that
Vµ(H)⊗ Vλ(G) has a nonzero H-invariant. Then,
Theorem 4.4.1 The set S(H,G) is a finitely generated semigroup with re-
spect to addition.
When G = H ×H and the embedding H ⊆ G is diagonal, this special-
izes to the Brion-Knop result mentioned above. The proof follows by an
extension the technique therein.
Proof: Let B be a Borel subgroup of G, U the unipotent radical of B and
T the maximal torus in B. Similarly, let B′ be a Borel subgroup of H, U ′
the unipotent radical of B′ and T ′ the maximal torus in B′. Without loss
of generality, we can assume that B′ ⊆ B, U ′ ⊆ U , T ′ ⊆ T . Let A = C[G]U
be the algebra of regular functions on G that are invariant with respect to
the right multiplication by U . It is known to be finitely generated [El]. The
groups G and T act on A via left and right multiplication, respectively. As
a G× T -module,
A = ⊕λVλ(G), (4.7)
where the torus T acts on Vλ(G) via multiplication by the highest weight
λ∗ of the dual module. Similarly,
A′ = C[H]U
= ⊕λVµ(H), (4.8)
where the torus T ′ acts on Vµ(H) via multiplication by the highest weight
µ∗ of the dual module.
Now A⊗A′ is finitely generated since A and A′ are. Let X = (A⊗A′)H
be the ring of invariants of H acting diagonally on A⊗A′. The torus T ×T ′
acts on X from the right. Since H is reductive, X is finitely generated [PV].
Hence, the semigroup of the weights of the right action of T × T ′ on X is
finitely generated. We have
X = (A⊗A′)H = ((⊕Vλ(G)) ⊗ (⊕Vµ(H)))
H = ⊕(Vλ(G)⊗ Vµ(H))
and the weights of the algebra X are of the form (λ∗, µ∗) such that Vλ(G)⊗
Vµ(H) contains a nontrivial H-invariant. Therefore these pairs form a
finitely generated semigroup. Q.E.D.
For the sake of simplicity, assume that G and H are semisimple in what
follows. Let TR(H,G) denote the polyhedral convex cone in the weight
space of H ×G generated by T (H,G), as defined in Theorem 3.4.2. This is
a generalization of the Littlewood-Richardson cone (Section 2.2.2).
The following generalization of Corollary 3.2.3 is a consequence of The-
orem 3.1.1 and its proof.
Theorem 4.4.2 Assume that the positivity hypothesis PH1 (Section 3.3)
holds for the subgroup restriction problem for the pair (H,G), where both H
and G are classical. Given dominant weights µ, λ of H and G, the polytope
Pµ,λ as in PH1 has a specification of the form
Ax ≤ b (4.9)
where A depends only on H and G, but not on µ or λ, and b depends
homogeneously and linearly on µ, λ. Let n be the total number of columns
in A.
Then, there exists a decomposition of TR(H,G) into a set of polyhedral
cones, which form a cell complex C(H,G), and, for each chamber C in this
complex, a set M(C) of O(n) modular equations, each of the form
aiµi +
biλi = 0 (mod d),
such that
1. Saturation hypothesis SH is equivalent to saying that: (µ, λ) ∈ T (H,G)
iff (µ, λ) ∈ TR(H,G) and (µ, λ) satisfies the modular equations in the
set M(Cµ,λ) associated with the smallest cone Cµ,λ ∈ C(H,G) contain-
ing (µ, λ).
2. Given (µ, λ), whether (µ, λ) ∈ TR(H,G) can be determined in polyno-
mial time.
3. If so, whether (µ, λ) satisfies the modular equations associated with
the smallest cone in C(H,G) containing it can also be determined in
polynomial time.
Proof: Given a point p = (µ′, λ′) in the weight space of H×G, where µ′ and
λ′ are arbitrary rational points, let S(p) denote the constraints (half-spaces)
in the sytem (4.9) whose bounding hyperplanes contain the polytope Pµ′,λ′ .
We can decompose TR(H,G) into a conical, polyhedral cell complex, so that
given a cone C in this complex, and a point p in its interior, the set S(p)
does not depend on p. We shall denote this set by S(C). Thus the affine
span of Pµ,λ, for any (µ, λ) ∈ C, is determined by the linear system
A′x = b′,
where [A′, b′] consists of the rows of [A, b] in (4.9) corresponding to the set
S(C). By finding the Smith normal form of A′, we can associate with C a set
of modular equations that the entries of b′ must satisfy for this affine span to
contain an integer point; see the proof of Theorem 3.1.1. Since the entries of
A′ depend only on H and G, these equations depend only on C. If (µ, λ) ∈
T (H,G), then (µ, λ) is integral, and hence these equations are satisfied.
Conversely, if (µ, λ) ∈ TR(H,G) and these equations are satisfied, then the
saturation property implies that (µ, λ) ∈ T (H,G), as seen by examining
the proof of Theorem 3.1.1. Furthermore, given (µ, λ), the algorithm in the
proof of Theorem 3.1.1 implicitly determines if (µ, λ) ∈ TR(H,G) and if
these modular equations are satisfied in polynomial time. Q.E.D.
4.5 Elementary proof of rationality
In this section we give an elementary proof of rationality in Theorem 3.4.1
(a), when H therein is connected–actually of a slightly stronger statement:
namely, the stretching function m̃π
(n) is asymptotically a quasipolynomial,
as n → ∞; cf. Remark 4.1.4. But this proof cannot be extended to prove
quasipolynomiality for all n. The proof here is motivated by the work of
Rassart [Rs], De Loera and McAllister on the stretching function associated
with a Littlewood-Richardson coefficient.
First, we recall some standard results that we will need.
Vector partition functions
Given an integral s×n matrix B and integral n-vector c, consider the vector
paritition function φB(c), which is the number of integer solutions to the
integer programming problem
By = c, y ≥ 0. (4.10)
For a fixed c, b, let
φB,c(n) = φB(nc)
φB,c,b(n) = φB(nc+ b).
(4.11)
By Sturmfels [Stm] and Szenes-Vergne residue formula [SV], φB(c) is a
piecewise quasipolynomial function of c. That is, Rn can be decomposed into
polyhedral cones, called chambers, so that the restriction of φB(c) to each
chamber R is a multivariate quasipolynomial function of the coordinates of c.
This implies that φB,c(n) is a quasipolynomial function of n. It also implies
that the function φB,c,b(n) is asymptotically a quasipolynomial function of
n, as n→ ∞, because the points nc+ b, as n→ ∞, lie in just one chamber.
The Szenes-Verne residue formula [SV] for vector partition functions also
implies that there is a constant d(B), depending only on B, such that the
period of φB,c(n), for any c, divides d(B).
Klimyk’s formula
Let H ⊆ G and mπλ be as in Theorem 3.4.1 (a), with H connected. Let us
assume that H is semisimple, the general case being similar. Let H and G
be the Lie algebras of H and G respectively. We recall Klimyk’s formula for
. Without loss of generality, we can assume that the Cartan subalgebra
C ⊆ H is a subalgebra of the Cartan subalgebra D ⊆ G. So we have a
restriction from D∗ to C∗, and we assume that the half-spaces determining
positive roots are compatible. We denote weights of H by symbols such as µ
and of G by symbols such as µ̄. To be consistent, we shall use the notation
instead of mπ
in this proof. We write µ̄ ↓ µ if the weight µ̄ of G restricts
to the weight µ of H. We denote a typical element of the Weyl group of
H by W , and a typical element of the Weyl group of G by W̄ . Given a
dominant weight π of G and a weight µ̄ of G, let nµ̄(λ̄) denote the dimension
of the weight space for µ̄ in Bλ̄ = Vλ̄(G).
We assume that:
(A): For any weight µ of H, the number of µ̄’s such that µ̄ ↓ µ is finite.
For example, this is so in the plethysm problem (Problem 1.1.2). We
shall see later how this assumption can be removed.
By Klimyk’s formula (cf. page 428, [FH]),
(−1)W
µ̄↓π−ρ−W (ρ)
nµ̄(Vλ̄), (4.12)
where ρ is half the sum of positive roots of H. We allow µ̄ in the inner sum
to range over all weights µ̄ of G such that µ̄ ↓ π − ρ −W (ρ) by defining
nµ̄(Vλ̄) to be zero if µ̄ does not occur in Vλ̄.
Proof of Theorem 3.4.1 (a)
The goal is to express m̃π
(n) as a linear combination of vector partition
functions φB,c,b(n)’s, for suitable B, c, b’s, using Klimyk’s formula for m
After this, we can deduce asymptotic quasipolynomiality of m̃π
(n) from
asymptotic quasipolynomiality of φB,c,b(n)’s.
By Kostant’s multiplicity formula (cf. page 421 [FH]),
nµ̄(Vλ̄) =
(−1)W̄P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)), (4.13)
where P (λ̄), for a weight λ̄ of G, denotes the Kostant partition function;
i.e., the number of ways to write λ̄ as a sum of positive roots of G. It is
important for the proof that Kostant’s formula (4.13) holds even if µ̄ is not
a weight that occurs in the representation Vλ̄–in this case, nµ̄(Vλ̄) = 0, and
the right hand side of (4.13) vanishes.
By eq.(4.12) and (4.13),
(−1)W (−1)W̄
µ̄↓π−ρ−W (ρ)
P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)). (4.14)
Let D denote the dominant Weyl chamber in the weight space of G. Let
C denote the Weyl chamber complex associated with the weight space of G.
The cells in this complex are closed polyhedral cones. Each cone is either
the chamber W̄ (D), for some Weyl group element W̄ , or a closed face of
W̄ (D) of any dimension.
Using Möbius inversion, the inner sum
µ̄↓π−ρ−W (ρ)
P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄))
in eq.(4.14) can be written as a linear combination
µ̄∈C:µ̄↓π−ρ−W (ρ)
P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)),
where C ranges over chambers in the Weyl chamber complex C, a(C) is an
appropriate constant for each C.
Hence,
(−1)W (−1)W̄
µ̄∈C:µ̄↓π−ρ−W (ρ)
P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)).
(4.15)
Now think of π and λ̄ as variables. But H and G are fixed, and hence
also the quantities such as ρ and ρ̄.
Claim 4.5.1 For fixed Weyl group elements W, W̄ and a fixed C, the sum
µ̄∈C:µ̄↓π−ρ−W (ρ)
P (W̄ (λ̄+ ρ̄)− (µ̄ + ρ̄)) (4.16)
can be expressed as a vector partition function associated with an appropriate
linear system
By = c, y ≥ 0, (4.17)
where the matrix
B = BH,G,C ,
depends only on C and the root systems of H and G, but not on π and λ̄,
and the coordinates of the vector
c = mW,W̄,C(λ̄, π, ρ, ρ̄),
depend on W, W̄ ,C, ρ, ρ̄, π, π, and furthermore, their dependence on π, λ̄, ρ, ρ̄
is linear.
Here assumption (A) is crucial. Without it, the sum (4.16) can diverge. Of
course, without assumption (A), we can still make the sum finite, by requir-
ing that µ̄ lie within the convex hull Hλ̄ generated by the points {W̄ (λ̄)},
where W̄ ranges over all Weyl group elements. This means we have to add
constraints to the system (4.17) corresponding to the facets of Hλ̄. But
the entries of the resulting B would depend on λ̄, and the theory of vector
partition functions will no longer apply.
Proof of the claim: Let µ̄i’s denote the integer coordinates of µ̄ in the basis
of fundamental weights. We denote the integer vector (µ̄1, µ̄2, · · · ) by µ̄
again. The Kostant partition function P (ν) is a vector partition function
associated with an integer programming problem:
BP v = ν, v ≥ 0,
where the columns of BP correspond to positive roots of G. The sum in
(4.16) is equal to the number of integral pairs (µ̄, v) such that
1. µ̄ ∈ C,
2. µ̄ ↓ π − ρ−W (ρ),
3. BP v = W̄ (λ̄+ ρ̄)− (µ̄ + ρ̄), v ≥ 0.
The first two condititions here can be expressed in terms of linear con-
straints (equalities and inequalities) on the coordinates µ̄i’s. Thus the three
conditions together can be expressed in terms of linear constraints on (µ̄, v).
By the finiteness assumption (A), the polytope determined by these con-
straints is a bounded polytope. The number of integer points in such a
polytope can be expressed as a vector partition function (cf. [BBCV]). This
proves the claim.
Let us denote the vector partition associated with the integer program-
ming problem (4.17) in the claim by φW,W̄ ,C(c(λ̄, π, ρ, ρ̄)). Then
(−1)W (−1)W̄
a(C)φW,W̄ ,C(c(λ̄, π, ρ, ρ̄)). (4.18)
Hence,
(n) = mnλ̄nπ =
(−1)W (−1)W̄
a(C)φW,W̄ ,C(c(nλ̄, nπ, ρ, ρ̄)).
(4.19)
It follows from Claim 4.5.1 and the standard results on vector partition
functions mentioned in the begining of this section that
gW,W̄ ,C(n) = φW,W̄ ,C(c(nλ̄, nπ, ρ, ρ̄)),
is asymptitically a quasipolynomial function of n. Hence, m̃π
(n) is also
asymptotically a quasipolynomial function of n. This implies (cf. [St1])
(t) =
(n)tn (4.20)
is rational function of t.
This proves Theorem 3.4.1 (a) under the finiteness assumption (A).
It remains to remove the assumption (A). Let G′ ⊇ H be the smallest
Levi subalgebra of G containing H. Then
mππ′ , (4.21)
where π′ ranges over dominant weights of G′, mπ
denotes the multiplicity of
Vπ′(G
′) in Vλ̄(G), and m
the multiplicity of Vπ(H) in Vπ′(G
′). Furthermore,
1. the finiteness asssumption (A) is now satisfied for the pair (G′,H): i.e.,
for any weight µ of H, the number of weights µ′’s of G′ such that µ′ ↓ µ
is finite.
2. There is a polyhedral expression for mπ
; this follows from [Li, Dh].
By the first condition and the argument above, we get an expression for
akin to (4.18). Substituting this expression and the polyhedral expres-
sion for mπ
in (4.21), leads to a formula for m̃π
(n) as a linear combination
of φB,c,b(n)’s for appropriate B, c, b’s. After this, we proceed as before.
This proves Theorem 3.4.1 (a). Q.E.D.
We also note down the following consequence of the proof.
Proposition 4.5.2 There is a constant D depending only G and H, such
that for any λ̄, π, orders of the poles of Mπ
(t) (cf. (4.20), as roots of unity,
divide D.
A bound onD provided by the proof below is very weak: D = O(2O(rank(G))).
Proof: It suffices to to bound the period of the quasipolynomial m̃π
(n). For
this, it suffices to let n→ ∞. For a fixed W, W̄ ,C, the chamber containing
c(nλ̄, nπ, ρ, ρ̄)) is completely determined by λ̄ and π as n→ ∞. Under these
conditions, the degree of φW,W̄ ,C(c(nλ̄, nπ, ρ, ρ̄)) is equal to the dimension of
the polytope associated with this vector partition function. This dimension
is clearly O(rank(G)2).
By Szenes-Vergne residue formula [SV], there is a constant D depending
on only G,H,W, W̄ , C, such that the period of the quasipolynomial h(n) =
φW,W̄ ,C(c(nλ̄, nπ, 0, 0)) divides D for every λ̄, π; here we are putting ρ and
ρ̄ equal to zero, since we are interested in what happens as n→ ∞. Q.E.D.
Chapter 5
Parallel and PSPACE
algorithms
In this chapter we give PSPACE algorithms (cf. Theorem 3.4.3) for com-
puting the various structural constants under consideration . We shall only
prove Theorem 3.4.3, when H is therein is either a complex, semisimple
group, or a symmetric group, or a general linear group over a finite field,
the extension to the general case being routine.
We recall two standard results in parallel complexity theory [KR], which
will be used repeatedly.
Let NC(t(N), p(N) denote the class of problems that can be solved
in O(t(N)) parallel time using O(p(N)) processors, where N denotes the
bitlength of the input. Let
NC = ∪iNC(log
i(N),poly(N)).
This is the class of problems having efficient parallel algorithms.
Proposition 5.0.3 [Cs, KR] Let A be an n × n-matrix with entries in a
ring R of characteristic zero. Then the determinant of A, and A−1, if A
is nonsingular, can be computed in O(log2 n) parallel steps using poly(n)
processors; here each operation in the ring is considered one step. Hence, if
R = Q, the problems of computing the determinant, the inverse and solving
linear systems belong to NC.
Proposition 5.0.4 The class NC(t(N), 2t(N)) ⊆ SPACE(O(t(N))). In
particular, NC(poly(N), 2O(poly(N))) ⊆ PSPACE.
5.1 Complex semisimple Lie group
In this section we prove a special case of Theorem 3.4.3 for the general-
ized plethym problem (Problem 1.1.2). Accordingly, let H be a complex,
semisimple, simply connected Lie group, G = GL(V ), where V = Vµ(H) is
an irreducible representation of H with dominant weight µ, ρ : H → G the
homomorphism corresponding to the representation, and mπ
the multiplic-
ity of Vπ(H) in Vλ(G), considered as an H-module via ρ; cf. Problem 1.1.3.
Then:
Theorem 5.1.1 The multiplicitymπ
can be computed in poly(〈λ〉, 〈µ〉, 〈π〉,dim(H))
space.
Here it is assumed that the partition λ = λ1 ≥ λ2 ≥ · · ·λr > 0 is rep-
resented in a compact form by specifying only its nonzero parts λ1, . . . , λr.
This is important since dim(G) can be exponential in dim(H) and 〈µ〉. A
compact representation allows 〈λ〉 to be small, say poly(dim(H), 〈µ〉), in this
case.
We begin with a simpler special case.
Proposition 5.1.2 If dim(V ) = poly(dim(H)), then mπ
can be computed
in PSPACE; i.e., in poly(〈λ〉, 〈µ〉, 〈π〉,dim(H)) space.
This implies that the Kronecker coefficient (Problem 1.1.1) can be computed
in PSPACE.
Proof: Let us use the notation λ̄ instead of λ to be consistent with the
notation used in Klimyk’s formula (4.12). By the latter,mπ
can be computed
in PSPACE if nµ̄(Vλ̄) in that formula can be computed in PSPACE for every
µ̄ and λ̄. In type A, this is just the number of Gelfand-Tsetlin tableau with
the shape λ̄ and weight µ̄. If dim(V ) = poly(dim(H)), the size of such
a tableau is O(dim(V )2) = poly(dim(H)). So we can count the number
of such tableu in PSPACE as follows: Begin with a zero count, and cycle
through all tableaux of shape λ̄ in polynomial space one by one, increasing
the count by one everytime the tableau satisfies all constraints for Gelfand-
Tsetlin tableau and has weight µ̄. In general, the role of Gelfand-Tsetlin
tableaux is played by Lakshmibai-Seshadri (LS) paths [Li, Dh]. Q.E.D.
The argument above does not work if dim(V ) is not poly(dim(H)), as
in the plethym problem (Problem 1.1.2), where dim(V ) = dim(Vµ) can
be exponential in n = dim(H) and the bitlength of µ. In this case, the
algorithm cannot even afford to write down a tableau since its size need not
be polynomial.
Next we turn to Theorem 5.1.1. For the sake of simplicity, we shall
prove it only for H = SLn(C), or rather GLn(C)–i.e., the usual plethysm
problem. This illustrates all the basic ideas. The general case is similar. We
shall prove a slightly stronger result in this case:
Theorem 5.1.3 The plethysm constant aπ
can be can be computed in
poly(〈λ〉, 〈µ〉, 〈π〉) space.
Here the dependence on n = dim(H) is not there. This makes a difference
if the heights of µ and π are less than n = dim(H)–remember that we are
using a compact representation of a partition in which only nonzero parts
are specified. This is really not a big issue. Because aπ
depends only on
the partitions λ, µ, π and not n. Hence, without loss of generality, we can
assume that n is the maximum of the heights of µ and π. It is possible to
strengthen Theorem 5.1.1 similarly.
To prove Theorem 5.1.3, we shall give an efficient parallel algorithm to
compute ãπλ,µ that works in poly(〈λ〉, 〈µ〉, 〈π〉) parallel time usingO(2
poly(〈λ〉,〈µ〉,〈π〉))
processors. This will show that the problem of computing ãπλ,µ is in the com-
plexity class NC(poly(〈λ〉, 〈µ〉, 〈π〉), 2poly(〈λ〉,〈µ〉,〈π〉)), which is contained in
PSPACE by Proposition 5.0.4. The basic idea is to parallelize the classical
character-based algorithm for computing aπ
by using efficient parallel algo-
rithm for inverting a matrix and solving a linear system (Proposition 5.0.3).
We begin by recalling the standard facts concerning the characters of
the general linear group. Given a representation W of GLm(C), let ρ :
GLm(C) → GL(W ) be the representation map. Let χρ(x1, . . . , xm) de-
note the formal character of this representation W . This is the trace of
ρ(diag(x1, . . . , xm)), where diag(x1, . . . , xn) denotes the generic diagonal ma-
trix with variable entries x1, . . . , xm on its diagonal. If W is an irreducible
representation Vλ(GLm(C)), then χρ(x1, . . . , xm) is the Schur polynomial
Sλ(x1, . . . , xm). By the Weyl character formula,
λi+m−i
|xm−ij |
, (5.1)
where |aij | denotes the determinant of anm×m-matrix a. The Schur polyno-
mials form a basis of the ring of symmetric polynomials in x1, . . . , xm. The
simplest basis of this ring consists of the complete symmetric polynomials
Mβ(x1, . . . , xm) defined by
Mβ(x1, . . . , xm) =
where γ ranges over all permutations of β and tγ =
i . Schur polyno-
mials are related to Mβ by:
Mβ , (5.2)
where k
is the Kostka number. This is the number of semistandard tableau
of shape λ and weight β.
If the representation W is reducible, its decomposition into irreducibles
is given by:
m(π)Vπ(GLn(C)), (5.3)
where m(π)’s are the coefficients of the formal character χρ(x1, . . . , xm) in
the Schur basis:
m(π)Sπ.
Proof of Theorem 5.1.3
Let λ, µ, π be as in Theorem 5.1.3. Let H = GLn(C), V = Vµ(H), G =
GL(V ). Let sλ(x1, . . . , xm) be the formal character of the representation
Vλ(G) of G. Here m = dim(Vµ) can be exponential in n and 〈µ〉. The basis
of Vµ(H) is indexed by semistandard tableau of shape µ with entries in [1, n].
Let us order these tableau, say lexicographically, and let Ti, 1 ≤ i ≤ m,
denote the i-th tableau in this order. With each tableau T , we associate a
monomial
t(T ) =
wi(T )
where wi(T ) denotes the number of i’s in T . Given a polynomial f(x1, . . . , xm),
let us define fµ = fµ(t1, . . . , tn) to be the polynomial obtained by substi-
tuting xi = t(Ti) in f(x1, . . . , xm). Then the formal character of Vλ(G),
considered as an H-representation of via the homomorphism H → G =
GL(Vµ(H)), is the symmetric polynomial Sλ,µ(t1, . . . , tn) = (Sλ)µ. The
plethysm constant aπλ,µ is defined by:
Sλ,µ(t1, . . . , tn) =
aπλ,µSπ(t1, . . . , tn). (5.4)
An efficient parallel algorithm to compute aπλ,µ is as follows. Here by an
efficient parallel algorithm, we mean an algorithm that works in poly(〈λ〉, 〈µ〉, 〈π〉)
time using 2poly(〈λ〉,〈µ〉,〈π〉) processors. We will repeatedly use Proposi-
tion 5.0.3.
Algorithm
(1) Compute Sλ,µ(t1, . . . , tn). By the Weyl character formula (5.1),
Sλ,µ(t1, . . . , tn) =
Aλ,µ(t1, . . . , tn)
Bλ,µ(t1, . . . , tn)
where Aλ(x1, . . . , xm) and Bλ(x1, . . . , xm) denote the numerator and denom-
inator in (5.1), and Aλ,µ = (Aλ)µ, and Bλ,µ = (Bλ)µ. Let R = C[t1, . . . , tn].
Aλ,µ(t1, . . . , tn) = |t(Tj)
λi+m−i|.
This is the determinant of an m×m matrix with entries in R, where m =
dim(V ) can be exponential in n and 〈µ〉. It can be evaluated in O(log2m)
parallel ring operations using poly(m) processors. Each ring element that
arises in the course of this algorithm is a polynomial in t1, . . . , tn of total
degree O(|λ|m), where |λ| denotes the size of λ. The total number of its
coefficients is r = O((|λ|m)n). Hence each ring operation can be carried
out efficiently in O(log2(r)) parallel time using poly(r) processors. Since
logm = poly(n, 〈µ〉) and log r = poly(n, 〈λ〉, 〈µ〉), it follows that Aλ,µ can
be evaluated in poly(n, 〈µ〉, 〈λ〉) parallel time using 2poly(n,〈µ,λ〉) processors.
The determinant Bλ,µ can also be computed efficiently in parallel in a similar
fashion. To compute Sλ,µ, we have to divide Aλ,µ by Bλ,µ. This can be done
by solving an r × r linear system, which, again, can be done efficiently in
parallel. This computation yields representation of Sλ,µ in the monomial
basis {Mβ} of the ring of symmetric polynomials in t1, . . . , tn.
(2) To get the coefficients aπλ,µ, we have to get the representation of Sλ,µ(t)
in the Schur basis. This change of basis requires inversion of the matrix
in the linear system (5.2). The entries of the matrix K occuring in this
linear system are Kostka numbers. Each Kostka number can be computed
efficiently in parallel. Hence, all entries of this matrix can be computed
efficiently in parallel. After this, the matrix can be inverted efficiently in
parallel, and the coefficients aπλ,µ’s of Sλ,µ in the Schur basis can be computed
efficiently in parallel. Finally, we use Proposition 5.0.4 to conclude that aπ
can be computed in PSPACE. Q.E.D.
5.2 Symmetric group
Next we prove Theorem 3.4.3 when H = Sm. Let X = Vµ(Sm) be an
irreducible representation (the Specht module) of Sm corresponding to a
partition µ of size m. Let ρ : H → G = GL(X) be the corresponding
homomorphism.
Theorem 5.2.1 Given partitions λ, µ, π, where µ and π have size m, the
multiplicity mπ
of the Specht module Vπ(Sm) in Vλ(G) can be computed in
poly(m, 〈λ〉) space.
Remark 5.2.2 The bitlengths 〈µ〉 and 〈π〉 are not mentioned in the com-
plexity bound because they are bounded by m.
For this, we need three lemmas.
Lemma 5.2.3 The character of a symmetric group can be computed in
PSPACE. Specifically, given a partition π of size m, and a sequence
i = (i1, i2, . . .) of nonnegative integers such that
jij = m, the value of
the character χπ of Sm on the conjugacy class Ci of permutations indexed
by i can be computed in poly(m) parallel time using 2poly(m) processors.
Hence it can be computed in poly(m) space (cf. Proposition 5.0.4).
Here the conjugacy class Ci consists of those permutations that have i1
1-cycles, i2 2-cycles, and so on.
Proof: Let k be the height of the partition π. Let x = (x1, . . . , xk) be the
tuple of variables xi’s. Given a formal series f(x) and a tuple (l1, . . . , lk) of
nonnegative integers, let [f(x)](l1,...,lk) denote the coefficient of x
1 · · · x
By the Frobenius character formula [FH],
χλ(Ci) = [f(x)](l1,...,lk), (5.5)
where
l1 = π1 + k − 1, l2 = π2 + k − 2, . . . , lk = πk,
f(x) = ∆(x)
Pj(x)
∆(x) =
i<j(xi − xj),
Pj(x) = x
1 + · · ·+ x
(5.6)
Since deg(f) = poly(m) and k ≤ m, the total number of coefficients of
f(x) is 2poly(m). Hence, we can evaluate f(x) in PSPACE by setting up
appropriate recurrence relations.
Alternatively, we can easily evaluate f(x) in poly(m) parallel time using
2poly(m) processors, and then extract its required coefficient. After this, the
result follows from Proposition 5.0.4. Q.E.D.
Lemma 5.2.4 Suppose φ is a character of Sm whose value on any conju-
gacy class Ci can be computed in O(s) space, for some parameter s. Then,
the multiplicity of the representation Vπ(Sm) in the representation Vφ(Sm)
corresponding φ can be computed in O(poly(m) + s) space.
Proof: The multiplicity is given by the inner product
〈φ, χπ〉 =
¯φ(σ)χπ(σ). (5.7)
By assumption, φ(σ) can be computed in O(s) space, and by Lemma 5.2.3,
χπ(σ) can be computed in poly(m) space. Hence, the result follows from
the preceding formula. Q.E.D.
Given an irreducible representation X = Vµ(Sm) and an irreducible rep-
resentation W = Vλ(G) of G = GL(X)), let ρµ denote the representation
map Sm → G, ρλ the representation map G→ GL(W ), and
ρ : Sm → G→ GL(W )
their composition. This is a representation of Sm. Let χρ be the character
of ρ.
Lemma 5.2.5 For any σ ∈ Sm, χρ(σ) can be computed in poly(m, 〈λ〉) in
poly(m, 〈λ〉) space.
The bitlength 〈µ〉 is not mentioned in the complexity bound because it
is bounded by m.
Proof: Let r = dim(X). The formal character of the representation Vλ(G)
of G = GL(X) is the Schur polynomial Sλ(x1, . . . , xr), r = dim(X). Hence,
χρ(σ) = Sλ(α)
where α = (α1, . . . , αr) is the tuple of eigenvalues of ρµ(σ). We shall compute
the right hand side fast in parallel–i.e., in poly(m, 〈λ〉) parallel time using
2poly(m,〈λ〉) processors–and then use Proposition 5.0.4 to conclude the proof.
This is done as follows.
(1) Let χµ denote the character of the representation ρµ. Let pi(α) = α
· · · + αir denote the i-th power sum of the eigenvalues. For any i,
pi(α) = χµ(σ
We can compute σi, for i ≤ |λ|, where |λ| denotes the size of λ, in poly(log i,m) =
poly(m, 〈λ〉) time using repeated squaring. After this χµ(σ
i) can be com-
puted fast in parallel in poly(m) time using Lemma 5.2.3. Thus each pi(α)
can be computed in poly(m, 〈λ〉) time in parallel using 2poly(m,〈λ〉) proces-
sors. We calculate pi(α) in parallel for all i ≤ |λ|, and all pγ(α) =
j pγj (α)
in parallel for all partitions γ of size at most m.
(2) After this, we calculate the complete symmetric function hi(α), for
each i ≤ |λ|, fast in parallel, by using the relation [Mc]:
|γ|=i
z−1γ pγ ,
where zγ =
i≥1 i
mimi!, and mi = mi(γ) denotes the number of parts of γ
equal to i. Thus we can calculate hγ(α) =
j hγj (α), for all partitions γ of
size m, fast in parallel.
(3) To compute Sλ(α), we recall that the transition matrix between
the Schur basis {Sλ} and the complete symmetric basis {hγ} of the ring
of symmetric functions is K∗, the transpose inverse of the Kostka matrix
K = [Kλ,γ ], where Kλ,γ denote the Kostka number; cf. [Mc]. As we noted in
the proof of Theorem 5.1.3, each Kostka number can be computed in fast in
parallel. Hence, K can be computed fast in parallel. After this, its inverse
K−1 can be computed fast in parallel by Proposition 5.0.3–this is the crux
of the proof–and finally K∗ as well. Thus Sλ(α) can be computed fast in
parallel, since each hγ(α) can be computed fast in parallel. Q.E.D.
Theorem 5.2.1 follows from Lemma 5.2.3,5.2.4 and 5.2.5. Q.E.D.
5.3 General linear group over a finite field
In this section we prove Theorem 3.4.3, when H therein is the general lin-
ear group GLn(Fpk) over a finite field Fpk . Irreducible representations of
H = GLn(Fpk) have been classified by Green [Mc]. They are labelled by
certain partition-valued functions. See [Mc] for a precise description of these
labelling functions. It is clear from the description therein that each labelling
function has a compact representation of bitlength O(n + k + 〈p〉), where
〈p〉 = log2 p; we specify a function by giving its partition values at the places
where it is nonzero. Let µ denote any such label. Let X = Vµ(H) be the
corresponding irreducible representation of H, and ρ : H → G = GL(X)
the corresponding homomorphism.
Theorem 5.3.1 Given a partition λ and labelling functions µ and π as
above, the multiplicity mπλ,µ of the irreducible representation Vπ(H) in Vλ(G)
can be computed in poly(n, k, 〈p〉, 〈λ〉) space.
The proof is similar to that of Theorem 5.2.1 for the symmetric group
with the following result playing the role of Lemma 5.2.3:
Lemma 5.3.2 Given a label γ of an irreducible character χγ of H = GLn(Fpk)
and a label δ of a conjugacy class in H, the value χγ(δ) can be computed
in poly(n, k, 〈p〉) parallel time using 2poly(n,k,〈p〉) processors, and hence by
Proposition 5.0.4, in poly(n, k, 〈p〉) space.
The label δ of a conjugacy class in H is also a partition valued function
[Mc], which admits a compact representation of bitlength poly(n, k, 〈p〉).
Proof: We shall parallelize Green’s algorithm [Mc] for computing the charac-
ter values, and then conclude by Proposition 5.0.3. Green shows that χγ(δ)’s
are entries of a transition matrix between a two polynomial bases: the first
constructed using Hall-Littlewood polynomials, and the second using Schur
polynomials. We have construct this transition matrix fast in parallel. We
shall only indicate here how the transition matrix between the basis of Hall-
Littlewood polynomials and the Schur basis for the ring symmetric functions
over Z[t] can be constructed fast in parallel. This idea can then be easily
extended to complete the proof.
First, we recall the definition of the Hall-Littlewood polynomial Pπ(x; t) =
Pπ(x1, . . . , xk; t) [Mc]. This is a symmetric polynomial in xi’s with co-
efficients in Z[t]. It interpolates between the Schur function sπ(x) and
the monomial symmetric function mπ(x) because Pπ(x; 0) = sπ(x) and
Pπ(x; 1) = mπ(x). The formal definition is as follows:
For a given partition π, let vπ(t) =
i≥0 vmi(t), where mi is the number
of parts of π equal to i, and
vm(t) =
1− ti
Pπ(x; t) =
Aπ(x, t)
Bπ(x, t)
, (5.8)
where
Aπ(x, t) =
sgn(σ)σ(xπ11 · · · x
i<j xi − txj ,
Bπ(x, t) = vπ(t)
i<j(xi − xj).
(5.9)
Here sgn(σ) denotes the sign of σ.
Let wπ,α(t)’s be the coeffcients of Pπ(x, t) in the Schur basis:
Pπ(x; t) =
wπ,α(t)sα(x). (5.10)
We want to calculate the matrix W = [wπ,α] fast in parallel. Using
formula (5.9), we calculate Aπ(x; t) fast in parallel; i.e., we calculate the
coefficients of Aπ(x; t) in the basis of monomials in x and t. We calculate
Bπ(x; t) similarly. After this the division in (5.8) can be carried out by
solving a an appropriate linear system. This can be done fast in parallel
by Proposition 5.0.3. Since, Pπ(x; t) is symmetric in xi’s, this yields its
coefficients in the monomial symmetric basis {mα(x)} with the coefficients
being in Z[t]. The transition matrix [Mc] from the monomial symmetric
basis to the Schur basis is given by the inverse of the Kostka matrix. This
inverse can be computed fast in parallel by Proposition 5.0.3. After this,
the coefficients wπ,α’s of Pπ(x; t) in the Schur basis can be computed fast in
parallel.
Furthermore, the inverse of W = [Wπ,α] can also be computed fast in
parallel by Proposition 5.0.3. Q.E.D.
5.3.1 Tensor product problem
Analogue of the Kronecker problem (Problem 1.1.1) for H = GLn(Fpk) is:
Problem 5.3.3 Given partition valued functions λ, µ, π, decide if the mul-
tiplicity bπλ,µ of Vπ(H) in the tensor product Vλ(H)⊗ Vµ(H) is nonzero.
In this context:
Theorem 5.3.4 The multiplicity mπ
can be computed in PSPACE; i.e.,
in poly(n, k, 〈p〉) space.
Proof: This follows from Lemma 5.3.2 and analogues of Lemmas 5.2.4 and
5.2.5 in this setting. Q.E.D.
A possible canditate for a stretching function assoociated with bπ
b̃πλ,µ(n) = b
nλ,nµ,
where nλ denotes the stretched partition-valed function obtained by stretch-
ing each partition value of λ by a factor of n. In other words b̃π
(n) is
the multiplicity of Vnπ(H(n)) in Vnλ(H(n)) ⊗ Vnµ(H(n)), where H(n) =
GLnm(Fpk) is the stretched group. Is it a quasi-polynomial? If so, we can
also ask for a good bound on its saturation and positivity indices.
5.4 Finite simple groups of Lie type
The work of Deligne-Lusztig [DL] and Lusztig[Lu5] yield an algorithm for
computing the character values for finite simple groups of Lie type.
Question 5.4.1 Can this algorithm be parallelized?
If so, Lemma 5.3.2, and hence Theorem 5.3.1, can be extended to finite
simple groups of Lie type.
Chapter 6
Experimental evidence for
positivity
In this chapter we give experimental evidence for positivity (PH2,3).
6.1 Littlewood-Richardson problem
Experimental evidence for PH2 in the context of the Littlewood-Richardson
problem (Problem 1.2.1) has been given in [DM2], and for PH3 in type A
in [KTT]. We give experimental evidence for PH3 in types B,C,D here.
Let Cλ
(t) be as in eq.(1.2). Its reduced positive form for various values
of α, β, λ is shown in Figure 6.1 for type B, in Figure 6.2 for type C, and
Figure 6.3 for type D. The rank of the Lie algebra is three in all cases.
In these types, the period of the stretching quasipolynomial c̃λ
(n) is at
most two. Accordingly, the period of every pole of Cλ
(t) is at most two.
The tables were computed from the tables in [DM2] for the stretching quasi-
polynomial c̃λ
(n) in these cases.
6.2 Kronecker problem, n = 2
Let kπλ,µ be the Kronecker coefficient in Problem 1.1.1. Let k̃
λ,µ(n) = k̃
nλ,nµ
be the associated stretching quasi-polynomial, and
Kπλ,µ(t) =
k̃πλ,µ(n)t
the associated rational function. An explicit formula (with alternating signs)
for the Kronecker coefficient, when n = 2, has given by Remmel and White-
head [RW] and Rosas [Ro], and a positive formula in [GCT9]. This case
turns out to be nontrivial. For example, the number of chambers (domains)
of quasi-polynomiality in this case turns out to be more than a million. Their
explicit description can be found out using the formula for the Kronecker
coefficient in [RW].
We implemented Rosas’ formula to check PH2 for the quasipolynomial
(n) for a few thousand values of µ, ν and λ with the help of a computer.
A large number of samples was chosen to ensure that a significant fraction
of the chambers were sampled. The quasi-polynomial k̃πλ,µ(n) and a positive
form of the rational function Cπλ,µ(t) are shown Figures 6.4 and 6.5 for few
sample values of λ = (λ1, λ2), µ = (µ1, µ2), and π = (π1, π2, π3, π4). It
may be noted that k̃π
(n) need not be a polynomial; this answers Kirillov’s
question [Ki] in the negative. But its period is at most two for n = 2. This
follows from the formula in [RW]. For the λ, µ and π that we sampled,
positivity index of k̃π
(n) is always zero. But it turns out [BOR] that
there are some λ, µ and π for which the saturation and positivity indices of
(n) are nonzero (one), but very small and thus consistent with SH and
PH2 (Hypothesis1.6.6) in this paper; in the earlier version of this paper, SH
and PH2 stipulated that the saturation and positivity indices are always
zero. These (λ, µ, π) escaped our random sampling, because their density is
extremely small [BOR].
6.3 G/P and Schubert varieties
Let V = Vλ(G) be an irreducible representation of G = SLk(C) correspond-
ing to a partition λ. Let vλ be the point in P (V ) corresponding to the
highest weight vector, and X = Gvλ ∼= G/Pλ its closed orbit. Let hk,λ(n)
be the Hilbert function of the homogeneous coordinate ring R of X. It is
a quasipolynomial since spec(R) has rational singularities. In fact, it is a
polynomial, since t = 1 is the only pole of the Hilbert series
Hk,λ(t) =
hk,λ(n)t
Figure 6.6 gives experimental evidence for strict positivity (PH2) of hk,λ(n)
(as discussed in Section 3.5.5) for a few sample values of k and λ. Figure 6.7
gives experimental evidence for strict positivity of the Hilbert polynomial
of the Schubert subvarieties of the Grassmanian; there Gn,k denotes the
Grassmannian of k-planes in V = Cn, and Ωa, a = (a(1), . . . , a(d)) its
Schubert subvariety consisting of the k-subspaces W such that dim(W ∩
Vn−k+i−a(i)) ≥ i for all i, where V = Vn ⊃ · · ·V1 ⊃ 0 is a complete flag of
subspaces in V . The Hilbert polynomials were computed using the explicit
polyhedral interpretation for them deduced from the theory of algebras with
straightening laws (Hodge algebras) [DEP2].
6.4 The ring of symmetric functions
Let V = Ck, G = GL(V ), H = Sk, with the natural embedding H →
G. Let us consider the spacial case of the subgroup restriction problem
(Problem 1.1.3), with Vλ(G) = V , and Vπ(H) the trivial representation of
H. Then s = mπ
, the multiplicity of the trivial representation in V , is
one. Though the decision problem (Problem 1.1.3) is trivial in this case, the
canonical model associated with s is nontrivial.
The canonical rings R = R(s) and S = S(s) associated with s in this
case coincide with C[V ] = C[x1, . . . , xk]. The ring T = T (s) = S
C[x1, . . . , xk]
Sk is the subring of symmetric functions. Its Hilbert function
h(n) is a quasipolynomial. PH1 and PH3 for Z = Proj(T ), as per Defini-
tion 3.5.2, follow easily, the latter from the well known rational generating
function for the partition function [St1]. But PH2 turns out to be nontriv-
ial. Figures 6.8-6.13 give experimental evidence for strict positivity of h(n)
(PH2). In these figures, the i-th row of the table for a given k shows hi(n),
where hi(n), 1 ≤ i ≤ l, are such that h(n) = hi(n), when n = i modulo the
period l of h(n).
α β λ Cλ
(0, 15, 5) (12, 15, 3) (6, 15, 6) 350 t
8+19121 t7+123576 t6+297561 t5+342064 t4+192779 t3+46992 t2+2641 t+1
(1−t)
(1−t2)
(4, 8, 11) (3, 15, 10) (10, 1, 3) 1+5 t+6 t
(1−t2)
(8, 1, 3) (11, 13, 3) (8, 6, 14) 2 t
8+45 t7+259 t6+591 t5+773 t4+522 t3+198 t2+29 t+1
(1−t)3(1−t2)
(8, 9, 14) (8, 4, 5) (1, 5, 15) 136 t
9+3422 t8+20204 t7+53608 t6+76076 t5+60986 t4+26674 t3+5568 t2+345 t+1
(1−t)3(1−t2)
(10, 5, 6) (5, 4, 10) (0, 7, 12) 219 t
8+12135 t7+79231 t6+193003 t5+223919 t4+127907 t3+31704 t2+1870 t+1
(1−t)6(1+t)3
Figure 6.1: Cλ
(t) for B3
α β λ Cλα,β(t)
(1, 13, 6) (14, 15, 5) (5, 11, 7) 18145 t
8+267151 t7+1070716 t6+1917716 t5+1735692 t4+778184 t3+144596 t2+5538 t+1
(1−t)4(1−t2)
(4, 15, 14) (12, 12, 10) (4, 9, 8) 2280 t
9+267658 t8+2746131 t7+9276935 t6+14682332 t5+11903923 t4+4746803 t3+751126 t2+21249 t+1
(1−t)4(1−t2)
(9, 0, 8) (8, 12, 9) (7, 7, 3) 3 t
2+4 t+1
(1−t)6
(10, 2, 7) (8, 10, 1) (7, 5, 5) 8984 t
8+132826 t7+534183 t6+960491 t5+873227 t4+394045 t3+74067 t2+2941 t+1
(1−t)4(1−t2)
(10, 10, 15) (11, 3, 15) (10, 7, 15) 7162 t
9+736327 t8+7178960 t7+23540366 t6+36359642 t5+28788904 t4+11166361 t3+1693696 t2+43515 t+1
(1−t)7(1+t)3
Figure 6.2: Cλα,β(t) for C3
α β λ Cλ
(0, 15, 5) (12, 15, 3) (6, 15, 6) 633 t
7+24259 t6+142236 t5+252113 t4+168220 t3+36131 t2+1414 t+1
(1−t)
(1−t2)
(4, 8, 11) (3, 15, 10) (10, 1, 3) 7962 t
8+503679 t7+4525372 t6+11944350 t5+12218255 t4+4879052 t3+586370 t2+10862 t+1
(1−t)
(1−t2)
(8, 1, 3) (11, 13, 3) (8, 6, 14) 81 t
7+19407 t6+211964 t5+513585 t4+426652 t3+110317 t2+4609 t+1
(1−t)
(1−t2)
(8, 9, 14) (8, 4, 5) (1, 5, 15) 9 t
2+8 t+1+2 t3
(1−t)
(10, 5, 6) (5, 4, 10) (0, 7, 12) 3647 t
7+111208 t6+570739 t5+920201 t4+560336 t3+106748 t2+3435 t+1
(1−t)
(1+t)
Figure 6.3: Cλ
(t) for D3
λ1 λ2 µ1 µ2 π1 π2 π3 π4 k̃
λ,µ(n); n odd k̃
λ,µ(n); n even K
λ,µ(t)
87 62 97 52 64 39 24 22 1/2 + 4n+ 11/2n2 1 + 4n+ 11/2n2 1+8 t+11 t
2+2 t3
(1−t)2(1−t2)
104 95 149 50 95 78 15 11 1/2 + 13/2n + 18n2 1 + 13/2n + 18n2 1+23 t+36 t
2+12 t3
(1−t)2(1−t2)
101 85 102 84 78 72 24 12 17/2n + 71
n2 1 + 17/2n + 71
n2 1+42 t+72 t
2+27 t3
(1−t)
(1−t2)
79 63 93 49 88 37 14 3 3/4 + 27
n+ 303
n2 1 + 27
n+ 303
n2 1+88 t+151 t
2+63 t3
(1−t)
(1−t2)
97 93 114 76 77 66 47 0 1/2 + 15/2n + 21n2 1 + 15/2n + 21n2 1+27 t+42 t
2+14 t3
(1−t)2(1−t2)
88 56 113 31 99 35 7 3 1/2 + 11/2n + 10n2 1 + 11/2n + 10n2 1+14 t+20 t
2+5 t3
(1−t)2(1−t2)
134 82 140 76 91 72 49 4 3/4 + 21n + 669
n2 1 + 21n + 669
n2 1+187 t+334 t
2+147 t3
(1−t)2(1−t2)
133 69 149 53 98 55 43 6 1 + 6n + 8n2 1 + 6n + 8n2 15 t
2+13 t+1+3 t3
(1−t)3
80 63 111 32 88 38 10 7 1 1 1+t
118 69 151 36 95 63 20 9 1 + 4n + 4n2 1 + 4n + 4n2 7 t
2+7 t+1+t3
(1−t)3
96 51 103 44 90 53 3 1 1/2 + 39
n+ 36n2 1 + 39
n+ 36n2 1+54 t+72 t
2+17 t3
(1−t)2(1−t2)
117 72 133 56 82 57 41 9 1 + 9n+ 18n2 1 + 9n + 18n2 35 t
2+26 t+1+10 t3
(1−t)
72 63 77 58 49 38 28 20 1/2 + 7n + 55
n2 1 + 7n+ 55
n2 1+33 t+55 t
2+21 t3
(1−t)
(1−t2)
48 37 49 36 34 24 16 11 1/2 + 6n + 37
n2 1 + 6n+ 37
n2 1+23 t+37 t
2+13 t3
(1−t)
(1−t2)
108 56 113 51 73 50 29 12 1 + 4n + 4n2 1 + 4n + 4n2 7 t
2+7 t+1+t3
(1−t)3
Figure 6.4: The quasipolynomial k̃π
and the rational function Kπ
(t) for the Kronecker problem, n = 2.
λ1 λ2 µ1 µ2 π1 π2 π3 π4 k̃
(n); n odd k̃π
(n); n even Kπ
77 40 78 39 58 29 24 6 1 + 19/2n + 57
n2 1 + 19/2n + 57
n2 56 t
2+37 t+1+20 t3
(1−t)3
153 81 157 77 96 63 61 14 1 + 3n+ 2n2 1 + 3n + 2n2 3 t
2+4 t+1
(1−t)3
90 89 102 77 90 42 30 17 1/2 + 13/2n + 6n2 1 + 13/2n + 6n2 1+11 t+12 t
(1−t)2(1−t2)
145 102 160 87 96 84 39 28 1 + 10n + 25n2 1 + 10n + 25n2 49 t
2+34 t+1+16 t3
(1−t)3
109 95 136 68 78 60 46 20 1 + 3n+ 2n2 1 + 3n + 2n2 3 t
2+4 t+1
(1−t)3
100 42 104 38 85 27 27 3 1 + 8n 1 + 8n 8 t+1+7 t
(1−t)2
74 51 86 39 52 34 26 13 1 1 1+t
98 90 124 64 92 67 22 7 1/2 + 23/2n + 60n2 1 + 23/2n + 60n2 1+70 t+120 t
2+49 t3
(1−t)2(1−t2)
57 38 75 20 52 25 17 1 1 + 3n+ 2n2 1 + 3n + 2n2 3 t
2+4 t+1
(1−t)
159 140 170 129 89 82 73 55 1 + 3/2n + 1/2n2 1 + 3/2n + 1/2n2 1+t
(1−t)3
144 122 157 109 88 86 74 18 3/4 + n+ 1/4n2 1 + n+ 1/4n2 1
(1−t)2(1−t2)
90 68 92 66 88 37 23 10 1/4 + 12n + 351
n2 1 + 12n+ 351
n2 1+98 t+176 t
2+76 t3
(1−t)
(1−t2)
89 42 100 31 76 28 19 8 1 + 6n+ 8n2 1 + 6n + 8n2 15 t
2+13 t+1+3 t3
(1−t)
88 56 107 37 71 39 20 14 1 + 9/2n + 9/2n2 1 + 9/2n + 9/2n2 8 t
2+8 t+1+t3
(1−t)3
124 111 133 102 98 89 27 21 1/2 + 7n+ 53
n2 1 + 7n+ 53
n2 1+32 t+53 t
2+20 t3
(1−t)2(1−t2)
Figure 6.5: Continuation of Figure 6.4
k λ hk,λ(n)
3 (21, 19) 399n3 + 35527969472513
137438953472
n2 + 4329327034365
137438953472
5 (21, 19) 3700378042361
4194304
n7 + 575575719967
524288
n6 + 2157156441
n5 + 266554253
n4 + 4643843
n3 + 1468423
n2 + 7619
3 (21, 9, 6) 270n3 + 40819369181185
274877906944
n2 + 3092376453119
137438953472
3 (12, 9, 5) 42n3 + 40132174413825
1099511627776
n2 + 11544872091645
1099511627776
3 (21, 9, 6) 27396522639355
536870912
n6 + 463063744509
8388608
n5 + 6265700353
262144
n4 + 5577375771
1048576
n3 + 84246529
131072
n2 + 20971505
524288
n+ 1048573
1048576
3 (21, 19, 16) 15n3 + 81363860455425
4398046511104
n2 + 8246337208319
1099511627776
4 (9, 7, 5) 7215545057279
17179869184
n6 + 4183298146289
4294967296
n5 + 247765925897
268435456
n4 + 1914699777
4194304
n3 + 4160749567
33554432
n2 + 587202553
33554432
n+ 67108863
67108864
4 (21, 12, 9) 16437913583613
268435456
n6 + 132498063359
2097152
n5 + 109509083155
4194304
n4 + 1462763527
262144
n3 + 171442179
262144
n2 + 10485755
262144
n+ 524287
524288
4 (21, 9, 5) 32469952757755
536870912
n6 + 129805320191
2097152
n5 + 108129157137
4194304
n4 + 2926313487
524288
n3 + 86638593
131072
n2 + 10616825
262144
n+ 262143
262144
4 (21, 9, 6) 27396522639355
536870912
n6 + 463063744509
8388608
n5 + 6265700353
262144
n4 + 5577375771
1048576
n3 + 84246529
131072
n2 + 20971505
524288
n+ 1048573
1048576
4 (31, 19, 5) 35969680015355
33554432
n6 + 1424674346311
2097152
n5 + 22705493343
131072
n4 + 46973953
n3 + 3423915
n2 + 65365
n+ 16383
16384
Figure 6.6: Hilbert polynomial for G/Pλ, G = SLk(C). There is a slight rounding error caused by interpolation–
e.g., the constant term of each polynomial should be one.
n k a
7 3 (1, 3, 5) 1/3n3 + 59373627899905
39582418599936
n2 + 28587302322173
13194139533312
7 3 (1, 2, 4) n+ 1
7 3 (1, 4, 6) 22265110462465
534362651099136
n5 + 4638564679679
11132555231232
n4 + 13
n3 + 105942526633
34359738368
n2 + 146028888073
51539607552
n+ 34359738361
34359738368
6 2 (1, 4, 5) 15637498706143
2251799813685248
n6 + 3665038759245
35184372088832
n5 + 1389660529559
2199023255552
n4 + 272014595421
137438953472
+230973796809
68719476736
n2 + 100215903571
34359738368
6 2 (1, 4, 6) 69578470195
25048249270272
n7 + 1217623228439
25048249270272
n6 + 372534725887
1043677052928
n5 + 30953963537
21743271936
n4 + 12044363351
3623878656
+683671553
150994944
n2 + 1335466297
402653184
n+ 268435457
268435456
7 3 (1, 4, 6) 23456248059223
562949953421312
n5 + 7330077518505
17592186044416
n4 + 1786706395137
1099511627776
n3 + 423770106525
137438953472
n2 + 24338148015
8589934592
n+ 17179869169
17179869184
6 3 (1, 3, 6) 1/8n4 + 16126170540715
17592186044416
n3 + 19
n2 + 710101259605
274877906944
8 3 (1, 3, 6) 171798691840001
1374389534720000
n4 + 31496426837333
34359738368000
n3 + 4080218931199
1717986918400
n2 + 443813287253
171798691840
Figure 6.7: Hilbert polynomial of the Schubert subvariety Ωa, a = (a(1), . . . , a(k)), of the Grassmannian Gn,k.
k = 2
1/2n + 1/2
1/2n + 1
k = 3
1/12n2 + 1/2n + 5
1/12n2 + 1/2n + 2/3
1/12n2 + 1/2n + 3/4
1/12n2 + 1/2n + 46912496118443
70368744177664
1/12n2 + 1/2n + 58640620148053
140737488355328
1/12n2 + 1/2n + 1
k = 4
n3 + 5
n2 + 61572651155457
140737488355328
n+ 15881834623431
35184372088832
n3 + 117281240296107
1125899906842624
n2 + 140737488355325
281474976710656
n+ 19
n3 + 234562480592215
2251799813685248
n2 + 123145302310909
281474976710656
n+ 19791209299969
35184372088832
n3 + 234562480592215
2251799813685248
n2 + 70368744177667
140737488355328
n+ 62549994824587
70368744177664
n3 + 5
n2 + 61572651155453
140737488355328
n+ 748278746681
2199023255552
n3 + 117281240296107
1125899906842624
n2 + 70368744177665
140737488355328
n+ 26388279066621
35184372088832
n3 + 117281240296107
1125899906842624
n2 + 7
n+ 7940917311717
17592186044416
n3 + 117281240296107
1125899906842624
n2 + 35184372088831
70368744177664
n+ 6841405683939
8796093022208
n3 + 29320310074027
281474976710656
n2 + 30786325577729
70368744177664
n3 + 58640620148053
562949953421312
n2 + 35184372088831
70368744177664
n+ 5619726097523
8796093022208
n3 + 117281240296105
1125899906842624
n2 + 7
n+ 2993114986727
8796093022208
n3 + 58640620148055
562949953421312
n2 + 1/2n + 1
Figure 6.8: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]
Sk ; k = 2, 3, 4.
n4 + 46912496118441
4503599627370496
n3 + 3787206717893
35184372088832
n2 + 469583091025
1099511627776
n+ 499743305817
1099511627776
n4 + 46912496118445
4503599627370496
n3 + 3787206717891
35184372088832
n2 + 503942829399
1099511627776
n+ 310001195055
549755813888
n4 + 46912496118441
4503599627370496
n3 + 3787206717897
35184372088832
n2 + 469583091031
1099511627776
n+ 30279519437
68719476736
400319966877379
1152921504606846976
n4 + 23456248059221
2251799813685248
n3 + 3787206717893
35184372088832
n2 + 503942829403
1099511627776
n+ 47340083975
68719476736
400319966877379
1152921504606846976
n4 + 5864062014805
562949953421312
n3 + 3787206717895
35184372088832
n2 + 117395772755
274877906944
n+ 89955703917
137438953472
400319966877379
1152921504606846976
n4 + 5864062014805
562949953421312
n3 + 3787206717893
35184372088832
n2 + 31496426837
68719476736
n+ 92771293595
137438953472
400319966877379
1152921504606846976
n4 + 46912496118447
4503599627370496
n3 + 1893603358947
17592186044416
n2 + 234791545515
549755813888
n+ 5661005505
17179869184
n4 + 23456248059223
2251799813685248
n3 + 1893603358949
17592186044416
n2 + 62992853675
137438953472
n+ 94680167945
137438953472
400319966877379
1152921504606846976
n4 + 46912496118441
4503599627370496
n3 + 3787206717891
35184372088832
n2 + 117395772757
274877906944
n+ 38869454029
68719476736
n4 + 46912496118441
4503599627370496
n3 + 3787206717897
35184372088832
n2 + 62992853675
137438953472
n+ 52494044729
68719476736
n4 + 5864062014805
562949953421312
n3 + 946801679473
8796093022208
n2 + 234791545515
549755813888
n+ 2830502753
8589934592
400319966877379
1152921504606846976
n4 + 23456248059219
2251799813685248
n3 + 1893603358949
17592186044416
n2 + 251971414695
549755813888
n+ 27487790695
34359738368
400319966877379
1152921504606846976
n4 + 23456248059223
2251799813685248
n3 + 1893603358945
17592186044416
n2 + 234791545509
549755813888
n+ 31233956605
68719476736
n4 + 11728124029611
1125899906842624
n3 + 946801679473
8796093022208
n2 + 251971414699
549755813888
n+ 19375074691
34359738368
n4 + 11728124029611
1125899906842624
n3 + 473400839737
4398046511104
n2 + 29348943189
68719476736
n+ 11005853695
17179869184
400319966877379
1152921504606846976
n4 + 23456248059219
2251799813685248
n3 + 473400839737
4398046511104
n2 + 251971414699
549755813888
n+ 5917510497
8589934592
n4 + 11728124029611
1125899906842624
n3 + 473400839737
4398046511104
n2 + 234791545511
549755813888
n+ 15616978311
34359738368
n4 + 23456248059223
2251799813685248
n3 + 1893603358947
17592186044416
n2 + 62992853673
137438953472
n+ 23192823403
34359738368
n4 + 11728124029611
1125899906842624
n3 + 946801679475
8796093022208
n2 + 117395772757
274877906944
n+ 2830502755
8589934592
400319966877379
1152921504606846976
n4 + 11728124029609
1125899906842624
n3 + 1893603358949
17592186044416
n2 + 31496426837
68719476736
n+ 30541989663
34359738368
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 946801679473
8796093022208
n2 + 117395772759
274877906944
n+ 9717363505
17179869184
n4 + 2932031007403
281474976710656
n3 + 1893603358945
17592186044416
n2 + 125985707353
274877906944
n+ 4843768673
8589934592
400319966877379
1152921504606846976
n4 + 23456248059223
2251799813685248
n3 + 59175104967
549755813888
n2 + 117395772755
274877906944
n+ 2830502755
8589934592
400319966877379
1152921504606846976
n4 + 23456248059221
2251799813685248
n3 + 946801679475
8796093022208
n2 + 62992853675
137438953472
n+ 3435973837
4294967296
400319966877379
1152921504606846976
n4 + 23456248059223
2251799813685248
n3 + 1893603358947
17592186044416
n2 + 58697886379
137438953472
n+ 11244462985
17179869184
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 946801679475
8796093022208
n2 + 15748213419
34359738368
n+ 9687537337
17179869184
n4 + 11728124029609
1125899906842624
n3 + 473400839737
4398046511104
n2 + 117395772753
274877906944
n+ 1892469965
4294967296
n4 + 23456248059221
2251799813685248
n3 + 946801679473
8796093022208
n2 + 7874106709
17179869184
n+ 11835020991
17179869184
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 473400839737
4398046511104
n2 + 117395772753
274877906944
n+ 3904244579
8589934592
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 946801679473
8796093022208
n2 + 125985707349
274877906944
n+ 1879048193
2147483648
Figure 6.9: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]
Sk , k = 5; the
first 30 rows.
n4 + 11728124029609
1125899906842624
n3 + 946801679473
8796093022208
n2 + 58697886375
137438953472
n+ 88453211
268435456
n4 + 11728124029611
1125899906842624
n3 + 946801679473
8796093022208
n2 + 15748213419
34359738368
n+ 2958755247
4294967296
400319966877379
1152921504606846976
n4 + 23456248059223
2251799813685248
n3 + 946801679475
8796093022208
n2 + 58697886375
137438953472
n+ 4858681755
8589934592
400319966877379
1152921504606846976
n4 + 11728124029609
1125899906842624
n3 + 473400839737
4398046511104
n2 + 62992853673
137438953472
n+ 151367771
268435456
n4 + 23456248059221
2251799813685248
n3 + 946801679473
8796093022208
n2 + 58697886375
137438953472
n+ 2274244835
4294967296
n4 + 11728124029611
1125899906842624
n3 + 236700419869
2199023255552
n2 + 62992853671
137438953472
n+ 858993459
1073741824
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 946801679473
8796093022208
n2 + 14674471595
34359738368
n+ 3904244571
8589934592
400319966877379
1152921504606846976
n4 + 23456248059219
2251799813685248
n3 + 59175104967
549755813888
n2 + 7874106709
17179869184
n+ 2421884337
4294967296
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 59175104967
549755813888
n2 + 29348943189
68719476736
n+ 3784939927
8589934592
400319966877379
1152921504606846976
n4 + 23456248059223
2251799813685248
n3 + 473400839737
4398046511104
n2 + 62992853673
137438953472
n+ 1908874355
2147483648
400319966877379
1152921504606846976
n4 + 5864062014805
562949953421312
n3 + 946801679473
8796093022208
n2 + 29348943189
68719476736
n+ 1952122291
4294967296
400319966877379
1152921504606846976
n4 + 11728124029609
1125899906842624
n3 + 946801679471
8796093022208
n2 + 62992853675
137438953472
n+ 2899102929
4294967296
400319966877379
1152921504606846976
n4 + 11728124029611
1125899906842624
n3 + 473400839735
4398046511104
n2 + 58697886381
137438953472
n+ 707625689
2147483648
400319966877379
1152921504606846976
n4 + 11728124029613
1125899906842624
n3 + 59175104967
549755813888
n2 + 15748213419
34359738368
n+ 2958755253
4294967296
n4 + 11728124029609
1125899906842624
n3 + 473400839737
4398046511104
n2 + 58697886377
137438953472
n+ 3288334339
4294967296
100079991719345
288230376151711744
n4 + 5864062014805
562949953421312
n3 + 59175104967
549755813888
n2 + 62992853675
137438953472
n+ 1210942169
2147483648
n4 + 11728124029611
1125899906842624
n3 + 946801679477
8796093022208
n2 + 29348943193
68719476736
n+ 1415251375
4294967296
n4 + 5864062014805
562949953421312
n3 + 59175104967
549755813888
n2 + 31496426839
68719476736
n+ 3435973835
4294967296
100079991719345
288230376151711744
n4 + 1466015503701
140737488355328
n3 + 473400839737
4398046511104
n2 + 29348943195
68719476736
n+ 976061147
2147483648
100079991719345
288230376151711744
n4 + 11728124029611
1125899906842624
n3 + 473400839737
4398046511104
n2 + 31496426839
68719476736
n+ 3280877793
4294967296
n4 + 5864062014805
562949953421312
n3 + 946801679473
8796093022208
n2 + 29348943191
68719476736
n+ 946234983
2147483648
100079991719345
288230376151711744
n4 + 2932031007403
281474976710656
n3 + 946801679475
8796093022208
n2 + 15748213419
34359738368
n+ 1479377623
2147483648
n4 + 5864062014805
562949953421312
n3 + 59175104967
549755813888
n2 + 29348943195
68719476736
n+ 122007643
268435456
100079991719345
288230376151711744
n4 + 2932031007403
281474976710656
n3 + 473400839737
4398046511104
n2 + 3937053355
8589934592
n+ 1449551461
2147483648
100079991719345
288230376151711744
n4 + 2932031007403
281474976710656
n3 + 236700419869
2199023255552
n2 + 29348943193
68719476736
n+ 71070151
134217728
n4 + 1466015503701
140737488355328
n3 + 59175104967
549755813888
n2 + 15748213419
34359738368
n+ 739688813
1073741824
100079991719345
288230376151711744
n4 + 11728124029611
1125899906842624
n3 + 473400839739
4398046511104
n2 + 14674471597
34359738368
n+ 607335219
1073741824
n4 + 2932031007403
281474976710656
n3 + 236700419869
2199023255552
n2 + 3937053355
8589934592
n+ 605471085
1073741824
100079991719345
288230376151711744
n4 + 11728124029611
1125899906842624
n3 + 473400839737
4398046511104
n2 + 29348943193
68719476736
n+ 88453211
268435456
100079991719345
288230376151711744
n4 + 1466015503701
140737488355328
n3 + 473400839737
4398046511104
n2 + 3937053355
8589934592
n+ 2147483647
2147483648
Figure 6.10: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]
Sk , k = 5;
the last 30 rows.
53375995583651
4611686018427387904
n5 + 21892498188609
36028797018963968
n4 + 418085902907
35184372088832
n3 + 115486898397
1099511627776
n2 + 26847522421
68719476736
n+ 8448724291
17179869184
53375995583651
4611686018427387904
n5 + 10946249094305
18014398509481984
n4 + 836171805815
70368744177664
n3 + 7396888121
68719476736
n2 + 15302809397
34359738368
n+ 9853503363
17179869184
106751991167299
9223372036854775808
n5 + 21892498188599
36028797018963968
n4 + 1672343611625
140737488355328
n3 + 57743449207
549755813888
n2 + 14060052663
34359738368
n+ 975427339
2147483648
13343998895913
1152921504606846976
n5 + 10946249094301
18014398509481984
n4 + 1672343611641
140737488355328
n3 + 59175104961
549755813888
n2 + 15302809423
34359738368
n+ 610309549
1073741824
106751991167305
9223372036854775808
n5 + 21892498188611
36028797018963968
n4 + 1672343611623
140737488355328
n3 + 115486898411
1099511627776
n2 + 13423761203
34359738368
n+ 4461910959
8589934592
13343998895913
1152921504606846976
n5 + 21892498188605
36028797018963968
n4 + 1672343611631
140737488355328
n3 + 29587552483
274877906944
n2 + 15939100865
34359738368
n+ 1927366573
2147483648
213503982334599
18446744073709551616
n5 + 10946249094305
18014398509481984
n4 + 418085902909
35184372088832
n3 + 57743449199
549755813888
n2 + 13423761217
34359738368
n+ 835973461
2147483648
106751991167299
9223372036854775808
n5 + 10946249094305
18014398509481984
n4 + 836171805821
70368744177664
n3 + 59175104963
549755813888
n2 + 7651404713
17179869184
n+ 320001575
536870912
106751991167299
9223372036854775808
n5 + 10946249094299
18014398509481984
n4 + 1672343611637
140737488355328
n3 + 115486898395
1099511627776
n2 + 14060052667
34359738368
n+ 255936429
536870912
213503982334597
18446744073709551616
n5 + 2736562273577
4503599627370496
n4 + 1672343611629
140737488355328
n3 + 7396888121
68719476736
n2 + 7651404703
17179869184
n+ 2859997491
4294967296
106751991167299
9223372036854775808
n5 + 342070284197
562949953421312
n4 + 104521475727
8796093022208
n3 + 57743449199
549755813888
n2 + 1677970153
4294967296
n+ 1790721319
4294967296
86400
n5 + 10946249094299
18014398509481984
n4 + 836171805815
70368744177664
n3 + 59175104959
549755813888
n2 + 3984775219
8589934592
n+ 987842475
1073741824
86400
n5 + 5473124547153
9007199254740992
n4 + 209042951453
17592186044416
n3 + 57743449193
549755813888
n2 + 3355940307
8589934592
n+ 884291849
2147483648
106751991167303
9223372036854775808
n5 + 10946249094301
18014398509481984
n4 + 836171805817
70368744177664
n3 + 118350209919
1099511627776
n2 + 1912851173
4294967296
n+ 264972307
536870912
213503982334605
18446744073709551616
n5 + 2736562273577
4503599627370496
n4 + 836171805827
70368744177664
n3 + 57743449201
549755813888
n2 + 7030026327
17179869184
n+ 1233125369
2147483648
53375995583651
4611686018427387904
n5 + 10946249094299
18014398509481984
n4 + 836171805819
70368744177664
n3 + 118350209933
1099511627776
n2 + 1912851177
4294967296
n+ 1478317113
2147483648
106751991167297
9223372036854775808
n5 + 1368281136787
2251799813685248
n4 + 836171805817
70368744177664
n3 + 57743449195
549755813888
n2 + 1677970151
4294967296
n+ 117959881
268435456
1667999861989
144115188075855872
n5 + 10946249094303
18014398509481984
n4 + 104521475727
8796093022208
n3 + 59175104961
549755813888
n2 + 3984775215
8589934592
n+ 109722991
134217728
106751991167297
9223372036854775808
n5 + 342070284197
562949953421312
n4 + 836171805815
70368744177664
n3 + 28871724597
274877906944
n2 + 1677970153
4294967296
n+ 332087389
1073741824
213503982334607
18446744073709551616
n5 + 10946249094307
18014398509481984
n4 + 836171805821
70368744177664
n3 + 59175104963
549755813888
n2 + 7651404709
17179869184
n+ 384426083
536870912
Figure 6.11: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]
Sk , k = 6; the first 20 rows.
213503982334615
18446744073709551616
n5 + 10946249094307
18014398509481984
n4 + 836171805813
70368744177664
n3 + 28871724597
274877906944
n2 + 7030026337
17179869184
n+ 640721865
1073741824
106751991167309
9223372036854775808
n5 + 10946249094297
18014398509481984
n4 + 209042951455
17592186044416
n3 + 29587552479
274877906944
n2 + 1912851179
4294967296
n+ 157275009
268435456
26687997791827
2305843009213693952
n5 + 342070284197
562949953421312
n4 + 836171805825
70368744177664
n3 + 57743449199
549755813888
n2 + 3355940301
8589934592
n+ 90445245
268435456
13343998895913
1152921504606846976
n5 + 5473124547153
9007199254740992
n4 + 104521475729
8796093022208
n3 + 29587552481
274877906944
n2 + 1992387605
4294967296
n+ 450971557
536870912
106751991167303
9223372036854775808
n5 + 10946249094307
18014398509481984
n4 + 836171805827
70368744177664
n3 + 28871724597
274877906944
n2 + 209746269
536870912
n+ 17843591
33554432
106751991167303
9223372036854775808
n5 + 10946249094297
18014398509481984
n4 + 836171805819
70368744177664
n3 + 924611015
8589934592
n2 + 239106397
536870912
n+ 5146825
8388608
1667999861989
144115188075855872
n5 + 5473124547153
9007199254740992
n4 + 418085902911
35184372088832
n3 + 7217931151
68719476736
n2 + 54922081
134217728
n+ 265331665
536870912
1667999861989
144115188075855872
n5 + 10946249094291
18014398509481984
n4 + 836171805825
70368744177664
n3 + 59175104963
549755813888
n2 + 478212795
1073741824
n+ 326629601
536870912
1667999861989
144115188075855872
n5 + 10946249094297
18014398509481984
n4 + 418085902909
35184372088832
n3 + 14435862303
137438953472
n2 + 3355940305
8589934592
n+ 24121261
67108864
53375995583655
4611686018427387904
n5 + 10946249094291
18014398509481984
n4 + 209042951453
17592186044416
n3 + 59175104973
549755813888
n2 + 3984775217
8589934592
n+ 503316475
536870912
26687997791827
2305843009213693952
n5 + 10946249094285
18014398509481984
n4 + 836171805825
70368744177664
n3 + 57743449207
549755813888
n2 + 3355940305
8589934592
n+ 115234099
268435456
106751991167303
9223372036854775808
n5 + 2736562273573
4503599627370496
n4 + 209042951459
17592186044416
n3 + 7396888121
68719476736
n2 + 3825702363
8589934592
n+ 42684549
67108864
106751991167301
9223372036854775808
n5 + 10946249094305
18014398509481984
n4 + 209042951453
17592186044416
n3 + 28871724605
274877906944
n2 + 1757506587
4294967296
n+ 277411267
536870912
106751991167303
9223372036854775808
n5 + 10946249094299
18014398509481984
n4 + 418085902905
35184372088832
n3 + 924611015
8589934592
n2 + 3825702353
8589934592
n+ 33950041
67108864
106751991167307
9223372036854775808
n5 + 1368281136787
2251799813685248
n4 + 52260737865
4398046511104
n3 + 28871724601
274877906944
n2 + 209746269
536870912
n+ 61328751
134217728
53375995583651
4611686018427387904
n5 + 2736562273575
4503599627370496
n4 + 104521475727
8796093022208
n3 + 29587552487
274877906944
n2 + 249048451
536870912
n+ 128849017
134217728
106751991167301
9223372036854775808
n5 + 5473124547145
9007199254740992
n4 + 209042951459
17592186044416
n3 + 7217931151
68719476736
n2 + 1677970157
4294967296
n+ 30318473
67108864
106751991167303
9223372036854775808
n5 + 342070284197
562949953421312
n4 + 418085902919
35184372088832
n3 + 14793776243
137438953472
n2 + 1912851181
4294967296
n+ 143223581
268435456
53375995583651
4611686018427387904
n5 + 5473124547153
9007199254740992
n4 + 209042951459
17592186044416
n3 + 1804482787
17179869184
n2 + 878753295
2147483648
n+ 13898875
33554432
106751991167303
9223372036854775808
n5 + 5473124547147
9007199254740992
n4 + 418085902907
35184372088832
n3 + 29587552479
274877906944
n2 + 956425585
2147483648
n+ 195527051
268435456
Figure 6.12: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]
Sk , k = 6; the middle 20 rows.
13343998895913
1152921504606846976
n5 + 5473124547145
9007199254740992
n4 + 418085902919
35184372088832
n3 + 7217931149
68719476736
n2 + 209746269
536870912
n+ 64348647
134217728
53375995583651
4611686018427387904
n5 + 5473124547143
9007199254740992
n4 + 104521475729
8796093022208
n3 + 14793776237
137438953472
n2 + 498096901
1073741824
n+ 28772927
33554432
106751991167301
9223372036854775808
n5 + 5473124547141
9007199254740992
n4 + 209042951457
17592186044416
n3 + 3608965575
34359738368
n2 + 1677970153
4294967296
n+ 11719905
33554432
13343998895913
1152921504606846976
n5 + 5473124547151
9007199254740992
n4 + 418085902905
35184372088832
n3 + 3698444059
34359738368
n2 + 478212797
1073741824
n+ 74631683
134217728
106751991167311
9223372036854775808
n5 + 5473124547147
9007199254740992
n4 + 418085902907
35184372088832
n3 + 28871724591
274877906944
n2 + 878753299
2147483648
n+ 10682367
16777216
106751991167313
9223372036854775808
n5 + 5473124547151
9007199254740992
n4 + 104521475729
8796093022208
n3 + 14793776237
137438953472
n2 + 478212797
1073741824
n+ 84006215
134217728
106751991167307
9223372036854775808
n5 + 2736562273573
4503599627370496
n4 + 209042951459
17592186044416
n3 + 28871724589
274877906944
n2 + 104873135
268435456
n+ 3161957
8388608
53375995583655
4611686018427387904
n5 + 5473124547149
9007199254740992
n4 + 209042951451
17592186044416
n3 + 29587552485
274877906944
n2 + 996193803
2147483648
n+ 118111589
134217728
106751991167309
9223372036854775808
n5 + 1368281136787
2251799813685248
n4 + 418085902907
35184372088832
n3 + 28871724601
274877906944
n2 + 419492539
1073741824
n+ 49899533
134217728
106751991167305
9223372036854775808
n5 + 2736562273573
4503599627370496
n4 + 52260737863
4398046511104
n3 + 29587552479
274877906944
n2 + 1912851173
4294967296
n+ 87717907
134217728
106751991167309
9223372036854775808
n5 + 1368281136787
2251799813685248
n4 + 104521475729
8796093022208
n3 + 28871724595
274877906944
n2 + 878753303
2147483648
n+ 8962701
16777216
106751991167305
9223372036854775808
n5 + 2736562273571
4503599627370496
n4 + 209042951453
17592186044416
n3 + 29587552499
274877906944
n2 + 956425585
2147483648
n+ 10878263
16777216
53375995583651
4611686018427387904
n5 + 5473124547157
9007199254740992
n4 + 104521475727
8796093022208
n3 + 3608965575
34359738368
n2 + 838985083
2147483648
n+ 53611225
134217728
13343998895913
1152921504606846976
n5 + 5473124547145
9007199254740992
n4 + 418085902907
35184372088832
n3 + 14793776249
137438953472
n2 + 498096903
1073741824
n+ 104354293
134217728
106751991167299
9223372036854775808
n5 + 2736562273575
4503599627370496
n4 + 209042951447
17592186044416
n3 + 3608965575
34359738368
n2 + 838985087
2147483648
n+ 7873221
16777216
106751991167303
9223372036854775808
n5 + 5473124547151
9007199254740992
n4 + 418085902899
35184372088832
n3 + 14793776243
137438953472
n2 + 956425599
2147483648
n+ 90737805
134217728
53375995583655
4611686018427387904
n5 + 5473124547155
9007199254740992
n4 + 104521475727
8796093022208
n3 + 14435862303
137438953472
n2 + 878753295
2147483648
n+ 18680385
33554432
106751991167305
9223372036854775808
n5 + 2736562273573
4503599627370496
n4 + 104521475725
8796093022208
n3 + 3698444063
34359738368
n2 + 478212799
1073741824
n+ 36634403
67108864
106751991167311
9223372036854775808
n5 + 684140568395
1125899906842624
n4 + 209042951463
17592186044416
n3 + 7217931153
68719476736
n2 + 52436567
134217728
n+ 19926951
67108864
26687997791827
2305843009213693952
n5 + 1368281136789
2251799813685248
n4 + 6532592233
549755813888
n3 + 14793776243
137438953472
n2 + 498096903
1073741824
n+ 67108871
67108864
Figure 6.13: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk]
Sk , k = 6; the last 20 rows.
Chapter 7
On verification and discovery
of obstructions
In this chapter we give applications of the results and positivity hypotheses
in this paper to the problem of verifying or discovering an obstruction, i.e.,
a “proof of hardness” [GCT2] in the context of the P vs. NP and the
permanent vs. determinant problems in characteristic zero.
7.1 Obstruction
An obstruction in an abstract setting of Problem 1.1.4 is defined as follows.
Let X and Y be H-varieties with compact specifications (Section 3.5),
H a connected reductive group. Let 〈X〉 and 〈Y 〉 denote the bit lengths of
their specifications (Section 3.5). Suppose we wish to show that X cannot
be embedded as an H-subvariety of Y . Pictorially:
X 6 →֒ Y. (7.1)
For example, in the context of the P vs. NP problem in characteristic
zero [GCT1, GCT2], X is a class variety XNP (n, l) associated with the
complexity class NP for the given input size parameter n and the circuit
size parameter l. The variety Y is the class variety XP (l) associated with
the class P for given l. And H is SLl(C). If NP ⊆ P (over C) to the
contrary, then it would turn out that
XNP (n, l) →֒ XP (l)
as an H-subvariety, for every l = poly(n). The goal is to show that this
embedding cannot exist when l = poly(n) and n→ ∞.
Let R(X) and R(Y ) be the homogeneous coordinate rings of X and
Y , respectively. Let R(X)d and R(Y )d denote their degree d-compoenents.
Suppose to the contrary that an H-embedding as in (7.1) exists. Then there
exists a degre preserving H-equivariant surjection from R(Y )d to R(X)d for
every d, and hence, a degree-preserving H-equivariant injection from R(X)∗
to R(Y )∗
. Hence, every irreducible H-module Vλ(G) that occurs in R(X)
also occurs in R(Y )∗
. This leads to:
Definition 7.1.1 An irreducible representation Vλ(H) is called an obstruc-
tion for the pair (X,Y) if it occurs (as an H-submodule) in R(X)∗
but not
in R(Y )∗
, for some d. We say that Vλ(H) is an obstruction of degree d.
Remark 7.1.2 The obstruction as defined here is dual to the obstrution as
defined in [GCT2].
Existence of such an obstruction implies that X cannot be embedded in
Y as an H-subvariety.
Let us assume that X and Y are H-subvarieties of P (V ), where V is an
H-module, and that we are given a point y ∈ Y ⊆ P (V ) that is distiguished
in the following sense. Let Hy ⊆ H be the stabilizer of y. Then Cy, the line
in V corresponding to y, is invariant under Hy. Let [y] be the set of points
in P (V ) stabilized by Hy. We say that y is characterized by its stabilizer Hy
if y = [y]; i.e., y is the only point in P (V ) stabilized by Hy. Let
H[y] = ∪z∈[y]Hz
be the union of the H-orbits of all points in [y]. We say that y is a distin-
guished point of Y if Y equals the projective closure of H[y] in P (V ). If y
is characterized by its stabilizer, this means Y is the projective closure of
the orbit Hy of y.
If Vλ(H) occurs in R(Y )
, then it can be shown (cf. Proposition 4.2 in
[GCT2]) that Vλ(H) contains an Hy-submodule isomorphic to (Cy)
d, the
d-th tensor power of Cy. This leads to the following stronger notion of
obstruction:
Definition 7.1.3 [GCT2] We say that Vλ(H) is a strong obstruction for
the pair (X,Y ) if, for some d, it occurs in R(X)∗
, but it does not contain
an Hy-module isomorphic to (Cy)
Existence of a strong obstruction also implies that X cannot be embedded
in Y as an H-subvariety. The results in [GCT2] suggest that strong obstruc-
tions exist in the context of the lower bound problems under cosideration.
The goal then is to show their existence.
7.2 Decision problems
The decisions problems that arise in this context are the following. Let sλ
be the multiplicity of Vλ(H) in R(X)
, and md
the multiplicity of the Hy-
module (Cy)d in Vλ(H), considered an Hy-module via the the embedding
ρ : Hy →֒ H. Thus λ is a strong obstruction of degree d iff s
is nonzero
and md
is zero.
Problem 7.2.1 (Decision Problems)
(a) Given d, λ and the specification of X, decide if sλ
is nonzero.
(b) Given d, λ and the specifications of H,Hy and ρ, decide if m
λ is nonzero.
(c) Given d, λ and the specifications of X,H,Hy and ρ, decide if λ is a
strong obstruction of degree d.
The first is an instance of the decision Problem 1.1.4 in geometric in-
variant theory, and the second of the subgroup restriction Problem 1.1.3.
By the results in Chapter 3-4, relaxed forms of the decision problems in (a)
and (b) belong to P assuming appropriate PH1 and SH; this implies that
a relaxed form of the decision problem in (c) also belongs to P assumming
PH1 and SH. We will only need a weak relaxed form of (c), for which the
weak form of SH that is implied by PH1 (cf. Theorem 3.3.5) will suffice.
7.3 Verification of obstructions
The relevant PH1 are as follows.
Assume that that singularities of spec(R(X)) are rational. By Theo-
rem 3.5.1, the stretching function s̃λ
(k) = skλ
is a quasipolynomial. Hence
PH1 for sλ
(Hypothesis 3.3.1, or rather its slight variant obtained by replac-
ing R(X)d with R(X)
) is well defined. It is:
Hypothesis 7.3.1 (PH1):
There exists a polytope P λ
such that:
1. The number of integer points in P λ
is equal to sλ
2. The Ehrhart quasi-polynomial of P λd coincides with the stretching quasi-
polynomial s̃λd(n) (cf. Theorem 3.5.1).
3. The polytope P λ
is given by a separating oracle, as in Section 2.3. Its
encoding bitlength 〈P λ
〉 is poly(〈d〉, 〈λ〉, 〈X〉), and the combinatorial
size ‖P λ
‖ is poly(ht(λ), ‖X‖), where ‖X‖ is the combinatorial size of
X (Section 3.5), and ht(λ) is the height of λ.
Similarly by Theorem 3.4.1 (or rather its slight variant which can be
proved similarly), the stretching function mkdkλ is a quasipolynomial. Hence
PH1 for mdλ (cf. Hypothesis 3.3.1 and Section 3.4) is also well defined. It is:
Hypothesis 7.3.2 (PH1:)
There exists a polytope Qd
such that:
1. The number of integer points in Qd
is equal to md
2. The Ehrhart quasi-polynomial of Qd
coincides with the stretching quasi-
polynomial m̃d
(n) (Theorem 3.5.1).
3. The polytope Qdλ is given by a separating oracle. Its encoding bitlength
〈Qdλ〉 is poly(〈d〉, 〈λ〉, 〈ρ〉, 〈Hy 〉, 〈H〉), and the combinatorial size ‖Q
is O(poly(ht(λ), 〈H〉, 〈Hy〉, 〈ρ〉)). Here 〈H〉, 〈Hy〉 and 〈ρ〉 denote the
bitlengths of H,Hy and ρ (Section 3.4).
Theorem 7.3.3 (Weak SH:)
(a) Assuming PH1 (Hypothesis 7.3.1), the saturation index of s̃λ
(n) is at
most apoly(‖P
‖), for some explicit constant a > 0.
(b) Assuming PH1 (Hypothesis 7.3.2), the saturation index of m̃dλ(n) is at
most bpoly(‖Q
‖), for some explicit constant b > 0.
This follows from Theorem 3.3.5.
Theorem 7.3.4 Assume PH1 (Hypotheses 7.3.1-7.3.2). Then, given d, λ,
the specifications of X,H,Hy and ρ, and a relaxation parameter c greater
than the explicit bounds on the saturation indices in Theorem 7.3.3, whether
cλ is an obstruction of degree d can be decided in
poly(〈d〉, 〈λ〉, 〈X〉, 〈H〉, 〈Hy 〉, 〈ρ〉, 〈c〉)
time.
This follows by applying Theorem 3.1.1 to the polytopes P λ
and Qd
the saturation index estimates in Theorem 7.3.3.
7.4 Robust obstruction
We now define a notion of obstruction that is well behaved with respect to
relaxation.
Definition 7.4.1 Assume PH1 for both sλd andm
λ (Hypotheses 7.3.1-7.3.2).
We say that Vλ(H) is a robust obstruction for the pair (X,Y ) if one of the
following hold:
1. Qd
is empty, and P λ
is nonempty.
2. Both Qdλ and P λ
are nonempty, the affine span of Qd
does not contain
an integer point and the affine span of P λ
contains an integer point.
If Vλ(H) is a robust obstruction, so is Vlλ(H), for all or most positive
integral l, hence the name robust.
Proposition 7.4.2 Assume PH1 for both sλd and m
λ as above. If Vλ(H)
is a robust obstruction for the pair (X,Y ), then for some positive integer
k–called a relaxation parameter–Vkλ(H) is a strong obstruction for (X,Y ).
In fact, this is so for most large enough k.
Proof:
(1) Suppose Qd
is empty, and P λ
is nonempty. Let k be a large enough
positive integer k such that kP λ
= P kλ
contains an integer point. Then skλ
is nonzero. But mkd
is zero since Qkd
= kQd
is empty. Thus kλ is a strong
obstruction.
(2) Suppose both Qdλ and P λd are nonempty, the affine span of Q
λ does not
contain an integer point and the affine span of P λd contains an integer point.
We can choose a positive integer k such that the affine span of kQdλ = Q
does not contain an integer point, but kP λ
= P kλ
contains an integer point;
most large enough k have this property. This means skλ
is nonzero, butmkd
is zero. Thus kλ is a strong obstruction. Q.E.D.
7.5 Verification of robust obstructions
Theorem 7.5.1 Assume that the singularities of spec(R(X)) are rational.
Assume PH1 for both sλ
and md
as above. Then, given λ, d and the speci-
fications of ρ : Hy →֒ H and X, whether Vλ(H) is a robust obstruction can
be verified in poly(〈ρ〉, 〈Hy〉, 〈H〉, 〈X〉, 〈d〉, 〈λ〉) time. Furthermore, a positive
integral relaxation parameter k such that Vkλ(G) is a strong obstruction can
also be found in the same time.
The crucial result used implicitly here is the quasipolynomiality theorem
(Theorem 4.1.1) because of which PH1 for both sλ
and md
are well defined.
Proof: By linear programming [GLS], whether Qd
is nonempty or not can
be determined in poly(〈Qdλ〉) = poly(〈ρ〉, 〈Hy〉, 〈H〉, 〈d〉, 〈λ〉) time. If it
is nonempty, the linear programming algorithm also gives its affine span.
Whether this contains an integer point can be determined in polynomial
time, using the polynomial time algorithm for computing the Smith normal
form, as in the proof of Theorem 3.1.1.
Similarly, whether P λ
is nonempty or not can be determined in poly(〈P λ
poly(〈X〉, 〈d〉, 〈λ〉) time. If it is nonempty, whether its affine span contains
an integer point can be determined in polynomial time similarly. Further-
more, the algorithm can also be made to return a vertex v of the polytope
P λd if it is nonempty.
Using these observations, whether Vλ(G) is a robust obstruction can be
determined in polynomial time.
As far as the computation of the relaxation parameter k is concerned,
let us consider the second case in Definition 7.4.1–when both Qdλ and P
d are
nonempty, the affine span of Qdλ does not contain an integer point and the
affine span of P λd contains an integer point–the first case being simpler. In
this case, by examining the Smith normal forms of the defining equations
of the affine spans of P λ
and Qd
and the rational coordinates of a vertex
v ∈ P λ
, we can find a large enough k so that the affine span of Qkd
does not
contain an integer point, the affine span of P kλ
contains an integer point,
and P kλ
contains an integer point that is some multiple of v. Q.E.D.
The value of the relaxation parameter k computed above is rather con-
servative. One may wish to compute as small value of k as possible for
which Vkλ(G) is a strong obstruction (though in our application this is not
necessary). If SH for holds for the structural constant sλ
(cf. Hypothe-
sis 3.3.2 and Section 3.5), then we can let k be the smallest integer larger
than the saturation index (estimate) for P λ
such that affine span of Qkd
nonempty) does not contain an integer point (as can be ensured by looking
at the Smith normal of the defining equations of the affine span).
7.6 Arithemetic version of the P#P vs. NC prob-
lem in characteristric zero
We now specialize the discussion in the preceding sections in the context
of the arithmetic form of the P#P vs. NC problem in characteristric zero
[V]. In concrete terms, the problem is to show that the permanent of an
n×n complex matrix X cannot be expressed as a determinant of an m×m
complex matrix, whose entries are (possibly nonhomogeneous) linear com-
binations of the entries of X.
7.6.1 Class varieties
The class varieties in this context are as follows [GCT1]. Let Y be an m×m
variable matrix, which can also be thought of as a variable l-vector, l = m2.
Let X be its, say, principal bottom-right n × n submatrix, n < m, which
can be thought of as a variable k-vector, k = n2. Let V = Symm(Y ) be the
space of homogeneous forms of degree m in the variable entries of Y . The
space V , and hence P (V ), has a natural action of G = GL(Y ) = GLl(C)
given by
(σf)(Y ) = f(σ−1Y ),
for any f ∈ V , σ ∈ G, and thinking of Y as an l-vector. Let W = Symn(X)
be the space of homogeneous forms of degree n in the variable entries of
X. The space W , and also P (W ), has a similar action of K = GL(X) =
GLk(C). We use any entry y of Y not in X as the homogenizing variable
for embedding W in V via the map φ : W → V defined by:
φ(h)(Y ) = ym−nh(X), (7.2)
for any h(X) ∈W . We also think of φ as a map from P (W ) to P (V ).
Let g = det(Y ) ∈ P (V ) be the determinant form, and f = φ(h), where
h = perm(X) ∈ P (W ). Let ∆V [g],∆V [f ] ⊆ P (V ) be the projective closures
of the orbits Gg and Gf , respectively, in P (V ). Let ∆W [h] ⊆ P (W ) be the
projective closure of the K-orbit Kh of h in P (W ). Then ∆V [g] is called the
class variety associated with NC and ∆V [f ] the class variety associated with
P#P ; ∆W [h] is called the base class variety associated with P
#P . (The base
class variety is not used in what follows. Rather its variant, called a reduced
class variety defined below, will be used.) These class varieties depend on
the lower bound parameters n and m. If we wish to make these explicit, we
would write ∆V [f, n,m] and ∆V [g,m] instead of ∆V [f ] and ∆V [g].
The class varieties ∆V [g] = ∆V [g,m] and ∆V [f ] = ∆V [f, n,m] are
G-subvarieties of P (V ), and their homogeneous coordinate rings RV [g] =
RV [g,m] and RV [f ] = RV [f, n,m] have natural degree-preserving G-action.
It is conjectured in [GCT1] that, if m = poly(n) and n → ∞, then
f 6∈ ∆V [g]; this is equivalent to saying that the class variety ∆V [f, n,m]
cannot be embedded in the class variety ∆V [g,m] (as a subvariety). This
implies the arithmetic form of the P#P 6= NC conjecture in characteristic
zero.
7.6.2 Obstructions
The obstruction in this context is defined as follows. A G-module Vλ(G) is
called an obstruction for the pair (f, g) if it occurs in RV [f, n,m]
d but not
RV [g,m]
for some d. It is called a strong obstruction if, for some d, it occurs
in RV [f, n,m]
but it does not contain (Cg)d as a Gg-submodule, where
(Cg) ⊆ V denotes the one dimensional line corresponding to g, and Gg ⊆ G
is the stabilizer of g = det(Y ) ∈ P (V ). If Vλ(G) is a (strong) obstruction
of degree d, then the size |λ| = dm; hence d is completely determined by λ
and m.
Existence of an obstruction or a strong obstruction implies that the
class variety ∆V [f, n,m] cannot be embedded in the class variety ∆V [g,m],
as sought. The main algebro-geometric results of [GCT1, GCT2] suggest
that strong obstructions should indeed exist for all n → ∞, assuming m =
poly(n); cf. Section 4, Conjecture 2.10 and Theorem 2.11 in [GCT2]. The
goal then is to prove existence of strong obstructions for all n.
The definition of a strong obstruction can be simplified further as follows.
Let X ′ denote the set of variables, which consists of the variable entries in
X and the homogenizing variable y above. Let W ′ = Symm(X ′) ⊆ V =
Symm(Y ) be the space of homogeneous forms of degree m in the variables
of X ′. We have a natural action of H = GL(X ′) = GLn2+1(C) on W
and hence on P (W ′). We have a natural map φ′ : W → W ′ given by
φ′(h)(X ′) = ym−nh(X). The map φ in (7.2) is φ′ followed by the inclusion
from W ′ to V . We also think of φ′ as a map from P (W ) to P (W ′).
Let f ′ = φ′(h), for h = perm(X) ∈ P (W ). Let ∆W ′ [f
′] ⊆ P (W ′) be
the orbit closure of Hf ′. It is an H-subvariety of P (W ′), and hence its
homogeneous coordinate ring RW ′ [f
′] has the natural degree preserving H-
action. We call ∆W ′ [f
′] the reduced class variety for P#P . It is known (cf.
Theorem 8.2 in [GCT2]) that Vλ(G) occurs in RV [f ]
iff Vλ(H) occurs in
RW ′ [f
. Here the dominant weight λ of G is considered a dominant weight
of H by restriction from G to H.
Hence Vλ(G) is a strong obstruction for the pair (f, g), iff for some d,
Vλ(H) occurs in RW ′ [f
as an H-submodule and Vλ(G) does not contain
(Cg)d as a Gg-submodule. In particular, we can assume without loss of
generality that the height of the Young diagram for λ is at most n2 + 1;
otherwise Vλ(H) would be zero.
7.6.3 Robust obstructions
It is known that the stabilizer Gg of g = det(Y ) ∈ P (V ) consists of lin-
ear transformations in G of the form Y → AY ∗B−1, thinking of Y as an
m×m matrix, where Y ∗ is either Y or Y T , A,B ∈ GLm(C). Thus the con-
nected component of Gg is essentially GLm(C)×GLm(C) ⊆ G = GLl(C) =
GLm2(C). This means the subgroup restriction problem for the embedding
ρ : Gg →֒ G is essentially the Kronecker problem (Problem 1.1.1).
Assume PH1 (Hypothesis 7.3.2) for the subgroup restriction ρ : Gg →֒ G;
which is essentially PH1 for the Kronecker problem. It now assumes the
following concrete form. Let md
denote the multiplicity of the Gg-module
(Cg)d in Vλ(G). Assume that the height of λ is at most n
2+1 for the reasons
give above.
Hypothesis 7.6.1 (PH1:)
There exists a polytope Qd
such that:
1. The number of integer points in Qd
is equal to md
2. The Ehrhart quasi-polynomial of Qdλ coincides with the stretching quasi-
polynomial m̃d
(n) (cf. Theorem 3.5.1).
3. The polytope Qd
is given by a separating oracle, and its encoding
bitlength 〈Qdλ〉 is poly(n, 〈m〉, 〈d〉, 〈λ〉) time.
We have to explain why 〈Qd
〉 is stipulated to depend polynomially on
n and 〈m〉, rather than m. After all, the bitlengths 〈G〉, 〈Gg〉 and 〈ρ〉 are
O(poly(m2)) as per the definitions in Section 3.4. So, as per PH1 for sub-
group restriction in Section 3.4.3, 〈Qdλ〉 should depend polynomially on m.
We are stipulating a stronger condition for the following reason. First, as we
already mentioned, the above hypothesis is essentially PH1 for the Kronecker
problem, which is obtained by specializing PH1 for the plethysm problem
(Hypothesis 1.6.4). In Hypothesis 1.6.4, the encoding bitlength of the poly-
tope depends polynomially on the bitlengths of the various partition param-
eters λ, π, µ of the plethysm constant aπ
, but is independent of the rank of
the group G therein. (As explained in the remarks after Hypothesis 1.6.4,
this is justified because the bound in Theorem 1.6.3 is also independent of
the rank of G). For the same reason, the encoding bitlength of the polytope
here should be independent of the rank of G (which is m2), but should de-
pend polynomiallly on the total bit length of the partitions parametrizing
the representations Vλ(G) and (Cy)
d. This is O(n+ 〈m〉+ 〈d〉+ 〈λ〉). (Note
that the one dimensional representation (Cy)d of Gg is essentially the d-th
power of the determinant representation of Gg, since the connected compo-
nent of Gg is isomorphic to GLm(C)×GLm(C). The Young diagram for the
partition corresponding to the d-th power of the determinant representation
of GLm(C) is a rectangle of height m and width d. It can be specified by
simply giving m and d–this specification has bit length 〈m〉+ 〈d〉.)
Next let us specialize PH1 as per Hypothesis 7.3.1. The class variety
∆V [f ] = ∆[f, n,m] will now play the role of X in Hypothesis 7.3.1. But,
for the reasons explained in the proof of Proposition 7.6.4 below, we shall
instead specialize Hypothesis 7.3.1 to the (simpler) reduced class variety
Z = ∆W ′[f
′]. It now assumes that following concrete form. Let sλ
denote
the multiplicity of Vλ(H) in RW ′ [f
. Putting Z in place of X in Hypothe-
sis 7.3.1, we get:
Hypothesis 7.6.2 (PH1):
There exists a polytope P λ
such that:
1. The number of integer points in P λ
is equal to sλ
2. The Ehrhart quasi-polynomial of P λ
coincides with the stretching quasi-
polynomial s̃λ
(n) (cf. Theorem 3.5.1).
3. The polytope P λd is given by a separating oracle, and its encoding
bitlength 〈P λd 〉 is
poly(〈d〉, 〈λ〉, 〈Z〉) = poly(〈d〉, 〈λ〉, n, 〈m〉). (7.3)
Here (7.3) follows because 〈Z〉 = n+〈m〉. To see why, let us observe that
Z = ∆W ′ [f
′] is completely specified once the point f ′ = ym−nh ∈ P (W ′) is
specified. To specify f ′, it sufficies to specify m,n and the point h ∈ P (W ).
It is known [GCT2] that the point h = perm(X) ∈ P (W ) is completely
characterized by its stabilizer Kh ⊆ K = GL(X) = GLk(C). Furthermore,
Kh is explicitly known [Mc]. It is generated by the linear transformation in
K of the form X → λXµ−1, thinking of X as an n×n matrix, where λ and
µ are either diagonal or permutation matrices. So to specifiy h, it suffices
to specify Kh,K and the embedding ρ
′ : Kh →֒ K. The bit length of this
specification is O(n) (cf. Section 3.4). To specify f ′, and hence Z, it suffices
to specify m,n,K,Kh and ρ
′. The total bit length of this specification is
O(n+ 〈m〉).
Assume PH1 for both md
and sλ
, i.e., Hypotheses 7.6.1 and 7.6.2.
Definition 7.6.3 We say that Vλ(G) is a robust obstruction for the pair
(f, , g) if one of the following hold:
1. Qdλ is empty, and P
d is nonempty.
2. Both Qdλ and P
d are nonempty, the affine span of Q
λ does not contain
an integer point and the affine span of P λd contains an integer point.
If the first condition holds, we say that Vλ(G) is a geometric obstruction.
If the second condition holds, it is called a modular obstruction.
Proposition 7.6.4 Assume PH1 for both mdλ and s
d (Hypotheses 7.6.1 and
7.6.2). If Vλ(G) is a robust obstruction for the pair (f, g), then for some
positive integral relaxation parameter k, Vkλ(G) is a strong obstruction for
(f, g). In fact, this is so for most large enough k.
Proof: This essentially follows from Proposition 7.4.2. It only remains to
clarify why we can use PH1 for the reduced class variety ∆W ′[f
′]–as we are
doing here– in place of PH1 for the class variety ∆V [f ]. This is because,
as already mentioned, Vλ(G) occurs in RV [f ]
d iff Vλ(H) occurs in RW ′ [f
′]∗d.
Q.E.D.
7.6.4 Verification of robust obstructions
Theorem 7.6.5 Assume that the singularities of spec(RW ′ [f
′]) are ratio-
nal. Assume PH1 for both md
and sλ
as above (Hypotheses 7.6.1 and 7.6.2).
Then, given n,m, λ and d, whether Vλ(H) is a robust obstruction can be
verified in poly(n, 〈m〉, 〈d〉, 〈λ〉) time. Furthermore, a positive integral relax-
ation parameter k such that kλ is a strong obstruction can also be computed
in this much time.
Once n andm are specified, the various class varieties andK,Kh, ρ
′, G,Gg, ρ
above are automatically specified implicitly.
Proof: This follows from Theorem 7.5.1; cf. also the remark following its
proof. Q.E.D.
Theorem 7.3.4 can be similarly specialized in this context; we leave that
to the reader.
7.6.5 On explicit construction of obstructions
Theorem 7.6.6 Assume that m = poly(n) or even 2polylog(n), and:
1. (RH) [Rationality Hypothesis]: The singularities of spec(RW ′ [f
′]) are
rational.
2. PH1 for both md
and sλ
(Hypotheses 7.6.1 and 7.6.2).
3. OH [Obstruction Hypothesis]: For every (large enough) n, there exists
λ of poly(n) bit length such that |λ| is divisible by m and one of the
following holds (with d = |λ|/m):
(a) Qd
is empty, and P λ
is nonempty.
(b) Both Qdλ and P λ
are nonempty, the affine span of Qd
does not
contain an integer point and the affine span of P λ
contains an
integer point.
Then there exists an explicit family {λn} of robust obstructions.
Here we say that {λn} is an explicit family of robust obstructions if each
λn is short and easy to verify. Short means 〈λn〉 is O(poly(n)). Easy to verify
means whether λn is a robust obstruction can be verified in O(poly(n)) time.
The poly(n) bound here and in OH is meant to be independent of m,
as long as m << 2n; i.e., it should hold even when m = 2polylog(n). In
other words, the family {λn} should continue to remain an explicit robust
obstruction family, as we vary m over all values ≤ 2polylog(n), and perhaps
even values ≤ 2o(n), but will cease to be an obstruction family for some large
enough m = 2Ω(n). This is an important uniformity condition.
Proof: OH basically says that there exists a short robust obstruction λn for
every n. By Theorem 7.6.5, it is easy to verify. Q.E.D.
7.6.6 Why should robust obstructions exist?
The main question now is: why should OH hold? That is, why should
(short) robust obstructions exist?
As we already mentioned, the results in [GCT1, GCT2] indicate that
strong obstructions should exist for every n, assuming m = poly(n). We
shall give a heuristic argument for existence of robust obstructions assuming
that strong obstructions exist. This will crucially depend on the following SH
formd
, which is essentially SH for the Kronecker problem (i.e. specialization
of Hypothesis 1.6.5 to the Kronecker problem), good experimental evidence
for which is provided in [BOR].
Hypothesis 7.6.7 (SH:) (a): The saturation index of m̃d
(k) is bounded by
a polynomial in m. (Observe that the rank of G is poly(m) and the height of
λ is at most n2 + 1). (b): The quasi-polynomial m̃d
(n) is strictly saturated,
i.e. the saturation index is zero, for almost all λ (and d).
If Vλ(G) is a strong obstruction, s
is nonzero but md
is zero. Thus,
assumming PH1, there are three possibilities:
1. Qd
is empty, and P λ
is nonempty and contains an integer point.
2. Both Qdλ and P λd are nonempty, the affine span of Q
λ does not contain
an integer point and P λd contains an integer point.
3. Both Qd
and P λ
are nonempty. The affine span of Qd
contains an
integer point, but Qd
does not. And P λ
contains an integer point.
In the first two cases, λ is a robust obstruction. As per SH (Hypothe-
sis 7.6.7), for almost all λ, the Ehrhart quasipolynomial of Qd
is saturated:
this means (cf. the proof of Theorem 3.1.1), if the affine span of Qd
contains
an integer point then Qd
also contains an integer point. And hence, with
a high probability, the third case should not occur. In other words, strong
obstructions can be expected to be robust with a high probability.
Let us call a strong obstruction λ fragile if it is not robust; this means
the affine span of Qdλ contains an integer point, but Q
λ does not. By SH
(Hypothesis 7.6.7), if λ is fragile, then for some k = poly(m), Qkdkλ contains
an intger point, and hence, kλ is not obstruction. Thus fragile obstructions
are close to not being obstructions, and furthermore, are expected to be
rare, as argued above. This is why we are focussing on robust obstructions.
It may be remarked that the only SH needed in the argument above
is the one (Hypothesis 7.6.7) for the structural constant md
. This is a
special case of the SH for the subgroup restriction problem (cf. Section 3.4)
specialized to the embedding Gg →֒ G. In particular, we do not need SH
for the structural constant sλd ; i.e., for the more difficult decision problem in
geometric invariant theory (cf. Problem 1.1.4 and Section 3.5).
7.6.7 On discovery of robust obstructions
It may be conjectured that not just the verification (cf. Theorem 7.6.5)
but also the discovery of robust obstructions is easy for the problem under
consideration. In this section we shall give an argument in support of this
conjecture for geometric (robust) obstructions (which may be conjectured to
exist in the problem under consideration). For this we need to reformulate
the notions of strong and robust obstructions (Definition 7.6.3) as follows.
Let TZ be the set of pairs (d, λ) such that s
d is nonzero and SZ the set
of pairs (d, λ) such that mdλ is nonzero.
Proposition 7.6.8 Assuming PH1 above (Hypotheses 7.6.1 and 7.6.2), TZ
and SZ are finitely generated semigroups with respect to addition.
These semi-groups are analogues of the Littlewood-Richardson semigroup
(Section 2.2.2) in this setting.
Proof: The proof is similar to that for the Littlewood-Richardson semigroup
For given d and λ, the polytope P λ
in PH1 for sλ
(Hypothesis 7.6.2) has
a specification of the form
Ax ≤ b (7.4)
where A depends only the variety Z = ∆W ′ [f
′], but not on d or λ, and
b depends homogeneously and linearly on d and λ. Let P be the polytope
defined by the inequalities (7.4) where both d and λ are treated as variables.
Then P is a polyhedral cone (through the origin) in the ambient space
containing P with the coordinates x, d and λ. Let PZ be the set of integer
points in P . It is a finitely generated semigroup since P is a polyhedral cone.
Let TR be the orthogonal projection of P on the hyperplane corresponding
to the coordinates d and λ. Now TZ is simply the projection of PZ. Hence
it is a finitely generated semigroup as well.
The proof for SZ is similar, with SR defined similarly. Q.E.D.
The polyhedral cones TR and SR here are analogues of the Littlewood-
Richardson cone (Section 2.2.2) in this setting. Note that (d, λ) ∈ TR iff P
is nonempty; similarly for SR.
A Weyl module Vλ(G) is a strong obstruction for the pair (f, g) of degree
d iff (d, λ) occurs in TZ but not in SZ. It is a robust obstruction iff it occurs
in TR but not in SZ. It is a geometric obstruction iff it occurs in TR but not
in SR. It is a modular obstruction iff it occurs in TR and also in SR but not
in SZ.
Assuming PH1 (Hypothesis 7.6.2), whether (d, λ) belongs to TR can be
determined in polynomial time by linear programming, since (d, λ) ∈ TR
iff P λ
is nonempty. Similarly, assuming PH1 (Hypothesis 7.6.1), whether
(d, λ) ∈ SR can be determined in polynomial time.
The following is a stronger complement to PH1.
Hypothesis 7.6.9 (PH1*)
Whether TR\SR is nonempty can be determined in polynomial time; i.e.,
poly(n, 〈m〉) time. If so, the algorithm can also output (d, λ) ∈ TR \ SR of
polynomial bit length.
Proposition 7.6.10 Assuming PH1*, given n and m, the problem of decid-
ing if a geometric obstruction exists for the pair (f, g), and finding one if one
exists, belongs to the complexity class P ; i.e., it can be done in poly(n, 〈m〉)
time.
This immediately follows from Hypothesis 7.6.9 since (d, λ) is a geometric
obstruction iff (d, λ) ∈ TR \ SR.
Hypothesis 7.6.9 is supported by the following:
Proposition 7.6.11 Assuming PH1 (Hypotheses 7.6.1 and 7.6.2), Hypoth-
esis 7.6.9 holds if TR and SR have polynomially many explicitly given con-
straints with the specification of polynomial bit length; here polynomial means
poly(n, 〈m〉).
The proposition holds even if the polytope SR has exponentially many
constraints, as long as it is given by a separation oracle that works in poly-
nomial time.
Proof: It suffices to check if SR satisfies each constraint of TR. This can be
done in polynomial time using the linear programming algorithm in [GLS].
Specifically, let l(y) ≥ 0 be a constraint of TR. Then we just need to minimize
l(y) on SR and check if the minimum exceeds zero. Q.E.D.
But this method does not work when the number of constraints of TR is
exponential, as expected in the context of the lower bound problems under
consideration. In fact, no generic black-box-type algorithm, like the one in
[GLS] based on just a membership or separation oracle for TR, can be used
to prove (4) when the number of constraints of TR is exponential.
Fortunately, this is not a serious problem. A basic principle in combi-
natorial optimization, as illustrated in [GLS], is that a complexity theoretic
property that holds for polytopes with polynomially many constraints will
also hold for polytopes with exponentially many constraints, provided these
constraints are sufficiently well-behaved. For example, Edmond’s perfect
matching polytope for nonbipartite graphs has complexity-theoretic proper-
ties similar to the perfect matching polytope for bipartite graphs, though it
can have exponentially many constraints. We have already remarked that
TR and SR are analogues of the Littlewood-Richardson cones. The facets of
the Littlewood-Richardson cone have a very nice explicit description [Kl, Z].
The cones TR, SR here are expected to have similar nice explicit descrip-
tion. This is why Hypothesis 7.6.9 can be expected to hold even if the
number of constraints of TR is exponential, just as it holds even when SR
has exponentially many constraints. But a polynomial-time algorithm as in
Hypothesis 7.6.9 would have to depend crucially on the specific nature of
the facets (constraints) of TR in the spirit of the linear-programming-based
algorithm for the construction of a maximum-weight perfect matching in
nonbipartite graphs [Ed], where too the number of constraints is exponen-
tial but the algorithm still works because of the structure theorems based
on the specific nature of the constraints.
7.7 Arithmetic form of the P vs NP problem in
characteristic zero
We turn now to the arithemetic form of the P vs. NP problem in character-
istic zero. The arguments are essentially verbatim translations of those for
the arithmetic form of the P#P vs. NC problem in the preceding section.
Hence we shall be brief.
In the preceding section h(X) was perm(X) and g(Y ) was det(Y ). Now
h(X) and g(Y ) would be explicit (co)-NP-complete and P -complete func-
tions E(X) and H(Y ) constructed in [GCT1]. They can be thought of as
points in suitable W = Symk(X) and V = Syml[Y ], k = O(n2), l = O(m2),
with the natural action of GL(X) and G = GL(Y ), where n denotes the
number of input parameters and m denotes the circuit size parameter in
the lower bound problem. These functions are extremely special like the
determinant and the permanent in the sense that they are “almost” char-
acterized by their stabilizers as explained in [GCT1]–and this is enough for
our purposes.
We again have a natural embedding φ : P (W ) → P (V ), which lets us
define f = φ(h). The class variety for NP is defined to be ∆V [f ] ⊆ P (V ),
the projective closure of the orbit Gf . The class variety for P is ∆̃V [g] ⊆
P (V ), which is defined to be the projective closure of G[g], where [g] denotes
the set of points in P (V ) that are stabilized by Gg ⊆ G, the stabilizer of g.
An explicit description of Gg is given in [GCT1]; cf. Section 7 therein. To
show P 6= NP in characteristic zero, it suffices to show that ∆V [f ] is not a
subvariety of ∆̃V [g] for all large enough n, if m = poly(n) (cf. Conjecture
7.4. in [GCT1]). For this, in turn, it suffices to show existence of strong
obstructions, defined very much as in Section 7.6, for all n, assumming
m = poly(n).
We can then formulate PH1 for the new h(X) and g(Y ) just as in Hy-
potheses 7.6.1 and 7.6.2, and the notion of a robust obstruction as in Defi-
nition 7.6.3. We then have:
Theorem 7.7.1 (Verification of obstructions)
Analogues of Theorems 7.6.5 and 7.6.6 holds for h(X) = E(X) and
g(Y ) = H(Y ).
Furthermore, even discovery of robust obstructions can be conjectured
to be easy (poly-time)–this would follow from the obvious analogue of Hy-
pothesis 7.6.9 here.
Heuristic argument for existence of robust obstructions is very similar
to the one in Section 7.6.6. It needs SH for the special case of the subgroup
restriction problem for the embedding Gg →֒ G. The group Gg, as described
in [GCT1], is a product of some copies of the algebraic torus and the sym-
metric group. The subgroup restriction problem in this case is akin to but
harder than the plethysm problem.
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	Introduction
	The decision problems
	Deciding nonvanishing of Littlewood-Richardson coefficients
	Back to the general decision problems
	Saturated and positive integer programming
	Quasi-polynomiality, positivity hypotheses, and the canonical models
	The plethysm problem
	Towards PH1, SH, PH2,PH3 via canonial bases and canonical models
	Basic plan for implementing the flip
	Organization of the paper
	Notation
	Preliminaries in complexity theory
	Standard complexity classes
	Example: Littlewood-Richardson coefficients
	Convex #P
	Littlewood-Richardson coefficients
	Littlewood-Richardson cone
	Eigenvalues of Hermitian matrices
	Separation oracle
	Saturation and positivity
	Saturated and positive integer programming
	A general estimate for the saturation index
	Extensions
	Is there a simpler algorithm?
	Littlewood-Richardson coefficients again
	The saturation and positivity hypotheses
	The subgroup restriction problem
	Explicit polynomial homomorphism
	Input specification and bitlengths
	Stretching function and quasipolynomiality
	The decision problem in geometric invariant theory
	Reduction from Problem 1.1.3 to Problem 1.1.4
	Input specification
	Stretching function and quasi-polynomiality
	Positivity hypotheses
	G/P and Schubert varieties
	PH3 and existence of a simpler algorithm
	Other structural constants
	Quasi-polynomiality and canonical models
	Quasi-polynomiality
	The minimal positive form and modular index
	The rings associated with a structural constant
	Canonical models
	From PH0 to PH1,3
	On PH0 in general
	Nonstandard quantum group for the Kronecker and the plethysm problems
	The cone associated with the subgroup restriction problem
	Elementary proof of rationality
	Parallel and PSPACE algorithms
	Complex semisimple Lie group
	Symmetric group
	General linear group over a finite field
	Tensor product problem
	Finite simple groups of Lie type
	Experimental evidence for positivity
	Littlewood-Richardson problem
	Kronecker problem, n=2
	G/P and Schubert varieties
	The ring of symmetric functions
	On verification and discovery of obstructions
	Obstruction
	Decision problems
	Verification of obstructions
	Robust obstruction
	Verification of robust obstructions
	Arithemetic version of the P#P vs. NC problem in characteristric zero
	Class varieties
	Obstructions
	Robust obstructions
	Verification of robust obstructions
	On explicit construction of obstructions
	Why should robust obstructions exist?
	On discovery of robust obstructions
	Arithmetic form of the P vs NP problem in characteristic zero
