New versions of Schur-Weyl duality
NEW VERSIONS OF SCHUR-WEYL DUALITY
STEPHEN DOTY
Abstract. After reviewing classical Schur-Weyl duality, we present
some other contexts which enjoy similar features, relating to Brauer
algebras and classical groups.
1. Classical Schur-Weyl duality
1.1. Schur’s double-centralizer result. Consider the vector space
V = Cn. The symmetric group Sr acts naturally on its r-fold tensor
power V ⊗r, by permuting the tensor positions. This action obviously
commutes with the natural action of GLn = GLn(C), acting by matrix
multiplication in each tensor position. So we have a CGLn-CSn bimod-
ule structure on V ⊗r. (Here CG denotes the group algebra of a group
G.) In 1927, Schur [Sc] proved that the image of each group algebra
under its representation equals the full centralizer algebra for the other
action. More precisely, if we name the representations as follows
(1) CGLn
−−−→ End(V ⊗r)
←−−− CSr
then we have equalities
ρ(CGLn) = EndSr(V
⊗r)(2)
σ(CSr) = EndGLn(V
⊗r).(3)
(Here, for a given set S operating on a vector space T through linear
endomorphisms, EndS(T ) denotes the set of linear endomrphisms of T
commuting with each endomorphism coming from S.)
Results of Carter-Lusztig [CL] and J.A. Green [G] (and others) show
that all the above statements remain true if one replaces C by an
arbitrary infinite field K.
These notes are based on a lecture, various versions of which I have given in
the past year, in a number of locations, including Stuttgart, Birmingham, Queen
Mary (London), Lancaster, Manchester, Oxford, and Cambridge. I’m grateful to
the organizers of those events for the opportunity to present these ideas.
http://arxiv.org/abs/0704.1877v1
2 STEPHEN DOTY
1.2. Schur algebras. The finite-dimensional algebra in (2) above, for
any K, is known as the Schur algebra, and often denoted by SK(n, r)
or simply S(n, r). The Schur algebra “sees” the part of the ratio-
nal representation theory of the algebraic group GLn(K) occurring (in
some appropriate sense) in V ⊗r. More precisely, there is an equiva-
lence between r-homogeneous polynomial representations of GLn(K)
and SK(n, r)-modules. In characteristic 0, those representations (as r
varies) determine all finite-dimensional rational representations, while
in positive characteristic they still provide a tremendous amount of
information.
The representation σ in (1) is faithful if n > r, so σ induces an
isomorphism
(4) KSr ≃ EndGLn(V
⊗r) = EndSK(n,r)(V
⊗r) (n > r).
This leads to intimate connections between polynomial representations
of GLn(K) and representations of KSr, a theme that has been ex-
ploited by many authors in recent years. Perhaps the most dramatic
example of this is the result of Erdmann [E] (building on previous work
of Donkin [Do3] and Ringel [R]) which shows that knowing decompo-
sition numbers for all symmetric groups in positive characteristic will
determine the decomposition numbers for general linear groups in the
same characteristic. Conversely, James [Ja] had already shown that
the decomposition matrix for a symmetric group is a submatrix of the
decomposition matrix for an appropriate Schur algebra. Thus the (still
open) general problem of determining the modular characters of sym-
metric groups is equivalent to the similar problem for general linear
groups (over infinite fields).
1.3. The enveloping algebra approach. Return to the basic setup,
over C. One may differentiate the action of the Lie group GLn(C) to
obtain an action of its Lie algebra gln. Replacing the representation ρ
in (1) by its derivative representation dρ : U(gln)→ End(V
⊗r) leads to
the following alternative statement of Schur’s result:
dρ(U(gln)) = EndSr(V
⊗r)(5)
σ(CSr) = Endgln(V
⊗r).(6)
In particular, the Schur algebra (over C) is a homomorphic image of
U(gln). All of this works over an arbitrary integral domain K if we
replace U(gln) by its “hyperalgebra” UK := K⊗ZUZ obtained by change
of ring from a suitable Z-form of U(gln); see [Do1]. (One can adapt
the Kostant Z-form, originally defined for the enveloping algebra of a
semisimple Lie algebra, to the reductive gln.)
NEW VERSIONS OF SCHUR-WEYL DUALITY 3
1.4. The quantum case. Jimbo [Ji] extended the results of 1.3 to the
quantum case (where the quantum parameter is not a root of unity).
One needs to replace Sr by the Iwahori-Hecke algebra H(Sr) and re-
place U(gln) by the quantized enveloping algebra U(gln). The analogue
of the Schur algebra in this context is known as the q-Schur algebra, of-
ten denoted by S(n, r) or Sq(n, r). Dipper and James [DJ] have shown
that q-Schur algebras are fundamental for the modular representation
theory of finite general linear groups.
As many authors have observed, the picture in 1.1 can also be quan-
tized. For that one needs a suitable quantization of the coordinate
algebra of the algebraic group GLn.
There is a completely different (geometric) construction of q-Schur
algebras given in [BLM].
1.5. Integral forms. The Schur algebras SC(n, r) admit an integral
form SZ(n, r) such that SK(n, r) ≃ K ⊗Z SZ(n, r) for any field K. In
fact SZ(n, r) is simply the image of UZ (see 1.3) under the surjective
homomorphism U(gln) → SC(n, r). Similarly, the quantum Schur al-
gebra SQ(v)(n, r) admits an integral form defining all specializations
via base change. One needs to replace Z by A = Z[v, v−1]; then the
integral form SA(n, r) is the image of the Lusztig A-form UA under
the surjection U(gln) → SQ(v)(n, r). (To match this up with various
specializations in the literature, one often has to take q = v2.)
1.6. Generators and relations. Recently, in joint work with Gi-
aquinto (see [DG]), a very simple set of elements generating the kernel
of the surjection U(gln) → SC(n, r) was found. A very similar set of
elements generates the kernel of the surjection U(gln) → SQ(v)(n, r).
These elements are expressible entirely in terms of the Chevalley gener-
ators for the zero part of U(gln) or U(gln). Thus we obtain a presenta-
tion of SC(n, r) and SQ(v)(n, r) by generators and relations, compatible
with the usual Serre (Drinfeld-Jimbo) presentation of U(gln) (resp.,
U(gln)). As a result, we find a certain subset of the integral PBW-
basis for U(gln) or U(gln) the image of which gives an integral basis
for SZ(n, r) or SA(n, r). This basis yields a similar basis in any spe-
cialization. Moreover, a subset of it provides a new integral basis of
H(Sn).
2. The Brauer algebra
From now on I will assume, unless stated otherwise, that the under-
lying field is C (it could just as well be any field of characteristic zero).
4 STEPHEN DOTY
One expects that many statements will be valid over an arbitrary infi-
nite field, via some appropriate integral form, similar to what happens
in type A.
2.1. The algebra B
r . Let R be a commutative ring, and consider
the free R[x]-module B
r with basis consisting of all r-diagrams. An
r-diagram is an (undirected) graph on 2r vertices and r edges such that
each vertex is incident to precisely one edge. One usually thinks of the
vertices as arranged in two rows of r each, the top and bottom rows.
(See Figure 1.) Edges connecting two vertices in the same row (different
rows) are called horizontal (resp., vertical). We can compose two such
diagrams D1, D2 by identifying the bottom row of vertices in the first
diagram with the top row of vertices in the second diagram. The result
is a graph with a certain number, δ(D1, D2), of interior loops. After
removing the interior loops and the identified vertices, retaining the
edges and remaining vertices, we obtain a new r-diagram D1 ◦D2, the
composite diagram.
Multiplication of r-diagrams is defined by the rule
D1 ·D2 = x
δ(D1,D2)(D1 ◦D2).
One can check that this multiplication makes B
r into an associative
algebra; this is the Brauer algebra. (See Figures 1–3 for an illustration
of the multiplication in the Brauer algebra.)
Note that if we take x = 1 then the set of r-diagrams is a monoid
under diagram composition, and B
r is simply the semigroup algebra
of that monoid.
Figure 1. Two Brauer diagrams D1, D2 for r = 5.
For any x the group algebra R[x]Sr may be identified with the sub-
algebra of B
r spanned by the diagrams containing only vertical edges.
Such Brauer diagrams provide a graphical depiction of permutations.
The group algebra R[x]Sr of Sr also appears as a quotient of B
r , the
quotient by the two-sided ideal spanned by all diagrams containing at
least one horizontal edge.
NEW VERSIONS OF SCHUR-WEYL DUALITY 5
Figure 2. Computing the composite of D1 and D2.
Figure 3. The composite diagram D1 ◦D2.
Label the vertices in each row of an r-diagram by the indices 1, . . . , r.
For any 1 6 i 6= j 6 r let ci,j be the r-diagram with horizontal edges
connecting vertices i, j on the top and bottom rows. All other vertices
in the diagram ci,j are vertical, connecting vertex k on the top and
bottom rows, for all k 6= i, j. Brauer observed that B
r is generated
by the permutation diagrams together with just one of the ci,j.
2.2. Schur-Weyl duality. Brauer [Br] introduced the algebra B
1936 to describe the invariants of symplectic and orthogonal groups
acting on V ⊗r. (Brauer’s conventions were slightly different; we are
here following the approach of Hanlon and Wales [HW], who pointed
out that B
r is isomorphic with the algebra defined by Brauer to deal
with the symplectic case.) Let G be Spn or On, where n is even in the
first instance. By restricting the action ρ considered in 1.1 we have an
action of G on V ⊗r. One can extend the action of Sr to an action of
r (over C) on V
⊗r, where ǫ = −1 if G = Spn and ǫ = 1 if G = On.
To do this, it is enough to specify the action of the diagram ci,j. This
acts on V ⊗r as one of Weyl’s contraction maps contracting in tensor
positions i and j. So we have (commuting) representations
(7) CG
−−−→ End(V ⊗r)
←−−− B
6 STEPHEN DOTY
which satisfy Schur-Weyl duality; i.e., the image of each representation
equals the full centralizer algebra of the other action:
ρ(CG) = End
(V ⊗r)(8)
σ(B(ǫn)r ) = EndG(V
⊗r).(9)
The algebras in equality (8) are the symplectic and orthogonal Schur
algebras (see [Do2], [D1], [D2]).
If n > r − 1 the representation σ in (7) is faithful [Bro]; thus it
induces an isomorphism B
r ≃ EndG(V
2.3. Schur-Weyl duality in type D. In type Dn/2 (n even) the
orthogonal group On is not connected, and contains the connected
semisimple group SOn (special orthogonal group) as subgroup of in-
dex 2. In order to handle this situation, Brauer (see also [Gr]) defined
a larger algebra D
r , spanned by the usual r-diagrams previously de-
fined, together with certain partial r-diagrams on 2r vertices and r−n
edges, in which n vertices in each of the top and bottom rows are not
incident to any edge, and showed that the action of B
r can be ex-
tended to an action of this larger algebra D
r on V
⊗r. Thus we have
representations
(10) CSOn
−−−→ End(V ⊗r)
←−−− D
Brauer showed that the actions of SOn and D
r on V
⊗r satisfy Schur-
Weyl duality:
ρ(CSOn) = EndD(ǫn)r
(V ⊗r)(11)
σ(D(ǫn)r ) = EndSOn(V
⊗r).(12)
The algebra in (11) is a second Schur algebra in type D, a proper
subalgebra of the algebra End
(V ⊗r) appearing in (8) above.
2.4. Generators and relations. One can formulate the above state-
ments of Schur-Weyl duality using enveloping algebras, analogous to
1.3. This leads to a presentation (see [DGS]) of the symplectic and
orthogonal Schur algebras which is compatible with (a slight modifi-
cation of) the usual Serre presentation of the enveloping algebra U(g),
where g = spn (n even) or son.
2.5. The quantum case. There is a q-version of the Schur-Weyl du-
ality considered in this section, although not as developed as in type A.
One needs to replace the Brauer algebra by its q-analogue, the Birman-
Murakami-Wenzl (BMW) algebra (see [BW], [Mu]), and replace the
NEW VERSIONS OF SCHUR-WEYL DUALITY 7
enveloping algebra by a suitable quantized enveloping algebra. One
can think of the BMW algebra in terms of Kauffman’s tangle monoid;
see [Ka], [HR], [MW]. (Roughly speaking, tangles are replacements for
Brauer diagrams, in which one keeps track of under and over cross-
ings, subject to certain natural relations.) There are applications of
the BMW algebra to knot theory, as one might imagine.
This leads to a q-analogue of the symplectic Schur algebras, in par-
ticular, which have been studied by Oehms [Oe].
To the best of my knowledge, a q-analogue of the larger algebra D
(n even) considered in 2.3 remains to be formulated.
3. The walled Brauer algebra
3.1. The algebra B
r,s . This algebra was defined in 1994 in [BCHLLS].
It is the subalgebra of B
r+s spanned by the set of (r, s)-diagrams. By
definition, an (r, s)-diagram is an (r+ s)-diagram in which we imagine
a wall separating the first r from the last s columns of vertices, such
that:
(a) all horizontal edges cross the wall;
(b) no vertical edges cross the wall.
An edge crosses the wall if its two vertices lie on opposite sides of the
wall. The multiplication in B
r,s is that of B
Label the vertices on the top and bottom rows of an (r, s)-diagram
by the numbers 1, . . . , r to the left of the wall and −1, . . . ,−s to the
right of the wall. Let ci,−j (1 6 i 6 r; 1 6 j 6 s) be the diagram
with a horizontal edge connecting vertices i and −j on the top row
and the same on the bottom row, and with all other edges connecting
vertex k (k 6= i,−j) in the top and bottom rows. It is easy to see that
the walled Brauer algebra is generated by the permutations it contains
along with just one of the ci,−j. (Note that ci,−j is the (r+ s)-diagram
denoted by ci,r+j in 2.1.)
3.2. Dimension. What is the dimension of B
r,s ? One way to answer
that question is to consider the map, flip, from (r + s)-diagrams to
(r+s)-diagrams, defined by interchanging the top and bottom vertices
to the right of the imaginary wall. For example, Figure 4 shows a
(4, 2)-diagram (to the left) and its corresponding 6-diagram, obtained
from the left diagram by applying flip. Note that flip is involutary:
applying it twice gives the original diagram back again.
8 STEPHEN DOTY
Figure 4. A (4, 2)-diagram and its corresponding per-
mutation, after applying flip.
One easily checks that the map flip carries (r, s)-diagrams bijectively
onto the set of (r+ s)-diagrams with all edges vertical. Such diagrams
correspond with permutations of r+s objects, so the dimension of B
is (r + s)!.
3.3. Another view of B
r,s . The above correspondence between (r, s)-
diagrams and permutations gives another way to think of the multipli-
cation in B
r,s . Given two (r, s)-diagrams D1, D2 let D
2 be their
corresponding permutations obtained by applying flip. Define a new
(rather bizarre) composition on permutations as follows. Given any two
permutation diagrams D′1, D
2 (with r+ s columns of vertices) identify
the first r vertices of the bottom row of D′1 with the first r vertices of
the top row of D′2, and identify the last s vertices of the top row of
D′1 with the last s vertices of the bottom row of D
2. After removing
loops and identified vertices this gives a new permutation diagram D′3
in which the vertices in the top (resp., bottom) row are the remaining
top (bottom) row vertices from the original diagrams.
Let δ(D′1, D
2) be the number of loops removed in computing the com-
posite permutation diagram D′3. Define multiplication of permutation
diagrams by the rule
D′1 ·D
2 = x
δ(D′1,D
2)D′3
In other words, we are multiplying permutations by composing maps
“on the right” on one side of the wall, and “on the left” on the other
side (roughly speaking). For example, Figure 5 below shows the com-
putation of the composite diagram in the walled Brauer algebra (left
column) and the computation in terms of the corresponding permuta-
tions (right column). Figure 6 shows the resulting diagrams after the
single loop and identified vertices have been removed.
One can check that D′3 coincides with (D1 ◦ D2)
′ and δ(D1, D2) =
δ(D′1, D
2). In other words, flip defines an algebra isomorphism between
NEW VERSIONS OF SCHUR-WEYL DUALITY 9
Figure 5. Composition of diagrams and permutations.
The diagrams on the left correspond under flip with the
permutations on the right.
Figure 6. The corresponding diagrams resulting from
Figure 5. The two diagrams correspond under flip.
the algebra B
r,s and the algebra B̃
r,s spanned by permutation diagrams
with the multiplication defined above. Note that in particular B̃
r,0 ≃
R[x]Sr and B̃
0,s ≃ (R[x]Ss)
3.4. Schur-Weyl duality. Consider mixed tensor space V r,s := V ⊗r⊗
V ∗⊗s, where V ∗ is the usual linear dual space of V . Mixed tensor space
is naturally a module for GLn, and one obtains an action of B
r,s on
V r,s simply by restricting the action of B
r+s, which acts the same on
V r,s as it does on V ⊗(r+s), since on restriction to On we have V ≃ V
Thus we have the following commutative diagram
−−−→ End(V r,s)
←−−− B
yι′
−−−→ End(V ⊗(r+s))
←−−− B
10 STEPHEN DOTY
in which the vertical maps ι, ι′ are inclusion. By [BCHLLS], the actions
of GLn and B
r,s on V
r,s in the first row of the diagram satisfy Schur-
Weyl duality:
ρ(CGLn) = EndB(n)r,s
(V r,s)(14)
σ(B(n)r,s ) = EndGLn(V
r,s).(15)
The algebra in (14) is another Schur algebra S(n; r, s) in type A, stud-
ied in [DD]. These Schur algebras provide us with a new family of
quasihereditary algebras, generalizing the classical Schur algebras, since
S(n; r, 0) ≃ S(n, r). In fact, the S(n; r, s) provide a new class of gen-
eralized Schur algebras in the sense of Donkin [Do1]. For fixed n, the
family of S(n; r, s)-modules as r, s vary constitutes the family of all
rational representations of GLn. Whence the name rational Schur alge-
bras for the S(n; r, s).
When n > r + s, the representation σ in the top row of (13) above
is faithful, so induces an isomorphism B
r,s ≃ EndGLn(V
r,s).
3.5. The quantum case. Quantizations of the walled Brauer algebra
have been defined and studied in work of Halverson [Ha], Leduc [Le],
Kosuda-Murakami [KM], and Kosuda [K].
4. The deranged algebra
4.1. The problem. One might wonder if there are versions of Schur-
Weyl duality in which the natural module V is replaced by some other
representation. Perhaps the first choice would be to replace V with
the adjoint module, i.e., an algebraic group acting on its Lie algebra
via the adjoint representation. The simplest instance of this would be
type A, where we consider the module sl⊗rn as an SLn-module, and ask
for its centralizer algebra EndSLn(sl
n ) = EndGLn(sl
n ). (It makes no
difference whether we regard sln as module for SLn or for GLn. We
look at sl⊗rn rather than gl
n since gln is not simple as a GLn-module
or SLn-module.)
4.2. Relation with Brauer algebras. Even though this is a ques-
tion about type A, its solution is intimately connected with the walled
Brauer algebra. Here is a brief outline of the solution to this problem,
recently obtained in [BD]. The main idea is to utilize the decomposition
gln = sln ⊕ C (as SLn or GLn module) to write
(16) gl⊗rn = sl
06t<r sl
NEW VERSIONS OF SCHUR-WEYL DUALITY 11
where the left-hand-side identifies with V r,r = V ⊗r ⊗ V ∗⊗r via the
natural isomorphism V ⊗ V ∗ ≃ End(V ) ≃ gln. Thus it follows that
our desired tensor space sl⊗rn is isomorphic with a direct summand (as
a GLn or SLn-module) of the mixed tensor space V
r,r, and for n > 2r
the centralizing algebra C = EndGLn(sl
n ) is obtainable as a certain
subalgebra eEndGLn(V
r,r)e where e is the idempotent corresponding to
the projection onto the summand sl⊗rn . Thus
(17) C ≃ eB(n)r,r e (n > 2r)
where e =
i(1 − n
−1ci,−i) (notation of 3.1). The algebra in (17) has
a basis consisting of all elements of the form eDe as D ranges over
the set of (r, r)-diagrams with no horizontal edges connecting i to −i.
Such diagrams correspond under flip, after inverting the signs labeling
the bottom row, with derangements of 2r objects. (A derangement
is a permutation having no fixed points.) Let N(k) = the number of
derangements of k objects, then we have by the Inclusion-Exclusion
Principle (see [St, 2.2.1 Example])
(18) N(k) =
(−1)k−j
which is the nearest integer to k!/e (e = 2.7182818 . . . ). Thus the
dimension of the centralizer algebra eB
r,r e is given by N(2r). Because
of this connection with derangements, the algebra eB
r,r e is known as
the deranged algebra.
In [BD] explicit formulas are obtained for the number of times a
given simple module L appears as a summand of sl⊗rn . In particular, it
is shown that
N(r) = the multiplicity of the trivial module in sl⊗rn
N(r − 1) = the multiplicity of sln in sl
demonstrating that the combinatorics of derangement numbers is in-
herent in this theory.
4.3. Schur-Weyl duality. At least when n > 2r, the actions of GLn
and C on sl⊗rn satisfy Schur-Weyl duality:
ρ(CGLn) = EndeB(n)r,r e
(sl⊗rn )(19)
σ(eB(n)r,r e) = EndGLn(sl
n ).(20)
This will almost certainly hold for all n, r.
12 STEPHEN DOTY
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Uq(gl(n,C)), Osaka J. Math. 30 (1993), 475–507.
[Le] Leduc, R., A two-parameter version of the centralizer algebra of the
mixed tensor representations of the general linear group and quantum
general linear group, Ph.D. Dissertation, University of
Wisconsin-Madison 1994.
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algebra, Preprint 1989, with foreword and references updated Jan.
2000. [http://www.liv.ac.uk/∼su14/knotprints.html]
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http://www.liv.ac.uk/~su14/knotprints.html
14 STEPHEN DOTY
[Sc] Schur, I., Über die rationalen Darstellungen der allgemeinen linearen
Gruppe, (1927). Reprinted in Schur, I., Gesammelte Abhandlungen,
Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973.
[St] Stanley, R.P., Enumerative combinatorics Vol. 1, Cambridge Studies
in Advanced Math., 49, Cambridge Univ. Press, Cambridge, 1997.
Mathematics and Statistics, Loyola University Chicago, Chicago,
Illinois 60626 U.S.A.
E-mail address : doty@math.luc.edu
	1. Classical Schur-Weyl duality
	1.1. Schur's double-centralizer result
	1.2. Schur algebras
	1.3. The enveloping algebra approach
	1.4. The quantum case
	1.5. Integral forms
	1.6. Generators and relations
	2. The Brauer algebra
	2.1. The algebra Br(x)
	2.2. Schur-Weyl duality
	2.3. Schur-Weyl duality in type D
	2.4. Generators and relations
	2.5. The quantum case
	3. The walled Brauer algebra
	3.1. The algebra Br,s(x)
	3.2. Dimension
	3.3. Another view of Br,s(x)
	3.4. Schur-Weyl duality
	3.5. The quantum case
	4. The deranged algebra
	4.1. The problem
	4.2. Relation with Brauer algebras
	4.3. Schur-Weyl duality
	References
