A Diagrammatic Category for the Representation Theory of U_q(sl_n)
A Diagrammatic Category for the Representation Theory of Uq(sln)
Scott Edward Morrison
B.Sc. (Hons) (University of New South Wales) 2001
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
Mathematics
in the
GRADUATE DIVISION
of the
UNIVERSITY of CALIFORNIA, BERKELEY
Committee in charge:
Professor Vaughan Jones, Chair
Professor Richard Borcherds
Professor Steve Evans
Spring 2007
http://arxiv.org/abs/0704.1503v1
A Diagrammatic Category for the Representation Theory of Uq(sln)
Copyright 2007
Scott Edward Morrison
Abstract
A Diagrammatic Category for the Representation Theory of Uq(sln)
Scott Edward Morrison
Doctor of Philosophy in Mathematics
University of California, Berkeley
Professor Vaughan Jones, Chair
This thesis provides a partial answer to a question posed by Greg Kuperberg in [24] and
again by Justin Roberts as problem 12.18 in Problems on invariants of knots and 3-manifolds
[28], essentially:
Can one describe the category of representations of the quantum group Uq(sln)
(thought of as a spherical category) via generators and relations?
For each n ≥ 0, I define a certain tensor category of trivalent graphs, modulo iso-
topy, and construct a functor from this category onto (a full subcategory of) the category
of representations of the quantum group Uq(sln). One would like to describe completely
the kernel of this functor, by providing generators for the tensor category ideal. The re-
sulting quotient of the diagrammatic category would then be a category equivalent to the
representation category of Uq(sln).
I make significant progress towards this, describing certain elements of the ker-
nel, and some obstructions to further elements. It remains a conjecture that these elements
really generate the kernel. The argument is essentially the following. Take some trivalent
graph in the diagrammatic category for some value of n, and consider the morphism of
Uq(sln) representations it is sent too. Forgetting the full action of Uq(sln), keeping only a
Uq(sln−1) action, the source and target representations branch into direct sums, and the
morphism becomes a matrix of maps of Uq(sln−1) representations. Arguing inductively
now, we attempt to write each such matrix entry as a linear combination of diagrams for
n − 1. This gives a functor dGT between diagrammatic categories, realising the forgetful
functor at the representation theory level. Now, if a certain linear combination of diagrams
for n is to be in the kernel of the representation functor, each matrix entry of dGT applied
to that linear combination must already be in the kernel of the representation functor one
level down. This allows us to perform inductive calculations, both establishing families of
elements of the kernel, and finding obstructions to other linear combinations being in the
kernel.
This thesis is available electronically from the arXiv, at arXiv:0704.1503, and at
http://tqft.net/thesis.
http://arxiv.org/abs/0704.1503
http://tqft.net/thesis
Contents
1 Introduction 1
1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Temperley-Lieb algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Kuperberg’s spiders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 The ‘diagrammatic’ category Symn 5
2.1 Pivotal categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Quotients of a free tensor category . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Flow vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Polygonal webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Just enough representation theory 16
3.1 The Lie algebra sln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The quantum groups Uq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Strictifying RepUq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Generators for FundRepUq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 The representation functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 The diagrammatic Gel‘fand-Tsetlin functor 31
4.1 Definition on generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Descent to the quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Calculations on small webs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4 A path model, and polygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Describing the kernel 41
5.1 The I = H relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 The square-switch relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 The Kekulé relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.4 More about squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.5 A conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.6 Examples: Uq(sln), for n = 2, 3, 4 and 5. . . . . . . . . . . . . . . . . . . . . . 46
5.7 Proofs of Theorems 5.1.1, 5.2.1 and 5.3.2 . . . . . . . . . . . . . . . . . . . . . 52
6 Relationships with previous work 66
6.1 The Temperley-Lieb category . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2 Kuperberg’s spider for Uq(sl3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.3 Kim’s proposed spider for Uq(sl4) . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.4 Tags and orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.5 Murakami, Ohtsuki and Yamada’s trivalent graph invariant . . . . . . . . . 68
6.6 Jeong and Kim on Uq(sln) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.7 Other work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Future directions 71
Bibliography 74
A Appendices 77
A.1 Boring q-binomial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A.2 The Uq(sln) spider cheat sheet. . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Acknowledgements
First I’d like to thank Vaughan for being such a great advisor. I’m glad he took a chance on
me, and I hope he’s not too disappointed by the complete absence of subfactors in what fol-
lows! I’m grateful for so many things; good advice, lots of mathematics, constant interest
and encouragement, gentle reminders to keep climbing and work in balance, windsurfing
lessons, and a great friendship.
Thanks also to Dror Bar-Natan and Kevin Walker. I’ve learnt a ton from each of
them, thoroughly enjoyed working with them, and look forward to more! And further
thanks to Greg Kuperberg, for introducing me to the subject this thesis treats, and taking
interest in my work. Conversations with Joel Kamnitzer, Mikhail Khovanov, Ari Nieh, Ben
Webster, Noah Snyder and Justin Roberts helped me along the way.
Thanks to my family, Pam, Graham and Adele Morrison, for endless love and
support. And finally, thanks to my friends Nina White, Yossi Farjoun, Rahel Wachs, Erica
Mikesh, and Carl Mautner.
Chapter 1
Introduction
1.1 Summary
The eventual goal is to provide a diagrammatic presentation of the representation theory
of Uq(sln). The present work describes a category of diagrams, along with certain relations
amongst these diagrams, and a functor from these diagrams to the representation theory.
Further, I conjecture that the relations given are in fact all of them—that this functor is an
equivalence of categories.
We begin by defining a ‘freely generated category of diagrams’, and show that
there’s a well-defined functor from this category to the category of representations of
Uq(sln). Essentially, this is a matter of realising that the representation category is a piv-
otal category, and, as a pivotal category, it is finitely generated. It’s then a matter of trying
to find the kernel of this functor; if we could do this, the quotient by the kernel would give
the desired diagrammatic category equivalent to the representation category.
This work extends Kuperberg’s work [24] on RepUq(sl3), and agrees with a pre-
viously conjectured [21] description of RepUq(sl4). Some, but not all, of the relations have
been presented previously in the context of the quantum link invariants [11, 27]. There’s a
detailed discussion of connections with previous work in §6.
For each n ≥ 0 we have a category of (linear combinations of) diagrams
Pivn =
 , , ,
a+ b+ c = n
1 ≤ k ≤ n− 1
pivotal
with edges labelled by integers 1 through n−1, generated by two types of trivalent vertices,
and orientation reversing ‘tags’, as shown above. We allow arbitrary planar isotopies of
the diagrams. This category has no relations; it is a free pivotal category.
We can construct a functor from this category into the representation theory
Repn : Pivn → RepUq(sln) .
This functor is well-defined, in that isotopic diagrams give the same maps between rep-
resentations. Our primary goal is thus to understand this functor, and to answer two
questions:
1. Is Repn full? That is, do we obtain all morphisms between representations?
2. What is the kernel of Repn? When do different diagrams give the same maps of rep-
resentations? Can we describe a diagrammatic quotient category which is equivalent
to the representation theory?
The first question has a relatively straightforward answer. We do not get all of
RepUq(sln), but if we lower our expectations to the subcategory containing only the fun-
damental representations, and their tensor products, then the functor is in fact full. Ku-
perberg gave a proof of this fact for n = 3, by recognising the image of the functor using a
Tannaka-Krein type theorem. This argument continues to work with only slight modifica-
tions for all n. I’ll also give a direct proof using quantum Schur-Weyl duality, in §3.5.
The second question has proved more difficult. Partial answers have been known
for some time. I will describe a new method for discovering elements of the kernel, based
on branching. This method also gives us a limited ability to find obstructions for further
relations.
The core of the idea is that there is a forgetful functor
GT : RepUq(sln) → RepUq(sln−1) ,
which forgets the full Uq(sln) action but does not change the underlying linear maps, and
that this should be reflected somehow in the diagrams. A diagram in Pivn ‘represents’
some morphism in RepUq(sln); thinking of this as a morphism in RepUq(sln−1) via GT ,
we can hope to represent it by diagrams in Pivn−1. This hope is borne out—in §4 I con-
struct a functor dGT : Pivn → Mat(Pivn−1) (and explain what a ‘matrix category’ is), in
such a way that the following diagram commutes:
// RepUq(sln)
Mat(Pivn−1)
Mat(Repn−1)
// RepUq(sln−1)
With this functor on hand, we can begin determining the kernel of Repn. In particular,
given a morphism in Pivn (that is, some linear combination of diagrams) we can consider
the image under dGT . This is a matrix of (linear combinations of) diagrams in Pivn−1.
Then the original diagrammatic morphism becomes zero in the Uq(sln) representation the-
ory exactly if each entry in this matrix of diagrams is zero in the Uq(sln−1) representa-
tion theory. Thus, if we understand the kernel of Repn−1, we can obtain quite strong
restrictions on the kernel of Repn. Of course, the kernels of Rep2 and Rep3 are well
known, given by the relations in the Temperley-Lieb category and Kuperberg’s spider for
sl3. Moreover, the kernel of Rep1 is really easy to describe. The method described allows
us to work up from these, to obtain relations for all Uq(sln).
In §5 I use this approach to find three families of relations, and to show that these
relations are the only ones of certain types. It remains a conjecture that the proposed rela-
tions are in fact complete.
The first family are the I = H relations, essentially correspond to 6− j symbols:
= (−1)(n+1)a .
For a given boundary, there are two types of squares, and each can be written as a linear
combination of the others. I call these relations the ‘square-switch’ relations. When n +
Σa− Σb ≥ 0, for max b ≤ l ≤ min a+ n we have
min b
m=max a
n+Σa− Σb
m+ l − Σb
and when n+Σa− Σb ≤ 0, for max a ≤ l ≤ min b we have
n+mina
m=max b
Σb− n− Σa
m+ l − Σa− n
Finally, there are relations amongst polygons of arbitrarily large size (but with a cutoff for
each n), called the ‘Kekulé ’ relations. For each Σb ≤ j ≤ Σa+ n− 1,
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
1.2 The Temperley-Lieb algebras
The n = 2 part of this story has, unsurprisingly, been understood for a long time. The
Temperley-Lieb category gives a diagrammatic presentation of the morphisms between
tensor powers of the standard representation of Uq(sl2). The objects of this category are
natural numbers, and the morphisms from n to m are Z[q, q−1]-linear combinations of dia-
grams drawn in a horizontal strip consisting of non-intersecting arcs, with n arc endpoints
on the bottom edge of the strip, and m on the top edge. (Notice, in particular, that I’m an
optimist, not a pessimist; time goes up the page.) Composition of morphisms is achieved
by gluing diagrams, removing each closed circle in exchange for a factor of [2]q.
1.3 Kuperberg’s spiders
The n = 3 story, dates back to around Kuperberg’s paper [24]. There he defines the notion
of a ‘spider’ (in this work, we use the parallel notion of a pivotal category), and constructs
the spiders for each of the rank 2 Lie algebras A2 = su(3), B2 = sp(4) and G2 and their
quantum analogues. Translated into a category, his A2 spider has objects words in (+,−),
and morphisms (linear combinations of) oriented trivalent graphs drawn in a horizontal
strip, with orientations of boundary points along the top and bottom edges coinciding with
the target and source word objects, and each trivalent vertex either ‘oriented inwards’ or
‘oriented outwards’, subject to the relations
= [3]q = q
2 + 1 + q−2 (1.3.1)
= − [2]q (1.3.2)
= + . (1.3.3)
(Note that there’s a ‘typo’ in Equation (2) of [24], corresponding to Equation (1.3.3) above;
the term has been replaced by another copy of the term.)
Kuperberg proves that this category is equivalent to a full subcategory of the
category of representations of the quantum group Uq(sl3); the subcategory with objects
arbitrary tensor products of the two 3-dimensional representations. It is essentially an
equinumeration proof, showing that the number of diagrams (modulo the above relations)
with a given boundary agrees with the dimension of the appropriate Uq(sl3) invariant
space. I’m unable to give an analogous equinumeration argument in what follows.
Note that the n = 3 special case of my construction will not quite reproduce
Kuperberg’s relations above; the bigon relation will involve a + [2]q, not a − [2]q. This is
just a normalisation issue, resolved by multiplying each vertex by
Chapter 2
The ‘diagrammatic’ category Symn
Just as permutations form groups, planar diagrams up to planar isotopy form
pivotal categories. In what follows, we’ll define a certain ‘free (strict) pivotal category’,
Pivn along with a slight modification called Symn obtained by adding some symmetries
and some relations for degenerate cases. Essentially, Pivn will be the category of trivalent
graphs, with edges carrying both orientations and labels 1 through n − 1, up to planar
isotopy.
For lack of a better place, I’ll introduce the notion of a matrix category here; given
any category C in which the Hom spaces are guaranteed to be abelian groups, we can
form a new category Mat (C), whose objects are formal finite direct sums of objects in C,
and whose morphisms are matrices of appropriate morphisms in C. Composition of mor-
phisms is just matrix multiplication (here’s where we need the abelian group structure on
Hom spaces). If the category already had direct sums, then there’s a natural isomorphism
C ∼= Mat (C).
2.1 Pivotal categories
I’ll use the formalism of pivotal categories in the following. This formalism is essentially
interchangeable with that of spiders, due to Kuperberg [24], or of planar algebras, due to
Jones [13].1
A pivotal category is a monoidal category2 C equipped with
1. a cofunctor ∗ : Cop → C, called the dual,
2. a natural isomorphism τ : 1C → ∗∗,
3. a natural isomorphism γ : ⊗ ◦ (∗ × ∗) → ∗ ◦ ⊗op,
1These alternatives are perhaps more desirable, as the pivotal category view forces us to make a artificial
distinction between the domain and codomain of a morphism. I’ll have to keep reminding you this distinction
doesn’t matter, in what follows. On the other hand, the categorical setup allows us to more easily incorporate
the notion of direct sum.
2For our purposes, we need only consider strict monoidal categories, where, amongst other things, the
tensor product is associative on the nose, not just up to an isomorphism. The definitions given here must be
modified for non-strict monoidal categories.
4. an isomorphism e → e∗, where e is the neutral object for tensor product, and
5. for each object c ∈ C, a ‘pairing’ morphism pc : c∗ ⊗ c → e.
with the natural isomorphisms satisfying certain coherence conditions, and the pairing
morphisms certain axioms, all given in [2].3 The primary example of a pivotal category
is the category of representations of an involutory or ribbon Hopf algebra [2]. A pivotal
functor between two pivotal categories should intertwine the duality cofunctors, commute
with the isomorphisms e → e∗, and take pairing morphisms to pairing morphisms. It
should also intertwine the natural isomorphisms τ ; that is, for a functor F : C → D, we
require τD
= F(τCa ). There’s a similar condition for γ.
In the case that τ and γ are simply the identity on each object (and e = e∗), we
say the pivotal category is strict. Unfortunately, representations of a Hopf algebra do not
generally form a strict pivotal category; τ cannot be the identity. However, every pivotal
category is equivalent to a strict pivotal category [2]. We’ll take advantage of this later!
In a pivotal category with e = e∗ and γ the identity (but not necessarily also τ ),
the axioms satisfied by the natural isomorphism τ simplify to the requirements that it’s a
tensor natural transformation: τa⊗b = τa ⊗ τb, and that τ∗a : a∗∗∗ → a∗ and τa∗ : a∗ → a∗∗∗
are inverse morphisms.4 In this same situation, the axioms for the pairing morphisms
simplify to:
1. For each object a,
= . (2.1.1)
2. For each morphism f : a → b,
= . (2.1.2)
3. For all objects a and b,
= . (2.1.3)
3Actually, [6] is an earlier reference for the strict case (see below), and they cite a preprint of [15] for the
full version; however, the published version of that paper ended up introducing a slightly different notion, of
‘autonomy’ for duals.
4This condition might also be stated as τ commuting with the cofunctor ∗.
From these, we can derive
Lemma 2.1.1. The dual of a morphism can be written in terms of the original morphism, compo-
nents of τ , and the pairing morphisms as
Proof.
using by the naturality of τ , which becomes
by Equation (2.1.2), and finally
by Equation (2.1.1).
Note that the category of matrices over a pivotal category is still pivotal, in an
essentially obvious way.
2.2 Quotients of a free tensor category
To begin with, let’s just define a free (strict) monoidal category on some generating mor-
phisms, which we’ll call Tn. We’ll then add some relations implementing planar isotopy
to obtain Pivn, and some more relations to obtain Symn. The objects of Tn form a monoid
under tensor product, with neutral object 0 (sometimes also called n), generated by the set
{1, . . . , n− 1, 1∗, . . . , (n− 1)∗}. The ‘generating morphisms’ are diagrams
, , ,
for each a = 1, . . . , n− 1 (but not for the ‘dual integers’ 1∗, . . . , (n − 1)∗), along with
, , , and (2.2.1)
again for each a = 1, . . . , n − 1, and finally
and (2.2.2)
for each a, b, c = 0, 1, . . . , n − 1 such that a + b + c = n. We’ll sometimes speak of the
‘type’ of a vertex; the first, outgoing, vertex here is of +-type, the second, incoming, vertex
is of −-type. We say a vertex is ‘degenerate’ if one of its edges is labelled with 0. Notice
there are no nondegenerate trivalent vertices for n = 2, and exactly one of each type for
n = 3. To read off the source of such a morphism, you read across the lower boundary of
the diagram; each endpoint of an arc labelled a gives a tensor factor of the source object,
either a if the arc is oriented upwards, or a∗ if the arc is oriented downwards. To read the
target, simply read across the upper boundary. Thus the source of is 0, and the target
is a ⊗ b ⊗ c. All morphisms are then generated from these, by formal tensor product and
composition, subject only to the usual identities of a tensor category.
Next Pivn. This category has exactly the same objects. The morphisms, however,
are arbitrary trivalent graphs drawn in a strip, which look locally like one of the pictures
above, up to planar isotopy fixing the boundary of the strip. (Any boundary points of the
graph must lie on the boundary of the strip.) Thus the graphs are oriented, with each edge
carrying a label 1 through n, and edges only ever meet bivalently as in Equation (2.2.1)
or trivalently as in Equation (2.2.2). Being a little more careful, we should ask that the
diagrams have product structure near the boundary, that this is preserved throughout the
isotopies, and that small discs around the trivalent vertices are carried around rigidly by
the isotopies, so we can always see the ordering (not just the cyclic ordering) of the three
edges incident at a vertex. (Note, though, that we’re not excited about being able to see
this ordering; we’re going to quotient it out in a moment.) The source and target of such a
graph can be read off from the graph exactly as described for the generators of Tn above.
Next, we make this category into a strict pivotal category. For this, we need to
define a duality functor, specify the evaluation morphisms, and then check the axioms of
§2.1. The duality functor on objects is defined by 0∗ = 0 and otherwise (k)∗ = k∗, (k∗)∗ = k.
On morphisms, it’s a π rotation of the strip the graph is drawn in. Clearly the double dual
functor ∗∗ is the identity on the nose. The evaluation morphisms for a = 1, . . . , n − 1 are
‘leftwards-oriented’ cap diagrams; the evaluation morphisms for a = 1∗, . . . , n − 1∗ are
‘rightwards-oriented’. The axioms in Equation (2.1.1) and (2.1.2) are then satisfied auto-
matically, because we allow isotopy of diagrams. The evaluation morphisms for iterated
tensor products are just the nested cap diagrams, with the unique orientations and labels
matching the required source object. This definition ensures the axiom of Equation (2.1.3)
is satisfied.
There’s a (tensor) functor from Tn to Pivn, which I’ll call Draw. Simply take a
morphism in Tn, which can be written as a composition of tensor products of generating
morphisms, and draw the corresponding diagram, using the usual rules of stacking boxes
to represent composition, and juxtaposing boxes side by side to represent tensor product.
The resulting diagram can then be interpreted as a morphism in Pivn. This is well-defined
by the usual nonsense of [14], that the identities relating tensor product and composition
in a tensor category correspond to ‘rigid’ isotopies (that is, isotopies which do not rotate
boxes). The functor is obviously full; or at least, obviously modulo some Morse theory.
The kernel of this functor is generated by the extra isotopies we allow in Pivn. Thus, as a
tensor ideal of Tn, kerDraw is generated by
= = , (2.2.3)
= = , (2.2.4)
= = ,
(2.2.5)
= = . (2.2.6)
There are other obvious variations of Equations (2.2.3) (rotating the vertex other way) and
(2.2.5) (tags pointing the other way), but these follow easily from the ones given here.
Whenever we want to define a functor on Pivn by defining it on generators, we need to
check these morphisms are in the kernel.
Finally, we can define the category we’re really interested in, which I’ll call Symn.
We’ll add just a few more relations to Pivn; these will be motivated shortly when we define
a functor from Tn to the representations category of Uq(sln). This functor will descend to
the quotient Pivn, and then to the quotient Symn, and the relations we add from Pivn to
Symn will be precisely the parts of the kernel of this functor which only involve a single
generator. The real work of this thesis is, of course, understanding the rest of that kernel!
We add relations insisting that the trivalent vertices are rotationally symmetric
= , = , (2.2.7)
that opposite tags cancel
= , (2.2.8)
that dual of a tag is a ±1 multiple of a tag
= (−1)(n+1)a , = (−1)(n+1)a , (2.2.9)
and that trivalent vertices ‘degenerate’ to tags
= = . (2.2.10)
Notice here we’re implicitly using the canonical identifications between the objects 0 ⊗ a,
a, and a⊗ 0, available because our tensor categories are strict.
Clearly the element of kerDraw in Equation (2.2.3) can be constructed by tensor
product and composition out of the briefer rotations in Equation (2.2.7), and so in checking
the well-definedness of a functor on Symn, we only need to worry about the latter.
2.3 Flow vertices
We’ll now introduce two new types of vertices. You could add them as diagrammatic gen-
erators, then impose as relations the formulas below, but it’s less cumbersome to just think
of them as a convenient notation. In each of these vertices, there will be some ‘incoming’
and some ’outgoing’ edges, and the sum of the incoming edges will be the same as the
sum of the outgoing edges.
Definition 2.3.1. The ‘flow vertices’ are
The convention here is that the ‘hidden tag’ lies on the ‘thick’ edge, and points
counterclockwise. These extra vertices will be convenient in what follows, hiding a profu-
sion of tags. ‘Splitting’ vertices are of +-type, ‘merging’ vertices are of −-type.
2.4 Polygonal webs
To specify the kernel of the representation functor, in §5, we’ll need to introduce some
notations for ‘polygonal webs’. These webs will come in two families, the ‘P’ family and
the ‘Q’ family. In each family, the vertices around the polygon will alternate in type. A
boundary edge which is connected to a +-vertex in a P-polygon will be connected to a
−-vertex in a Q-polygon.
For a, b ∈ Zk define5 the boundary label pattern
L(a, b) = (bk − a1,−)⊗ (bk − ak,+)⊗ · · · ⊗ (b2 − a2,+)⊗ (b1 − a2,−)⊗ (b1 − a1,+).
We’ll now define some elements of HomSymn(∅,L(a, b)), Pna,b;l for max b ≤ l ≤ min a + n
and Qna,b;l for max a ≤ l ≤ min b by
Pna,b;l =
= (2.4.1)
5I realise this definition is ‘backwards’, or at least easier to read from right to left than from left to right.
Sorry—I only realised too late.
Qna,b;l =
= . (2.4.2)
(The diagrams are for k = 3, but you should understand the obvious generalisation for any
k ∈ N.) You should consider the first of each pair of diagrams simply as notation for the
second. Each edge label is a signed sum of the ‘flows labels’ on either side, determining
signs by relative orientations. It’s trivial6 to see that for web diagrams with only ‘2 in, 1
out’ and ‘1 in, 2 out’ vertices, it’s always possible to pick a set of flow labels corresponding
to an allowable set of edge labels. Not every set of flow labels, however, gives admissible
edge labels, because the edge labels must be between 0 and n. The allowable flow labels
for the P- and Q-polygons are exactly those for which ai, ai+1 ≤ bi ≤ n + ai, n + ai+1.
Further, there’s a Z redundancy in flow labels; adding a constant to every flow label in a
diagram doesn’t actually change anything. Taking this into account, there’s a finite set of
pairs a, b for each n and k. The inequalities on the ‘internal flow label’ l for both Pna,b;l and
Qna,b;l simply demand that all the internal edges have labels between 0 and n, inclusive.
We denote the subspace of HomSymn(∅,L(a, b)) spanned by all the P-type poly-
gons by APna,b, and the subspace spanned by the Q-type polygons by AQna,b. The space
APna,b is min a −max b+ n − 1 dimensional (or 0 dimensional when this quantity is nega-
tive), and the space AQna,b is max a−min b+ 1 dimensional.
A word of warning; a and b each having k elements does not necessarily mean
that Pna,b;l or Qna,b;l are honest 2k-gons. This can fail in two ways. First of all, if some
ai = bi, ai + n = bi, ai+1 = bi or ai+1 + n = bi, then one of the external edges carries a
trivial label. Further, when l takes on one of its extremal allowed values, at least one of
the internal edges of the polygon becomes trivial, and the web becomes a tree, or a disjoint
union of trees and arcs. For example (ignoring the distinction between source and target
of morphisms; strictly speaking these should all be drawn with all boundary points at the
6Actually, perhaps only trivial after acknowledging that the disk in which the diagrams are drawn is simply
connected.
top of the diagram),
P4(0,0,0),(1,1,1);1 = P
(0,0,0),(1,1,1);2 =
P4(0,0,0),(1,1,1);3 = P
(0,0,0),(1,1,1);4 =
P5(0,1,1),(2,2,2);2 = P
(0,1,1),(2,2,2);3 =
P5(0,1,1),(2,2,2);4 = P
(0,1,1),(2,2,2);5 =
Finally, it’s actually possible for a P-type polygon and a Q-type polygon to be
equal in Symn. This can only happen in the case that a and b each have length 2, or at any
length, when either a or b is constant. This only involves polygons with extreme values of
the internal flow label l. Specifically
Lemma 2.4.1. If a and b are each of length 2,
Pna,b;max b = Qna,b;min b
Pna,b;min a+n = Qna,b;max a.
Further, even if a and b have length greater than 2, when a is a constant vector a = −→a
Pn−→a ,b;a+n = Q
−→a ,b;a
and when b is a constant vector b =
in Sym. Otherwise, the P- and Q-polygons are linearly independent in Symn. In particular,
dim(APna,b ∩ AQna,b) is 0, 1 or 2 dimensional, depending on whether neither a nor b are constant,
one is, or either both are or a and b have length 2.
For example
P3(0,0),(1,1);1 = = Q
(0,0),(1,1);1,
P3(0,0),(1,1);2 = ,
P3(0,0),(1,1);3 = = Q
(0,0),(1,1);0.
2.4.1 Rotations
It’s important to point out that the two types of polygons described above are actually
closely related. In fact, by a ‘rotation’, and adding some tags, we can get from one to the
other. This will be important in our later descriptions of the kernel of the representation
functor. Unsurprisingly, this symmetry between the types of polygons is reflected in the
kernel, and we’ll save half the effort in each proof, essentially only having to deal with one
of the two types.
First, by rotl : Zk → Zk we just mean ‘rotate left’:
rotl(a1, a2, . . . , ak) = (a2, . . . , ak, a1),
and by rotr : Zk → Zk ‘rotate right’.
Lemma 2.4.2. Writing the identity out by explicit tensor products and compositions would be
tedious; much easier is to write it diagrammatically first off:
= , (2.4.3)
although keep in mind that we intend the obvious generalisation to arbitrary size polygons, not just
squares.
Proof. It’s just a matter of using isotopy, the definition of the ‘flow vertices’ in §2.3 and
cancelling tags by Equation (2.2.8).
= = .
Notice that it’s actually not important which way the tags point in Equation
(2.4.3); we could reverse them, since by Equation (2.2.9) we’d just pick up an overall sign
of (−1)(n+1)(2Σb−2Σa) = 1. We’ll sometimes write Equation (2.4.3) as
Pnb−n,rotl(a);l = drotl
Qna,b;l
or equivalently, re-indexing
Pna,b;l = drotl
Qnrotr(b),a+n;l
where the operation drotl is implicitly defined by Equation (2.4.3). (You should read drotl
as ‘diagrammatic rotation’, and perhaps have the ‘d’ prefix remind you that we’re not just
rotating, but also adding a tag, called d in the representation theory, to each external edge.)
The corresponding identity writing a Q-polygon in terms of a P-polygon is
Qnrotr(b),a+n;l = drotl
Pna,b;l
Qna,b;l = drotl
Pnb−n,rotl(a);l
Chapter 3
Just enough representation theory
In this chapter, we’ll describe quite a bit of representation theory, but hopefully
only what’s necessary for later! There are no new results in this section, although possi-
bly Proposition 3.5.8, explaining roughly that ‘the fundamental representation theory of
Uq(sln) is generated by triple invariants’ has never really been written down.
3.1 The Lie algebra sln
The Lie algebra sln of traceless, n-by-n complex matrices contains the commutative Cartan
subalgebra h of diagonal matrices, spanned by Hi = Ei,i − Ei+1,i+1 for i = 1, . . . , n − 1.
(Here Ei,j is simply the matrix with a single nonzero entry, a 1 in the (i, j) position.) When
sln acts on a vector space V , the action of h splits V into eigenspaces called weight spaces,
with eigenvalues in h∗. The points
Λ = {λ ∈ h∗ | λ(Hi) ∈ Z} ∼= Zn−1
form the weight lattice. We call h∗+ = {λ ∈ h∗ | λ(Hi) ≥ 0} the positive Weyl chamber, and
the lattice points in it, Λ+ = Λ ∩ h∗+, the dominant weights. The positive Weyl chamber
looks like Rn−1≥0 , and the dominant weights like N
n−1, generated by certain fundamental
weights, λ1, . . . , λn−1, the dual basis to {Hi} ⊂ h.
Under the adjoint action of sln on itself, sln splits into h, as the 0-weight space, and
one dimensional root spaces, each spanned by a root vector Ei 6=j with weight [Hk, Ei,j] =
(δi,k + δj,k+1 − δi,k+1 − δj,k)Ei,j . In particular, the simple roots are defined E+i = Ei,i+1, for
i = 1, . . . , n− 1. Similarly define E−i = Ei+1,i. Together, E
i , E
i and Hi for i = 1, . . . , n− 1
generate sln as a Lie algebra.
The universal enveloping algebra U(sln) is a Hopf algebra generated as an asso-
ciative algebra by symbols E+i , E
i and Hi for i = 1, . . . , n− 1, subject only to the relations
that the commutator of two symbols agrees with the Lie bracket in sln. (We won’t bother
specify the other Hopf algebra structure, the comultiplication, counit or antipode. See the
next section for all the details for quantum sln.) Trivially, sln and U(sln) have the same rep-
resentation theory. We denote the subalgebra generated by Hi and E
i for i = 1, . . . , n − 1
by U± (sln).
3.2 The quantum groups Uq(sln)
We now recall the q-deformation of the Hopf algebra U(sln). Unfortunately we can’t
straightforwardly recover U(sln) by setting q = 1, but we will see in §3.3 that this is the
case at the level of representation theory.
Let A = Q(q) be the field of rational functions in an indeterminate q, with co-
efficients in Q. The quantum group Uq(sln) is a Hopf algebra over A, generated as an
associative algebra by symbols X+i ,X
i ,Ki and K
i , for i = 1, . . . , n − 1, subject to the
relations
i = 1 = K
KiKj = KjKi
j = q
±(i,j)X±j Ki where (i, j) =
2 if i = j,
−1 if |i− j| = 1, or
0 if |i− j| ≥ 2
X+i X
i = δij
Ki −K−1i
q − q−1
X±i X
j = X
i when |i− j| ≥ 2
and the quantum Serre relations
X±j − [2]q X
= 0 when |i− j| = 1.
The Hopf algebra structure is given by the following formulas for the comultiplication ∆,
counit ε and antipode S.
∆(Ki) = Ki ⊗Ki,
∆(X+i ) = X
i ⊗Ki + 1⊗X
∆(X−i ) = X
i ⊗ 1 +K
ε(Ki) = 1 ε(X
i ) = 0,
S(Ki) = K
i S(X
i ) = −X
i S(X
i ) = −KiX−i .
(See, for example, [4, §9.1] for verification that these do indeed fit together to form a Hopf
algebra.) There are Hopf subalgebras Uq(sln)
, leaving out the generators X∓i .
3.3 Representations
The finite-dimensional irreducible representations of both U(sln) and Uq(sln) can be suc-
cinctly catalogued.
A representation V of U(sln) is said to be a highest weight representation if there
is a weight vector v ∈ V such that V = U(sln)− (v). The representation is said to have
weight λ if v has weight λ. As with sln, representations of Uq(sln) split into the eigen-
spaces of the action of the commutative subalgebra generated by the Ki and K
i . These
eigenspaces are again called weight spaces. A representation V of Uq(sln) is called a high
weight representation if it contains some weight vector v so V = Uq(sln)
(v). The finite-
dimensional irreducible representations (henceforth called irreps) are then classified by:
Proposition 3.3.1. Every irrep of U(sln) or of Uq(sln) is a highest weight representation, and there
is precisely one irrep with weight λ for each λ ∈ Λ+, the set of dominant weights.
Proof. See [7, §23] for the classical case, [4, §10.1] for the quantum case.
Further, the finite dimensional representation theories are tensor categories and
the tensor product of two irreps decomposes uniquely as a direct sum of other irreps. The
combinatorics of the the tensor product structures in RepU(sln) and RepUq(sln) agree.
A nice way to say this is that the Grothendieck rings of RepU(sln) and RepUq(sln) are
isomorphic, identifying irreps with the same highest weight [4, §10.1].
3.3.1 Fundamental representations
Amongst the finite dimensional irreducible representations, there are some particularly
simple ones, whose highest weights are the fundamental weights. These are called the
fundamental representations, and there are n− 1 of them for U(sln) or Uq(sln).
We’ll write, in either case, Va<n for the fundamental representation with weight
λa. For U(sln) this is just the representation ∧aCn. It will be quite convenient to agree that
V0<n and Vn<n (poor notation, I admit!) both denote the trivial representation.
We can now define FundRepUq(sln); it’s the full subcategory of RepUq(sln),
whose objects are generated by tensor product and duality from the trivial representa-
tion A, and the fundamental representations Va<n. Note that there are no direct sums in
FundRepUq(sln).
All dominant weights are additively generated by fundamental weights, and this
is reflected in the representation theory; the irrep with high weight λ =
amaλa is con-
tained, with multiplicity one, as a direct summand in the tensor product ⊗aV ⊗maa<n .
This observation explains, to some extent, why it’s satisfactory to simply study
FundRepUq(sln), instead of the full representation theory. Every irrep, while not neces-
sarily allowed as an object of FundRepUq(sln), reappears in the Karoubi envelope [17, 37],
since any irrep appears as a subrepresentation of some tensor product of fundamental rep-
resentations. In fact, there’s a canonical equivalence of categories
Kar (FundRepUq(sln)) ∼= RepUq(sln) .
3.3.2 The Gel‘fand-Tsetlin basis
We’ll now define the Gel‘fand-Tsetlin basis [9, 10], a canonical basis which arises from the
nice multiplicity free splitting rules for Uq(sln) →֒ Uq(sln+1). (Here, and hereafter, this
inclusion is just Ki 7→ Ki,X±i 7→ X
Lemma 3.3.2. If V is an irrep of Uq(sln+1), then as a representation of Uq(sln) each irrep appearing
in V appears exactly once.
More specifically, if V is the Uq(sln+1) irrep with highest weight (λ1, λ2, . . . , λn+1), then
an Uq(sln) irrep W of weight (µ1, µ2, . . . , µn) appears (with multiplicity one) if and only if ‘µ fits
inside λ’, that is
λ1 ≤ µ1 ≤ λ2 ≤ µ2 ≤ · · · ≤ λn ≤ µn ≤ λn+1.
In particular, the fundamental irreps Va<n break up as
Va<n ∼= Va−1<n−1 ⊕ Va<n−1.
Proof. See [4, §14.1.A], [23].
This allows us to inductively define ordered bases for Uq(sln) irreps, at least pro-
jectively. This was first done in the quantum case in [12]. Choose, without much effort,
a basis for the only irreducible representation of Uq(sl1), the trivial representation C(q).
Now for any representation V of Uq(sln+1), decompose V over Uq(sln), as V ∼=
α Wα,
ordered lexicographically by highest weight, and define the Gel‘fand-Tsetlin basis of V to
be the concatenation of the inclusions of the bases for each Wα into V . These inclusions are
unique up to complex multiples, and we thus obtain a canonical projective basis.
We’ll call this forgetful functor the ‘Gel‘fand-Tsetlin’ functor, GT : RepUq(sln) →
RepUq(sln−1). Restricted to the fundamental part of the representation theory (see §3.3.1),
it becomes a functor GT : FundRepUq(sln) → Mat (FundRepUq(sln)).
Next, I’ll describe in gory detail the action of Uq(sln) on each of its fundamental
representations Va<n, using the Gel‘fand-Tsetlin decomposition. We introduce maps p−1
and p0, the Uq(sln−1)-linear projections of Va<n ։ Va−1<n−1 and Va<n ։ Va<n−1. We also
introduce the inclusions i−1 : Va−1<n−1 →֒ Va<n and i0 : Va<n−1 →֒ Va<n.
Proposition 3.3.3. We can describe the action of Uq(sln) on Va<n recursively as follows. On V0<n
and Vn<n, Uq(sln) acts trivially: X
i by 0, and Ki by 1. On the non-trivial representations, we
X+n−1|Va<n = i−1i0p−1p0, (3.3.1)
X−n−1|Va<n = i0i−1p0p−1,
Kn−1|Va<n = i−1(i−1p−1 + qi0p0)p−1 + i0(q
i−1p−1 + i0p0)p0, (3.3.2)
and for Z ∈ Uq(sln−1)
Z |Va<n = i−1Z |Va−1<n−1p−1 + i0Z |Va<n−1p0. (3.3.3)
Remark. Relative to the direct sum decomposition Va<n ∼= (Va−2<n−2 ⊕ Va−1<n−2) ⊕
(Va−1<n−2 ⊕ Va<n−2) under Uq(sln−2), we can write these as matrices, as
X+n−1 =
0 0 0 0
0 0 1 0
0 0 0 0
0 0 0 0
, X−n−1 =
0 0 0 0
0 0 0 0
0 1 0 0
0 0 0 0
Kn−1 =
1 0 0 0
0 q 0 0
0 0 q−1 0
0 0 0 1
, Z =
Z 0 0 0
0 Z 0 0
0 0 Z 0
0 0 0 Z
Proof. We need to check the relations in Uq(sln) involving X
n−1 and Kn−1. We’re induc-
tively assuming the others hold. To begin, the Ki’s automatically commute, as they’re
diagonal with respect to the direct sum decomposition. Further, it’s clear from the defini-
tions that X±n−1 and Kn−1 commute with the subalgebra Uq(sln−2). We then need to check
the relation KiX
j = q
±(i,j)X±j Ki, for |i− j| ≤ 1. Now
Kn−1X
n−1 = (i−1(i−1p−1 + qi0p0)p−1 + i0(q
−1i−1p−1 + i0p0)p0)i−1i0p−1p0
= qi−1i0p−1p0,
while
X+n−1Kn−1 = i−1i0p−1p0(i−1(i−1p−1 + qi0p0)p−1 + i0(q
−1i−1p−1 + i0p0)p0)
= q−1i−1i0p−1p0,
so Kn−1X
n−1 = q
2X+n−1Kn−1 and similarly Kn−1X
n−1 = q
−2X−n−1Kn−1. Further
n−1 = (i−1(i−1(i−1p−1 + qi0p0)p−1 + i0(q
i−1p−1 + i0p0)p0)p−1+
+ i0(i−1(i−1p−1 + qi0p0)p−1 + i0(q
−1i−1p−1 + i0p0)p0)p0)i−1i0p−1p0
= i−1i0(q
i−1p−1 + i0p0)p−1p0,
while
X+n−1Kn−2 = i−1i0p−1p0(i−1(i−1(i−1p−1 + qi0p0)p−1 + i0(q
i−1p−1 + i0p0)p0)p−1+
+ i0(i−1(i−1p−1 + qi0p0)p−1 + i0(q
−1i−1p−1 + i0p0)p0)p0)
= i−1i0(i−1p−1 + qi0p0)p−1p0,
so Kn−2X
n−1 = q
−1X+n−1Kn−2, and by similar calculations Kn−2X
n−1 = qX
n−1Kn−2,
Kn−1X
n−2 = q
−1X+n−2Kn−1 and Kn−1X
n−2 = qX
n−2Kn−1.
Next, we check X+i X
j − X
i = δij
q−q−1
, for i = j = n − 1, and for i =
n− 1, j = n− 2.
X+n−1X
n−1 = i−1i0p−1p0i0i−1p0p−1
= i−1i0p0p−1
X−n−1X
n−1 = i0i−1p0p−1i−1i0p−1p0,
= i0i−1p−1p0
while
Kn−1 −K−1n−1 = i−1(i−1p−1 + qi0p0)p−1 + i0(q−1i−1p−1 + i0p0)p0−
− i−1(i−1p−1 + q−1i0p0)p−1 − i0(qi−1p−1 + i0p0)p0
= (q − q−1)i−1i0p0p−1 + (q−1 − q)i0i−1p−1p0,
so X+n−1X
n−1 −X
n−1 =
Kn−1−K
q−q−1
, as desired, and
X+n−1X
n−2 = i−1i0p−1p0(i−1i0i−1p0p−1p−1 + i0i0i−1p0p−1p0)
= i−1i0p−1i0i−1p0p−1p0 = 0
X−n−2X
n−1 = (i−1i0i−1p0p−1p−1 + i0i0i−1p0p−1p0)i−1i0p−1p0
= i−1i0i−1p0p−1i0p−1p0 = 0
so X+n−1X
n−2 −X
n−2 = 0.
Finally, we don’t need to check the Serre relations. By a result of [16], if you
defined a quantum group without the Serre relations, you’d see them reappear in the ideal
of elements acting by zero on all finite dimensional representations.
Corollary 3.3.4. We’ll collect here some formulas for the generators acting on the dual representa-
tions. Abusing notation somewhat, we write i0 : V
a<n−1 →֒ V ∗a<n, i−1 : V ∗a−1<n−1 →֒ V ∗a<n, p0 :
V ∗a<n ։ V
a<n−1 and p−1 : V
a<n ։ V
a−1<n−1. With these conventions, (i0|Va<n−1)
∗ = p0|V ∗a<n ,
and so on. Then
X+n−1|V ∗a<n = −qi0i−1p0p−1,
X−n−1|V ∗a<n = −q
i−1i0p−1p0,
Kn−1|V ∗a<n = i−1(i−1p−1 + q
i0p0)p−1 + i0(q
−1p−1 + i0p0)p0,
and for Z ∈ Uq(sln−1)
Z |V ∗a<n = i−1Z |V
a−1<n−1
p−1 + i0Z |V ∗
a<n−1
3.4 Strictifying RepUq(sln)
We’ll now make something of a digression into abstract nonsense. (But it’s good abstract
nonsense!) It’s worth understanding at this point that while RepUq(sln) isn’t a strict piv-
otal category, it can be ‘strictified’. That is, the natural isomorphism τ : 1 → ∗∗ isn’t the
identity, but RepUq(sln) is equivalent to another pivotal category in which that ‘pivotal
isomorphism’ τ is the identity. This strictification will be a full subcategory of RepUq(sln)
(as a tensor category), but with a new, modified duality functor.
While this discussion may seem a little esoteric, it actually pays off! Our eventual
goal is a diagrammatic category equivalent to RepUq(sln), with diagrams which are only
defined up to planar isotopy. Such a pivotal category will automatically be strict. The
‘strictification’ we perform in this section bridges part of the inevitable gap between such
a diagrammatic category, and the conventionally defined representation category.
Of course, it would be possible to have defined RepUq(sln) in the first place in
a way which made it strict as a pivotal category, but this would have required a strange
and unmotivated definition of the dual of a morphism. Hopefully in the strictification we
describe here, you’ll see the exact origin of this unfortunate dual.
First of all, let’s explicitly describe the pivotal category structure on RepUq(sln),
in order to justify the statement that it is not strict1. We need to describe the duality functor
∗, pairing maps, and the ‘pivotal’ natural isomorphism τ : 1 → ∗∗.
The contravariant2 duality functor ∗ on RepUq(sln) is given by the usual dual-
ity functor for linear maps. We need to dress up duals of representations of Uq(sln) as
representations again, which we do via the the antipode S, just as we use the comultipli-
cation to turn tensor products of representations into representations. Thus for V some
representation of Uq(sln), Z ∈ Uq(sln), f ∈ V ∗ and v ∈ V , we say (Zf)(v) = f(S(Z)v).
The pairing maps pV : V
∗ ⊗ V → A are just the usual duality pairing maps for
duals of vector spaces. That these are maps of representations follows easily from the Hopf
algebra axioms. Be careful, however, to remember that the other vector space pairing map
V ⊗ V ∗ → A is not generally a map of representations of the Hopf algebra. Similarly, the
copairing cV : A → V ⊗ V ∗ is equivariant, while A → V ∗ ⊗ V is not.
The pivotal isomorphism is where things get interesting, and we make use of the
particular structure of the quantum group Uq(sln).
Definition 3.4.1. We’ll define the element τn = K
j=1 K
j(n−j)
j of Uq(sln).
Lemma 3.4.2. Abusing notation, this gives an isomorphism of Uq(sln) representations, τn : V
V ∗∗, providing the components of the pivotal natural isomorphism.
Proof. This relies on the formulas for the antipode acting on Uq(sln); we need to check
i = S
2(X±i )τn.
Since S2(X±i ) = q
±2X±i , this becomes the condition
n = q
±2X±i ,
1Not strict as a pivotal category, that is. We’ve defined it in such a way that it’s strict as a tensor category,
meaning we don’t bother explicitly reassociating tensor products.
2In fact, it’s doubly contravariant; both with respect to composition, and tensor product.
which follows immediately from the definition of τn and the commutation relations in the
quantum group.
We also need to check that τ satisfies the axioms for a pivotal natural isomor-
phism. Since the Ki are group-like in Uq(sln) (i.e. ∆(Ki) = Ki ⊗Ki), τ is a tensor natural
transformation. Further
τ∗V |V ∗∗∗ ◦ τV ∗ |V ∗ = (Kρ)∗|V ∗∗∗ ◦Kρ|V ∗
= (Kρ)S(Kρ)
so τ∗V and τV ∗ are inverses, as required.
To construct the strictification of RepUq(sln), written as SRepUq(sln), we be-
gin by defining it as a tensor category, taking the full tensor subcategory of RepUq(sln)
whose objects are generated by tensor product (but not duality) from the trivial represen-
tation A, Va<n and V ∗a<n, for each a = 1, . . . , n − 1. Before describing the pivotal struc-
ture on SRepUq(sln), we might as well specify the equivalence; already there’s only one
sensible choice. We can include SRepUq(sln) into RepUq(sln) in one direction. For the
other, we simply send V
a<n (that is, the n-th iterated dual) to Va<n or V
a<n, depending on
whether n is even or odd, and modify each morphism by pre- and post-composing with
the unique3 appropriate tensor product of compositions of τ and τ−1. Clearly the compo-
sition RepUq(sln) → SRepUq(sln) → RepUq(sln) is not the identity functor, but equally
easily it’s naturally isomorphic to the identity, again via appropriate tensor products of
compositions of τ . Here, we’re doing nothing more than identifying objects in a category
related by a certain family of isomorphisms, and pointing out that the result is equivalent
to what we started with!
The interesting aspect comes in the pivotal structure on SRepUq(sln). Of course,
we define the duality cofunctor on objects of SRepUq(sln) so it exchanges Va<n and V ∗a<n.
It turns out (inevitably, according to the strictification result of [2], but explicitly here)
that if we define pairing maps on SRepUq(sln) by taking the image of the pairing maps
for RepUq(sln), we make SRepUq(sln) a strict pivotal category (and of course, we thus
make the equivalence of categories from the previous paragraph an equivalence of pivotal
categories). Explicitly, then, the pairing morphisms on SRepUq(sln) are pSa : V ∗a<n ⊗
Va<n → A = pa and pSa∗ : Va<n ⊗ V ∗a<n → A = pa∗ ◦ (τa ⊗ 1a∗). We’ve yet to define
the duality cofunctor at the level of morphisms in SRepUq(sln); there’s in fact a unique
definition forced on us by Lemma 2.1.1. Thus
f∗ = (1a∗ ⊗ pb∗) ◦ (1a∗ ⊗ (τb ◦ f ◦ τ−1a )⊗ 1b∗)) ◦ (p∗a ⊗ 1b∗).
The processes of strictifying, and of limiting our attention to the fundamental
part of the representation theory, are independent; the discussion above applies exactly to
FundRepUq(sln). In that case, we obtain the full subcategory SFundRepUq(sln).
3It’s unique, given the axioms for τ described in §2.1, and established for our particular τ in Lemma 3.4.2.
3.5 Generators for FundRepUq(sln)
In this section we’ll define certain morphisms in FundRepUq(sln), which we’ll term ‘el-
ementary’ morphisms. Later, in Proposition 3.5.8 we’ll show that they generate all of
FundRepUq(sln) as a tensor category.
We already have the duality pairing and copairing maps, discussed above, pV :
V ∗ ⊗ V → A, and cV : A → V ⊗ V ∗, for each fundamental representation V = Va<n,
and for arbitrary iterated duals V = V
a<n . We also have the non-identity isomorphisms
τ identifying objects with their double duals. Beyond those, we’ll introduce some more
interesting maps:
• a map which identifies a fundamental representations with the dual of another fun-
damental representation, da,n : Va<n
=→ V ∗n−a<n, along with the inverses of these
maps,
• a ‘triple invariant’, living in the tensor product of three fundamental representations
vna,b,c : A → Va<n ⊗ Vb<n ⊗ Vc<n with a+ b+ c = n, and
• a ‘triple coinvariant’, wna,b,c : Va<n ⊗ Vb<n ⊗ Vc<n → A, again with a+ b+ c = n.
Definition 3.5.1. The map da,n : Va<n → V ∗n−a<n is specified recursively by
d0,n : 1 7→ 1∗,
dn,n : 1 7→ 1∗,
da,n = i0da−1,n−1p−1 + (−q)−ai−1da,n−1p0.
Lemma 3.5.2. The map da,n is a map of Uq(sln) representations.
Proof. Since da,n is defined in terms of Uq(sln−1) equivariant maps, we need only check
da,n commutes with X
n−1 and Kn−1. We’ll do one calculation explicitly:
da,nX
n−1 =
i0da−1,n−1p−1 + (−q)−ai−1da,n−1p0
i−1i0p−1p0
= i0da−1,n−1i0p−1p0
i0da−2,n−2p−1 + (−q)1−ai−1da−1,n−2p0
i0p−1p0
= (−q)1−ai0i−1da−1,n−2p−1p0,
while
X+n−1da,n = −qi0i−1p0p−1
i0da−1,n−1p−1 + (−q)−ai−1da,n−1p0
= (−q)1−ai0i−1p0da,n−1p0)
= (−q)1−ai0i−1p0
i0da−1,n−2p−1 + (−q)−ai−1da,n−2p0
= (−q)1−ai0i−1da−1,n−2p−1p0).
The other two cases (commuting with X−n−1 and Kn−1) are pretty much the same.
Proposition 3.5.3. The duals of the maps da,n satisfy:
d∗a,nτn = (−1)(n+1)adn−a,n. (3.5.1)
We need two lemmas before proving this.
Lemma 3.5.4.
j |Va<n = q
n−ai−1p−1 + q
−ai0p0.
Proof. Arguing inductively, and using the formula for Kn−1|Va<n from Equation (3.3.2), we
j |Va<n =
j |Va−1<n−1
 p−1 + i0
j |Va<n−1
Kn−1n−1 |Va<n
i−1(q
n−ai−1p−1 + q
−a+1i0p0)p−1 + i0(q
n−a−1i−1p−1 + q
−ai0p0)p0
i−1(i−1p−1 + q
i0p0)p−1 + i0(q
i−1p−1 + i0p0)p0
= qn−ai−1i−1p−1p−1 + q
n−ai−1i0p0p−1 + q
−ai0i−1p−1p0 + q
−ai0i0p0p0
= qn−ai−1p−1 + q
−ai0p0.
Lemma 3.5.5.
τn|Va<n = q
i−1τn−1|Va−1<n−1p−1 + q
i0τn−1|Va<n−1p0
Proof. Making use of Equation (3.3.3) and Lemma 3.5.4:
τn|Va<n = τn−1|Va<n ×
j |Va<n
i−1τn−1|Va−1<n−1p−1 + i0τn−1|Va<n−1p0
qn−ai−1p−1 + q
−ai0p0
= qn−ai−1τn−1|Va−1<n−1p−1 + q
i0τn−1|Va<n−1p0.
Proof of Proposition 3.5.3. The proposition certainly holds for a = 0 or a = n, where all
three of the maps d∗a,n, τn and dn−a,n are just the identity, and the sign is +1. Otherwise, we
proceed inductively. First, write
d∗a,n = i−1d
a−1,n−1p0 + (−q)−ai0d∗a,n−1p−1.
Using Lemma 3.5.5 we then have
d∗a,nτn =
a−1,n−1p0 + (−q)−ai0d∗a,n−1p−1
qai−1τn−1p−1 + q
i0τn−1p0
= (−q)−aqai0d∗a,n−1τn−1p−1 + qa−ni−1d∗a−1,n−1τn−1p0
= (−1)ai0(−1)nadn−1−a,n−1p−1 + qa−ni−1(−1)n(a−1)dn−a,n−1p0
= (−1)(n+1)ai0dn−1−a,n−1p−1 + qa−n(−1)n(a−1)i−1dn−a,n−1p0,
while
(−1)(n+1)adn−a,n = (−1)(n+1)a(i0dn−a−1,n−1p−1 + (−q)a−ni−1dn−a,n−1p0)
= (−1)(n+1)ai0dn−a−1,n−1p−1 + qa−n(−1)n(a−1)i−1dn−a,n−1p0).
Finally, we need to define the triple invariants vna,b,c and w
a,b,c.
Definition 3.5.6. When a + b + c = m, we define vna,b,c ∈ Va<n ⊗ Vb<n ⊗ Vc<n and wna,b,c ∈
V ∗a<n ⊗ V ∗b<n ⊗ V ∗c<n by the formulas
v00,0,0 = 1⊗ 1⊗ 1,
w00,0,0 = 1
∗ ⊗ 1∗ ⊗ 1∗,
vna,b,c = (−1)cqb+c(i−1 ⊗ i0 ⊗ i0)(vn−1a−1,b,c) + (3.5.2)
(−1)aqc (i0 ⊗ i−1 ⊗ i0)(vn−1a,b−1,c) +
(−1)b (i0 ⊗ i0 ⊗ i−1)(vn−1a,b,c−1),
wna,b,c = (−1)c (wn−1a−1,b,c)(p−1 ⊗ p0 ⊗ p0) + (3.5.3)
(−1)aq−a (wn−1a,b−1,c)(p0 ⊗ p−1 ⊗ p0) +
(−1)bq−a−b(wn−1a,b,c−1)(p0 ⊗ p0 ⊗ p−1).
Lemma 3.5.7. The maps vna,b,c and w
a,b,c are maps of Uq(sln) representations.
Proof. As in Lemma 3.5.2, we just need to check that these maps commute with X±n−1 and
Kn−1. We’ll do the explicit calculation for X
n−1; here we need to check that
X+n−1|Va<b⊗Vb<n⊗Vc<bv
a,b,c = 0
wna,b,cX
n−1|Va<b⊗Vb<n⊗Vc<b
First, we need to know how X+n−1 acts on the tensor product of three representations, via
∆(2)(X+n−1) = X
n−1 ⊗Kn−1 ⊗Kn−1 + 1⊗X
n−1 ⊗Kn−1 + 1⊗ 1⊗X
Next, let’s use two steps of the inductive definition of vna,b,c to write
vna,b,c = q
2b+2c(i−1i−1 ⊗ i0i0 ⊗ i0i0)(vn−2a−2,b,c) +
q2c (i0i0 ⊗ i−1i−1 ⊗ i0i0)(vn−2a,b−2,c) +
(i0i0 ⊗ i0i0 ⊗ i−1i−1)(vn−2a,b,c−2) +
(−1)a+bqc(−i0i0 ⊗ i−1i0 ⊗ i0i−1 + q−1i0i0 ⊗ i0i−1 ⊗ i−1i0)(vn−2a,b−1,c−1) +
(−1)b+cqb+c(i−1i0 ⊗ i0i0 ⊗ i0i−1 − q−1i0i−1 ⊗ i0i0 ⊗ i−1i0)(vn−2a−1,b,c−1) +
(−1)a+cqb+2c(−i−1i0 ⊗ i0i−1 ⊗ i0i0 + q−1i0i−1 ⊗ i−1i0 ⊗ i0i0)(vn−2a−1,b−1,c).
We’ll now show X+n−1 kills each term (meaning each line, as displayed above) separately.
In each case, it follows immediately from the formulas for X+n−1 and Kn−1 in Equations
(3.3.1) and (3.3.2). In the first three terms, we use X+n−1i−1i−1 = X
i0i0 = 0. In the fourth
term, we see
∆(2)(X+n−1)(−i0i0 ⊗ i−1i0 ⊗ i0i−1 + q−1i0i0 ⊗ i0i−1 ⊗ i−1i0)
= −i0i0 ⊗ i−1i0 ⊗ i−1i0 + q−1qi0i0 ⊗ i−1i0 ⊗ i−1i0
Here only one of the three terms of ∆(2)(X+n−1) acts nontrivially on each of the two terms.
The fifth and sixth terms are exactly analogous.
For wna,b,c we use the same trick; write it in terms of w
a−2,b,c, w
a,b−2,c, w
a,b,c−2,
wn−2a,b−1,c−1, w
a−1,b,c−1 and w
a−1,b−1,c, and show that each of these terms multiplied by X
gives zero separately.
Remark. Going through this proof carefully, you’ll see that it would still work with some
variation allowed in constants the definitions of vna,b,c and of w
a,b,c in Equations (3.5.2) and
(3.5.3). However, given the normalisation for da,n that we’ve chosen in Definition 3.5.1,
these normalisation constants are pinned down by Lemmas 4.2.4 and 4.2.6 below.
Later, we’ll discuss relationships between these elementary morphisms, but for
now we want to justify our interest in them.
Proposition 3.5.8. The elementary morphisms generate, via tensor product, composition and lin-
ear combination, all the morphisms in FundRepUq(sln).
Remark. Certainly, they can’t generate all of RepUq(sln), simply because the sources and
targets of elementary morphisms are all in FundRepUq(sln)!
Remark. A fairly abstract proof of this fact for n = 3 has been given by Kuperberg [24].4
Briefly, he specialises to q = 1, and considers the subcategory of FundRepU(sl3) ∼=
FundRepSU(3) generated by the elementary morphisms. He extends this by formally
adding kernels and cokernels of morphisms (that is, by taking the Karoubi envelope). This
extension being (equivalent to) all of RepSU(3) is enough to obtain the result. To see this,
he notes that the extension is the sort of category to which an appropriate Tannaka-Krein
theorem applies, allowing him to say that it is the representation category of some com-
pact Lie group. Some arguments about the symmetries of the ‘triple invariant’ morphisms
allow him to conclude that this group must in fact be SU(3). While I presume this proof
can be extended to cover all n, I prefer to give a more direct proof, based on the quantum
version of Frobenius-Schur duality.
Proof. It’s a sort of bootstrap argument. First, we notice that the action of the braid group
Bm on tensor powers of the standard representation V1<n can be written in terms of ele-
mentary morphisms. Next, we recall that the braid group action generates all the endo-
morphisms of V ⊗m1<n , and finally, we show how to map arbitrary tensor products of fun-
damental representations into a tensor power of the standard representation, using only
elementary morphisms.
4His other results additionally give a direct proof for n = 3, although quite different from the one here.
These ideas are encapsulated in the following four lemmas.
Lemma 3.5.9. The action of the braid group Bm on V
1<n , given by R-matrices, can be written in
terms of elementary morphisms.
Proof. We only need to prove the result for V ⊗21<n. There, we can be particularly lazy, taking
advantage of the fact that dimHomUq(sln)
V ⊗21<n, V
= 2. (There are many paths to see-
ing this, the path of least effort perhaps being that HomUq(sln)
V ⊗21<n, V
is isomorphic to
HomUq(sln)
V1<n ⊗ V ∗1<n, V ∗1<n ⊗ V1<n
, and that both the source and target representations
there decompose into the direct sum of the adjoint representation and the trivial represen-
tation.) The map
p = (wn1,1,n−2 ⊗ 1V1<n ⊗ 1V1<n) ◦ (1V1<n ⊗ 1V1<n ⊗ vnn−2,1,1)
is not a multiple of the identity, so every endomorphism of V ⊗21<n is a linear combination of
compositions of elementary morphisms, in particular the braiding.
Remark. In fact, the map associated to a positive crossing is qn−11
− qnp. It’s easy
to prove that this, along with the negative crossing (obtained by replacing q with q−1),
is the only linear combination of 1 and p which satisfies the braid relation. One could
presumably also check that this agrees with the explicit formulas given in [4, §8.3 and
§10.1]5.
Lemma 3.5.10. The image of Bm linearly spans HomUq(sln)
V ⊗m1<n , V
Proof. This is the quantum version of Schur-Weyl duality. See [4, §10.2B] and [12].
Lemma 3.5.11. For any tensor product of fundamental representations
i Vαi , there’s some nat-
ural number m and a pair of morphisms constructed out of elementary ones ι :
i Vαi → V
and π : V ⊗m1 →
i Vαi such that π ◦ ι = 1Ni Vαi .
Proof. We just need to do this for a single fundamental representation, then tensor together
those morphisms. For a single fundamental representation, first define ι′ : Va → V1 ⊗
Va−1 = (pVn−a ⊗ 1V1 ⊗ 1Va−1) ◦ (da ⊗ vnn−a,1,a−1) and π′ : V1 ⊗ Va−1 → Va = x(wn1,a−1,n−a ⊗
da) ◦ (1V1 ⊗ 1Va−1 ⊗ cVn−a). The composition π′ ◦ ι′ is an endomorphism of the irreducible
Va, which I claim is nonzero, and so for some choice of the coefficient x is the identity. Now
build the maps ι : Va → V ⊗a1 and π : V
→ Va as iterated compositions of these maps.
Lemma 3.5.12. In fact, given two such tensor products of fundamental representations
i Vαi
j Vβj , such that there is some nonzero Uq(sln) map between them, it’s possible to choose
ια, πα, ιβ and πβ as in the previous lemma, with the same value of m.
5Although be careful there — the formula for the universal R-matrix in §8.3.C is incorrect, although it
shouldn’t matter for such a small representation. The order of the product is backwards [5].
Proof. (This argument is due to Ben Webster. Thanks!) First just pick ια, πα, ι
β and π
as in the previous lemma, with possibly different values of m, say mα and mβ . Now,
if there’s some Uq(sln) map between
i Vαi and
j Vβj , then there’s also a Uq(sln) map
between V ⊗mα1 and V
1 . This can happen if and only if there’s some sln map, and in fact
some SL(n) map, between the corresponding classical representations,Cn⊗mα and Cn⊗mβ .
This SL(n) intertwiner guarantees that the center of SL(n) acts in the same way on each
representation; the element
1I acts by 1mα/n and 1mβ/n on the two representations, so
mα and mβ must be congruent mod n. Assuming, without loss of generality, that mβ ≤ mα
we now define ιβ = ι
β ⊗ ι⊗(mα−mβ)/n, and πβ = π′β ⊗ π⊗(mα−mβ)/n, where ι : A → V
and π : V ⊗n1 → A are diagrammatic morphisms satisfying π ◦ ι = 1A. (These certainly
exist!)
Putting all this together, we take an arbitrary morphism φ :
i Vαi →
j Vβj in
FundRepUq(sln), and pick ια, πα, ιβ and πβ as in the last lemma. Then
φ = πβ ◦ ιβ ◦ φ ◦ πα ◦ ια.
The middle of this composition, ιβ ◦ φ ◦πα, is an endomorphism of V ⊗m1 , so can be written
as a linear combination of braids, and hence as a linear combination of morphisms built
from elementary morphisms.
3.6 The representation functor
Finally, everything is in place for a definition of the representation functor Rep : Symn →
SFundRepUq(sln). (Notice that we’re defining the functor with target the strictification
SFundRepUq(sln), as discussed in §3.4. To obtain the functor to the more usual represen-
tation theory FundRepUq(sln), you need to compose with the equivalences of categories
described in that section; of course, this is just the inclusion of a full subcategory!) First
define Rep′ on the free tensor category Tn, by
′(a ∈ {1, . . . , n− 1}) = Va<n, Rep′(a∗) = V ∗a<n,
on objects, and
= pa, Rep
= pa∗ ◦ (τn ⊗ 1),
= ca, Rep
= (Id⊗ τ−1n ) ◦ ca∗ ,
= da,n, Rep
= d−1a,n,
= (−1)(n+1)ada,n, Rep′
= (−1)(n+1)ad−1a,n,
 = vna,b,c Rep
 = wna,b,c
on morphisms.
Proposition 3.6.1. The functor Rep′ descends to a functor defined on the quotient Symn →
SFundRepUq(sln).
This is proved in the next chapter; we could do it now, but the proof will read
more nicely once we have better diagrams available.
Chapter 4
The diagrammatic Gel‘fand-Tsetlin
functor
It’s now time to define the diagrammatic Gel‘fand-Tsetlin functor,
dGT : Symn → Mat
Symn−1
The first incarnation of the diagrammatic functor will be a functor defined (in §4.1) on the
free version of the diagrammatic category, dGT ′ : Tn → Mat
Symn−1
. We’ll need to
show that it descends to the quotient Symn (in §4.2). This definition is made so that the
perimeter of the diagram
dGT ′
// SFundRepUq(sln)
Symn−1
) Mat(Rep)
// Mat (SFundRepUq(sln−1))
(4.0.1)
commutes.
Happily, as an easy consequence of this, we’ll see that the representation functor
Rep : Tn → SFundRepUq(sln) also descends to the quotient Symn.
At the end of the chapter in §4.4 we describe how to compute the diagrammatic
Gel‘fand-Tsetlin functor, and perform a few small calculations which we’ll need later.
4.1 Definition on generators
The target category for the diagrammatic Gel‘fand-Tsetlin functor is the matrix category
over Symn−1. We thus need to send each object of Tn to a direct sum of objects of Symn−1,
and for each generating morphism of Tn, we need to pick an appropriate matrix of mor-
phisms in Symn−1. On objects, we use the obvious
dGT ′(a) = (a− 1)⊕ a (4.1.1)
dGT ′(a∗) = (a− 1)∗ ⊕ a∗
(omitting the nonsensical direct summand in the case that a = 0, 0∗, n or n∗), extending
to tensor products by distributing over direct sum. For morphisms we’ll take advantage
of the lack of ‘multiplicities’ in Equation 4.1.1 to save on some notation. An arbitrary
morphism in Mat
Symn−1
has rows and columns indexed by tensor products of the fun-
damental (and trivial) objects and their duals, 0, 1, . . . , n − 1, 0∗, 1∗, . . . , (n − 1)∗. Notice,
however, that morphisms in the image of dGT ′ have distinct labels on each row (and also
on each column). Moreover, a (diagrammatic) morphism in Symn−1 explicitly encodes its
own source and target (reading across the top and bottom boundary points). We’ll thus
abuse notation, and write a sum of matrix entries, instead of the actual matrix, safe in the
knowledge that we can unambiguously reconstruct the matrix, working out which term
should sit in each matrix entry. For example, if f : a → b is a morphism in Tn, then
dGT ′(f) is a matrix (a − 1) ⊕ a → (b − 1) ⊕ b, which we ought to write as
f11 f12
f21 f22
, but
will simply write as f11 + f12 + f21 + f22. This notational abuse is simply for the sake of
brevity when writing down matrices with many zero entries. When composing matrices
written this way, you simply distribute the composition over summation, ignoring any
non-composable terms.
That said, we now define dGT ′ on trivalent vertices by
dGT ′
 = (−1)cqb+c + (−1)aqc + (−1)b ,
dGT ′
 = (−1)c + (−1)aq−a + (−1)bq−a−b
(the right hand sides of these equations are secretly 8× 1 and 1× 8 matrices, respectively),
on the cups and caps by
dGT ′
= + dGT ′
= qn−a + q−a
dGT ′
= + dGT ′
= qa−n + qa ,
(4.1.2)
and on the tags by
dGT ′
= + (−1)aq−a
dGT ′
= (−1)n+a + q−a
dGT ′
= qa + (−1)n+a
dGT ′
= (−1)aqa + . (4.1.3)
The functor dGT ′ then extends to all of Tn as a tensor functor.
Proposition 4.1.1. The outer square of Equation (4.0.1) commutes.
Proof. Straight from the definitions; compare the definition above with the definition of
Rep′ from §3.6, and use the inductive Definitions 3.5.1 and 3.5.6 of the generating mor-
phisms over in the representation theory.
4.2 Descent to the quotient
Proposition 4.2.1. The functor dGT ′ descends from Tn to the quotient Symn, where we call it
simply dGT .
Proof. Read the following Lemmas 4.2.2, 4.2.3, 4.2.4, 4.2.5 and 4.2.6.
Lemma 4.2.2. We first check the relation in Equation (2.2.4).
dGT ′
 = dGT ′
= dGT ′
dGT ′
 = dGT ′
= dGT ′
Proof. This is direct from the definitions in Equation (4.1.2), and using the fact that we can
straighten strands in the target category Symn−1.
Lemma 4.2.3. Next, the relations in Equation (2.2.5).
dGT ′
= dGT ′
dGT ′
= dGT ′
Proof.
dGT ′
= qn−a + (−1)a
= qn−a + (−1)a
= dGT ′
making use of the definitions in Equations (4.1.2) and (4.1.3), and the identity appearing in
Equation (2.2.5) for Symn−1, and
dGT ′
= + (−1)aq−a
= + (−1)aq−a
= dGT ′
Lemma 4.2.4. We check the ‘rotation relation’ from Equation (2.2.7) is in the kernel of dGT ′, which
of course implies the ‘2π rotation relation from Equations (2.2.3) is also in the kernel.
dGT ′
(4.2.1)
= (−1)c + (−1)aq−a + (−1)bq−a−b
= dGT ′
dGT ′
(4.2.2)
= (−1)cqb+c + (−1)aqc + (−1)b
= dGT ′
Proof. The statement of the lemma essentially contains the proof; we’ll spell out Equation
(4.2.1) in gory detail:
dGT ′
= dGT ′
1c∗ ⊗ dGT ′
⊗ 1c
dGT ′
⊗ 1a⊗b⊗c
1c∗ ⊗
(−1)b + (−1)cq−c +
+ (−1)aq−c−a
qc−n + qc
⊗ 1a⊗b⊗c
= (−1)c + (−1)aq−a + (−1)bq−a−b
= dGT ′
Going from the third last to the second last line, we simply throw out all non-composable
cross terms.
The next two lemmas are similarly direct.
Lemma 4.2.5.
dGT ′
= + (−1)aq−a = (−1)(n+1)adGT ′
dGT ′
= qa + (−1)n+a = (−1)(n+1)adGT ′
Lemma 4.2.6.
dGT ′
 = dGT ′
dGT ′
 = dGT ′
We can now give the
Proof of Proposition 3.6.1. Somewhat surprisingly, the fact that dGT ′ descends from Tn to
Symn implies that Rep′ : Tn → SFundRepUq(sln) also descends to Symn, simply because
the outer square of Equation (4.0.1) commutes, as pointed out in Proposition 4.1.1.
4.3 Calculations on small webs
We’ll begin with some calculations of dGT on the flow vertices introduced in §2.3.
 = (−1)a
 + (−1)n+b + q−a
 = (−1)a
+ (−1)n+bqb +
= (−1)a
+ (−1)n+bqb +
= (−1)a
qa + + (−1)n+bqa−n
We’ll also need the following formulas for two vertex webs:
 = +
+ q−a + +
+ q−a + (−1)b+cqb (4.3.1)
= qb +
+ + +
+ qb + (−1)a+cqb+c−n (4.3.2)
4.4 A path model, and polygons.
Call a diagram in Symn in which all vertices are flow vertices and there are no tags (except
those implicitly hidden inside the vertices) a flow diagram. In particular, the flow vertices
themselves, and the P- and Q-polygons described in §2.4 are flow diagrams. A reduction
path on a flow diagram is some disjoint union of cycles and paths beginning and ending
at the boundary of the diagram, always matching the orientations on the edges of the
diagram. We then notice that if v is a flow vertex, the three terms of dGT (v) correspond
to the three reduction paths on the vertex. Specifically, for each reduction path, in the
corresponding term of dGT (v) the label on each traversed edge has been reduced by one.
Each term additionally has a coefficient ±qk for some integer k.
It’s easy to see that this condition must also hold for larger flow diagrams. For a
reduction path π on a diagram D ∈ Symn, write π(D) for the diagram in Symn−1 obtained
from D by reducing the label on each edge traversed by π. Further, write t(π) for the subset
of the tensor factors in the target of D which π reaches, and s(π) for the subset of the tensor
factors in the source.
Proposition 4.4.1. The result of applying dGT to a flow diagram D is a sum over the reduction
paths on D:
dGT (D) =
reduction
paths π
±q•π(D). (4.4.1)
Further, two terms in this sum lie in same matrix entry (remember, dGT is a functor Symn →
Symn−1
) iff the boundaries of the two reduction paths agree.
Proof. Certainly this is true for the generators which can appear in a flow diagram; caps,
cups, and flow vertices. The condition is also preserved under tensor product and compo-
sition.
To see this, suppose we have two flow diagrams D1 and D2 in Symn, with fami-
lies of reduction paths P1 and P2, so that dGT (Di) =
±q•π(Di). Now the reduction
paths P for D1 ⊗ D2 are exactly those of the form π1 ∪ π2 for some π1 ∈ P1, π2 ∈ P2
(remember tensor product of diagrams is side-by-side disjoint union). Thus
dGT (D1 ⊗D2) = dGT (D1)⊗ dGT (D2)
π1∈P1
(−1)•q•π1(D1)
π2∈P2
(−1)•q•π2(D2)
π1∈P1
π2∈P2
(−1)•q•(π1 ∪ π2)(D1 ⊗D2)
(−1)•q•π(D1 ⊗D2).
The argument for compositions is a tiny bit more complicated. Supposing D1 and D2 are
composable, the reduction paths P for D1 ◦ D2 are those of the form π1 ∪ π2, for some
π1 ∈ P1, π2 ∈ P2 such that s(π1) = t(π2). Thus
dGT (D1 ◦D2) = dGT (D1) ◦ dGT (D2)
π1∈P1
±q•π1(D1)
π2∈P2
±q•π2(D2)
π1∈P1
π2∈P2
±q•δs(π1),t(π2)(π1 ∪ π2)(D1 ◦D2)
±q•π(D1 ⊗D2).
We’ll now calculate dGT on the P- and Q-polygons. On a P-polygon, there are
two reduction paths which do not traverse any boundary points; the empty reduction
path, and the path around the perimeter of the polygon. We’ll call these πP;∅ and πP;◦.
Otherwise, there is a unique reduction path for each (cyclic) subsequence of boundary
points which alternately includes incoming and outgoing edges. We’ll write πP;s for this,
where s is the subset of {1, . . . , 2k} corresponding to the points on boundary traversed
by the path components. The reduction path has one component for each such pair of
incoming and outgoing edges, and traverses the perimeter of the polygon counterclock-
wise between them. On a Q-polygon, there are two reductions paths which traverse every
boundary point; in one, each component enters at an incoming edge, and traverse coun-
terclockwise, departing at the nearest outgoing edge, while in the other each component
traverses clockwise. We’ll call these πQ;	 and πQ;�. Otherwise, there is a unique reduction
path for each disjoint collection of adjacent pairs of boundary edges; the path connects
each pair, traversing a single edge of the polygon between them, and again, we’ll write
πQ;s, where s is the subset of {1, . . . , 2k} corresponding to the subset of the boundary con-
sisting of the union of all the pairs. All these statements about reduction paths should
be immediately obvious, looking at the definition of the P- and Q-polygons in Equations
(2.4.1) and (2.4.2).
πP;∅(Pna,b;l) = Pn−1a,b;l
πP;◦(Pna,b;l) = Pn−1a+−→1 ,b+−→1 ;l = P
a,b;l−1.
Further
πP;s(Pna,b;l) = Pn−1a+a′,b+b′;l,
where a′, b′ ∈ {0, 1}k is determined from s by 2i − 1 ∈ s iff a′i = 1 and b′i = 0, and 2i ∈ s
iff a′i+1 = 1 and b
i = 0. The condition that s contains alternately incoming and outgoing
edges translates to b′i ≤ a′i, a′i+1; we’ll call such pairs a′, b′ P-admissible. Note that πP;∅ and
πP;◦ correspond to two exceptional pairs, a
′ = b′ =
0 and a′ = b′ =
1 respectively.
On the Q-polygons, we have πQ;	(Qna,b;l) = Qna,b+−→−1;l = Q
1 ,b;l+1
, πQ;�(Qna,b;l) =
1 ,b;l
and πQ;s(Qna,b;l) = Q
a+a′,b+b′;l
, where a′ ∈ {0, 1, }k and b′ ∈ {−1, 0}k are deter-
mined by 2i − 1 ∈ s iff ai = 1 or bi = −1, and 2i ∈ s iff ai+1 = 1 or bi = −1. The
condition that the points in s come in disjoint adjacent pairs translates to the condition
i = 0 = a
i; such pairs a
′, b′ are called Q-admissible. Note that πQ;	 and πQ;� cor-
respond to two exceptional pairs, a′ =
0 , b′ =
−→−1 and a′ = −→1 , b′ = −→0 respectively. Also,
note that a pair a′, b′ is both P- and Q-admissible exactly if b′ = −→0 ; any a′ ∈ {0, 1}k is
allowed.
Thus we have
Pna,b;l
a′,b′ P-admissible
dGT a′,b′
Pna,b;l
Qna,b;l
a′,b′ Q-admissible
dGT a′,b′
Qna,b;l
where
dGT a′,b′
Pna,b;l
= (−1)b′·a+rotl(a)ql(Σb′−Σa′+1)qrotl(a′)·b−b′·a−a1−na′1Pn−1
a+a′,b+b′;l
, (4.4.2)
dGT a′,b′
Qna,b;l
= (−1)b′·a+rotl(a)ql(Σa′−Σb′−
+1)qΣb−nΣb
′+b′·rotl(a)−a′·b−a1−na
1Qn−1a+a′,b+b′;l.
(4.4.3)
The coefficients here are products of the coefficients appearing in the formulas for dGT on
the two vertex webs in Equations (4.3.1) and (4.3.2). We’ll often also use the notation dGT ∅
for the terms corresponding to reduction paths not traversing any boundary edges, and
dGT ∂ for the terms corresponding to reduction paths traversing all boundary edges. Thus
dGT ∅
Pna,b;l
= dGT −→
Pna,b;l
+ dGT −→
Pna,b;l
= ql−a1Pn−1a,b;l + q
l−a1+Σb−Σa−nPn−1a,b;l−1, (4.4.4)
dGT ∅
Qna,b;l
= dGT −→
Qna,b;l
= ql−a1qΣb−
2 Qn−1a,b;l , (4.4.5)
dGT ∂
Pna,b;l
= dGT −→
Pna,b;l
= ql−a1−nqΣb−
2 Pn−1
1 ,b;l
, (4.4.6)
dGT ∂
Qna,b;l
= dGT −→
Qna,b;l
+ dGT −→
Qna,b;l
= ql−a1+Σb−Σa−
2 Qn−1
1 ,b;l+1
+ ql−a1−nQn−1
1 ,b;l
. (4.4.7)
Chapter 5
Describing the kernel
Finally, in this chapter we’ll use the methods developed to this point to try to pin
down the kernel of the representation functor. In particular, Theorems 5.1.1, 5.2.1 and 5.3.2,
stated in the three subsequent sections, describe particular classes of relations amongst
diagrams. Theorems 5.2.1 and 5.3.2 additionally rule out any other similar relations.
5.1 The I = H relations
Theorem 5.1.1. For each a, b, c ∈ {0, . . . , n} with a+ b+ c ≤ n, there’s an identity
= (−1)(n+1)a , (5.1.1)
and another
= (−1)(n+1)a . (5.1.2)
The proof appears in §5.7. Remark. It’s somewhat unfortunate that there’s a sign
here in the first place, and that it depends unsymmetrically on the parameters. (See also
§6.1 and §6.3, for a discussion of sign differences between my setup and previous work.)
Replacing the vertices in Equation (5.1.1) with standard ones, we can equivalently write
this identity in the following four ways (obtaining each by flipping tags, per Equation
(2.2.9)):
= (−1)(n+1)a = (−1)(n+1)b (5.1.3)
= (−1)(n+1)c = (−1)(n+1)d ,
where here a+ b+ c+ d = n. These signs are special cases of the 6− j symbols for Uq(sln).
5.2 The square-switch relations
We can now state the ‘square switch’ relations, which essentially say that every P-type
square can be written as a linear combination of Q-type squares, as vice versa. (Refer back
to §2.4 for the definitions of P- and Q-polygons.)
Theorem 5.2.1. The subspace (SS for ‘square switch’)
SSna,b =
spanA
Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l − Σb
Qna,b;m
}mina+n
l=max b
if n+Σa− Σb ≥ 0
spanA
Qna,b;l −
n+mina
m=max b
Σb− n− Σa
m+ l − Σa− n
Pna,b;m
}min b
l=maxa
if n+Σa− Σb ≤ 0
of APna,b +AQna,b ⊂ HomSymn(∅,L(a, b)) lies in the kernel of the representation functor.
The proof appears in §5.7. Notice that we’re stating this result for all tuples a and
b, not just those for which SS is non-trivial. This is a minor point, except that it makes the
base cases of our inductive proof easy.
Remark. In the case n+Σa− Σb = 0, this becomes simply
SSna,b = spanA
Pna,b;l −Qna,b;Σb−l
}min a+n
l=max b
Remark. For n + Σa − Σb ≤ 0, at the maximal or minimal values of l, we simply get
Pna,b;max b − Qna,b;min b (since the only terms for which the q-binomial is nonzero are m ≥
min b) and Pna,b;mina+n − Qna,b;max a (again, the q-binomials vanish unless m ≤ max a). We
already knew these were identically zero in Symn, by the discussion at the end of §2.4 on
relations between P- and Q-polygons. A similar statement applies when n+Σa−Σb ≤ 0.
Loops, and let bigons be bygones
Over on the representation theory side, any loops or bigon must become a multiple of the
identity, by Schur’s lemma. We can see this as a special case of the SS relations. (Alterna-
tively, we’d be able to derive the same formulas from Theorem 5.3.2 in the next section.)
Consider the boundary flow labels a = (0, 0), b = (k, 0), so
L(a, b) = ((0,−), (0,+), (k,−), (k,+)).
Then APna,b = spanA
Pna,b;l
, while AQna,b = spanA
Qna,b;0
. By 5.2.1, we have
SSna,b = spanA
Pna,b;l −
l − k
Qna,b;0
Diagrammatically, this says that in the quotient by kerRep,
l − k
or, removing trivial edges, and cancelling tags
l − k
The special case k = 0 evaluates loops:
5.3 The Kekulé relations
Friedrich August Kekulé is generally thought to have been the first to suggest the cyclic
structure of the benzene molecule [38]. A simplistic description of the benzene molecule is
as the quantum superposition of two mesomers, each with alternating double bonds
The similarity to the relation between the two hexagonal diagrams in Sym4/ ker,
+ = + (5.3.1)
discovered by Kim [21], prompted Kuperberg to suggest the name ‘Kekulé relation’, even
though in my classification all of the relations for Uq(sl2) and Uq(sl3) are also of this type!
In the following, we’ll use the convenient notation
x = Σx − max x,
Σx−minx. (With the convention that
∅ = 0.)
Definition 5.3.1. We’ll say a set of flow labels (a, b) is n-hexagonal if there’s a sequence of
six elements of L(a, b), each non-trivial for n, i.e. between 1 and n− 1 inclusive, which are
alternately incoming and outgoing edges.
Remark. It’s not enough to simply say there are 6 non-trivial edges. For example, the
boundary labels
((5,+), (1,−), (0,+), (1,−), (0,+), (1,−), (0,+), (1,−), (0,+), (1,−), (0,+), (1,−))
seem to allow hexagonal webs, but modulo the I = H relations, the P-polygons are all
actually bigons in disguise.
Theorem 5.3.2. The subspaces APRna,b ⊂ APna,b ⊂ HomSymn(∅,L(a, b)) and AQRna,b ⊂
AQna,b ⊂ HomSymn(∅,L(a, b)) defined by
APRna,b = spanA
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
Pna,b;j+k
Σa+n−1
(5.3.2)
AQRna,b = spanA
(−1)j+k
j + k −max a
j − Σa
min b− j − k
Σb− n( |∂|
− 1)− 1− j
Qna,b;j+k
Σb−n(
−1)−1
(5.3.3)
are in the kernel of the representation functor. Moreover, when the representation functor is re-
stricted to APna,b, its kernel is exactly APRna,b, and similarly for AQna,b and AQRna,b. Further, in
the case that (a, b) is n-hexagonal, when the representation functor is restricted to APna,b +AQna,b,
its kernel is exactly APRna,b +AQRna,b.
The proof appears in §5.7.
Remark. It’s worth being a little careful here; remember from Lemma 2.4.1 that sometimes
APna,b∩AQna,b 6= 0; when a or b is constant, the intersection is 1 dimensional, and when both
are constant, but not equal, it’s 2 dimensional. Besides this, however, you can understand
the second half of the above Theorem as saying that there are no relations between P-type
polygons, and Q-type polygons, when those polygons are at least hexagons. In particular,
this contradicts the conjecture of [11], which expected to find relations looking like the
‘square-switch’ relations described in Theorem 5.2.1 amongst larger polygons.
The elements of APRna,b described above have ‘breadth’
a + 2; that is,
they have that many terms. We can generalise the definition of n-hexagonal to say that a
pair (a, b) has n-circumference at least 2k if there’s a sequence of 2k edges in L(a, b), which
are nontrivial for n and alternately incoming and outgoing. Then
Lemma 5.3.3. If (a, b) has n-circumference at least 2k, the breadth of the relations in APRna,b is
at least k + 1. In particular, the only place that relations of breadth 2 appear is when (a, b) has
n-circumference no more than 2.
The proof appears in §5.7, after Lemma 5.7.7.
5.4 More about squares
Theorem 5.4.1. Further, when n+ Σa− Σb ≥ 0, the space AQRna,b of relations amongst the Q-
squares becomes 0, and in fact APRna,b ⊂ SSna,b. (Conversely, when n+Σa−Σb ≤ 0, APRna,b = 0
and AQRna,b ⊂ SSna,b.) When n+Σa− Σb ≥ 0, defining
SS ′na,b = spanA
Qna,b;m −
n+mina
l=n+Σa−m
(−1)m+l+n+Σa
m+ l − 1− Σb
m+ l − n− Σa
Pna,b;l
}min b
m=max a
we find that SSna,b = APR
a,b ⊕ SS′na,b. When n+Σa− Σb ≤ 0, we define instead
SS ′na,b = spanA
Pna,b;m −
min b
l=Σb−m
(−1)m+l+Σb
m+ l − n− 1− Σa
m+ l − Σb
Qna,b;l
}min a+n
m=max b
and find that SSna,b = AQRna,b ⊕ SS′na,b.
Corollary 5.4.2. The kernel of Rep on APna,b +AQna,b, in the case of squares, is exactly SSna,b.
Proof. This follows from Theorems 5.3.2 and 5.4.1. Suppose x ∈ kerRepn ∩(APna,b +
AQna,b), we’ll show that it’s zero as an element of (APna,b +AQna,b)/SSna,b. In fact, (AP
a,b +
AQna,b)/SSna,b = AP
a,b/SS
a,b = AP
a,b/APRna,b, by Theorem 5.4.1, and by Theorem 5.3.2,
(kerRepn ∩APna,b)/APRna,b = 0.
5.5 A conjecture
I wish I could prove the following
Conjecture. The kernel of the representation functor is generated, as a pivotal category ideal, by
the elements described in Theorems 5.1.1, 5.2.1 and 5.3.2.
Evidence. First of all, we have Corollary 5.4.2, which confirms that the conjecture holds for
diagrams with at most four boundary points.
Second, Theorem 5.3.2 tells us more than simply that certain elements are in the
kernel; it tells us exactly the kernel of the representation functor when restricted to dia-
grams with a single (hexagonal or bigger) polygonal face. Any relations not generated by
those discovered here must then be ‘more nonlocal’, in the sense that they involve dia-
grams with multiple polygons.
Finally, there’s some amount of computer evidence. I have a Mathematica pack-
age, sadly unreleased at this point, which can explicitly produce intertwiners between rep-
resentations of arbitrary quantum groups Uqg. A program which can translate diagrams
into instructions for taking tensor products and compositions of elementary morphisms
(i.e. an implementation of the representation functor described in §3.6) can then auto-
matically look for linear dependencies amongst diagrams. For small values of n (up to 8
or 9), I used such a program to look for relations involving two adjacent hexagons, but
found nothing besides relations associated, via Theorem 5.3.2, to one of the individual
hexagons.
5.6 Examples: Uq(sln), for n = 2, 3, 4 and 5.
For each n, we can enumerate the finite list of examples of each of the three families of re-
lations. To enumerate APRn relations, we find each set of flow labels (a, b) such that every
element of L(a, b) is between 1 and n − 1 inclusive, and dimAPRna,b = n + Σa − Σb > 0.
When sets of flow labels (a, b) and (rotlk(a), rotlk(b)) differ by a cyclic permutation, we’ll
only discuss the lexicographically smaller one. There’s no need to separately enumerate
the AQRn relations; they’re just rotations of the APRn relations. To enumerate SSn re-
lations, we’ll do this same, but weakening the inequality to n + Σa − Σb ≥ 0. Recall that
we don’t need to look at the extreme values of l in the SS relations in Theorem 5.2.1; those
relations automatically hold already in Symn.
For n = 2 and n = 3, there are no APRna,b relations with a, b of length 3 or more;
the relations of length 2 or less follow from the SSn relations. We recover exactly the
Temperley-Lieb loop relation for n = 2, and Kuperberg’s loop, bigon and square relations
for n = 3, from SS3(0,0),(0,0),SS
(0,0),(0,1) and SS
(0,0),(1,1) respectively.
For n = 4, we have non-trivial APR4a,b spaces for
(a, b) =
(∅, ∅) when k = 0
((0), (1)), ((0), (2)), or ((0), (3)) when k = 1
((0, 0), (1, 1)), ((0, 0), (1, 2)), or ((0, 0), (2, 2)) when k = 2
and for k = 3
(a, b) = ((0, 0, 0), (1, 1, 1)).
Thus we have relations involving loops:
APR4∅,∅ = spanA
[4]q P
∅,∅;0 − P
∅,∅;1,− [3]q P
∅,∅;1 + [2]q P
∅,∅;2,
[2]q P
∅,∅;2 − [3]q P
∅,∅;3,−P
∅,∅;3 + [4]q P
∅,∅;4
= spanA
[4]q − ,− [3]q + [2]q ,
[2]q − [3]q ,− + [4]q
, (5.6.1)
and bigons:
APR4(0),(1) = spanA
− [3]q P
(0),(1);1 + P
(0),(1);2, [2]q P
(0),(1);2 − [2]q P
(0),(1);3,
−P4(0),(1);3 + [3]q P
(0),(1);4
= spanA
− [3]q + , [2]q − [2]q
, (5.6.2)
APR4(0),(2) = spanA
[2]q P
(0),(2);2 − P
(0),(2);3,−P
(0),(2);3 + [2]q P
(0),(2);4
= spanA
[2]q −
, (5.6.3)
APR4(0),(3) = spanA
−P4(0),(3);3 + P
(0),(3);4
= 0, (5.6.4)
(there’s a redundancy in each APR4
(0),(k)
space, since P4
(0),(k);k
(0),(k);4
) and then rela-
tions involving squares:
APR4(0,0),(1,1) = spanA
− [3]q P
(0,0),(1,1);1 + [2]q P
(0,0),(1,1);2 − P
(0,0),(1,1);3,
P4(0,0),(1,1);2 − [2]q P
(0,0),(1,1);3 + [3]q P
(0,0),(1,1);4
= spanA
− [3]q + [2]q − ,
− [2]q + [3]q
, (5.6.5)
APR4(0,0),(1,2) = spanA
P4(0,0),(1,2);2 − P
(0,0),(1,2);3 + P
(0,0),(1,2);4
= spanA
, (5.6.6)
APR4(0,1),(2,2) = spanA
P4(0,1),(2,2);2 − P
(0,1),(2,2);3 + P
(0,1),(2,2);4
= spanA
, (5.6.7)
and finally the eponymous Kekulé relation, involving hexagons:
APR4(0,0,0),(1,1,1) = spanA
−P4(0,0,0),(1,1,1);1 + P
(0,0,0),(1,1,1);2−
−P4(0,0,0),(1,1,1);3 + P
(0,0,0),(1,1,1);4
= spanA
− + − +
(5.6.8)
Here we’re telling some small lies; APR4a,b is always a subspace of HomSym4(∅,L(a, b)),
but I’ve drawn some of these diagrams as elements of other Hom spaces, via rotations.
The interesting SS4 spaces are
SS4(0,0),(1,1) = spanA
P4(0,0),(1,1);2 −Q
(0,0),(1,1);0 − [2]q Q
(0,0),(1,1);1,
P4(0,0),(1,1);3 − [2]q Q
(0,0),(1,1);0 −Q
(0,0),(1,1);1
= spanA
− − [2]q ,
− [2]q −
SS4(0,0),(1,2) = spanA
P4(0,0),(1,2);3 −Q
(0,0),(1,2);0 −Q
(0,0),(1,2);1
= spanA
SS4(0,0),(2,2) = spanA
P4(0,0),(2,2);3 −Q
(0,0),(2,2);1
= spanA
SS4(0,1),(2,2) = spanA
P4(0,1),(2,2);3 −Q
(0,1),(2,2);1 −Q
(0,1),(2,2);2
= spanA
Notice the redundancies here: SS4(0,0),(1,1) = APR
(0,0),(1,1), SS
(0,0),(1,2) = APR
(0,0),(1,2)
and SS4(0,1),(2,2) = APR
(0,1),(2,2), but SS
(0,0),(2,2) is independent of any of the APR rela-
tions.
For n = 5, we’ll just look at non-trivial APR5a,b spaces where a and b are each of
length at least 3; we know what the length 0 and 1 relations look like (loops and bigons),
and the length 2 relations all follow from SS relations. Thus we have
(a, b) = ((0, 0, 0), (1, 1, 1)), ((0, 0, 0), (1, 1, 2)), ((0, 0, 1), (1, 2, 2)), ((0, 1, 1), (2, 2, 2))
= ((0, 0, 0, 0), (1, 1, 1, 1)).
APR5(0,0,0),(1,1,1) =
= spanA
− [4]q P
+ [3]q P
− [2]q P
+ P5−→
− [2]q P
+ [3]q P
− [4]q P
= spanA
− [4]q + [3]q − [2]q + ,
− [2]q + [3]q − [4]q
(5.6.9)
APR5(0,0,0),(1,1,2) = spanA
P5(0,0,0),(1,1,2);2 − P
(0,0,0),(1,1,2);3+
+P5(0,0,0),(1,1,2);4 − P
(0,0,0),(1,1,2);5
= spanA
− + −
(5.6.10)
APR5(0,0,1),(1,2,2) = spanA
P5(0,0,1),(1,2,2);2 − P
(0,0,1),(1,2,2);3+
+P5(0,0,1),(1,2,2);4 − P
(0,0,1),(1,2,2);5
= spanA
− + −
(5.6.11)
APR5(0,1,1),(2,2,2) = spanA
P5(0,1,1),(2,2,2);2 − P
(0,1,1),(2,2,2);3+
+P5(0,1,1),(2,2,2);4 − P
(0,1,1),(2,2,2);5
= spanA
− + −
(5.6.12)
APR5(0,0,0,0),(1,1,1,1) = spanA
−P5(0,0,0,0),(1,1,1,1);1 + P
(0,0,0,0),(1,1,1,1);2 − P
(0,0,0,0),(1,1,1,1);3+
+P5(0,0,0,0),(1,1,1,1);4 − P
(0,0,0,0),(1,1,1,1);5
= spanA
− + − +
(5.6.13)
The interesting SS5 relations are
SS5(0,0),(1,1) = spanA
P5(0,0),(1,1);2 −Q
(0,0),(1,1);0 − [3]q Q
(0,0),(1,1);1,
P5(0,0),(1,1);3 − [3]q Q
(0,0),(1,1);0 − [3]q Q
(0,0),(1,1);1,
P5(0,0),(1,1);4 − [3]q Q
(0,0),(1,1);0 −Q
(0,0),(1,1);1
= spanA
− − [3]q ,
− [3]q − [3]q ,
− [3]q −
SS5(0,0),(1,2) = spanA
P5(0,0),(1,2);3 −Q
(0,0),(1,2);0 − [2]q Q
(0,0),(1,2);1,
P5(0,0),(1,2);4 − [2]q Q
(0,0),(1,2);0 −Q
(0,0),(1,2);1
= spanA
− − [2]q ,
− [2]q −
SS5(0,0),(1,3) = spanA
P5(0,0),(1,3);4 −Q
(0,0),(1,3);0 −Q
(0,0),(1,3);1
= spanA
SS5(0,0),(2,2) = spanA
P5(0,0),(2,2);3 −Q
(0,0),(2,2);1 −Q
(0,0),(2,2);2,
P5(0,0),(2,2);4 −Q
(0,0),(2,2);0 −Q
(0,0),(2,2);1
= spanA
− − ,
SS5(0,1),(2,2) = spanA
P5(0,1),(2,2);3 −Q
(0,1),(2,2);1 − [2]q Q
(0,1),(2,2);2,
P5(0,1),(2,2);4 − [2]q Q
(0,1),(2,2);1 −Q
(0,1),(2,2);2
= spanA
− − [2]q ,
− [2]q −
SS5(0,1),(2,3) = spanA
P5(0,1),(2,3);4 −Q
(0,1),(2,3);1 −Q
(0,1),(2,3);2
= spanA
SS5(0,1),(3,2) = spanA
P5(0,1),(3,2);4 −Q
(0,1),(3,2);1 −Q
(0,1),(3,2);2
= spanA
SS5(0,2),(3,3) = spanA
P5(0,2),(3,3);4 −Q
(0,2),(3,3);2 −Q
(0,2),(3,3);3
= spanA
5.7 Proofs of Theorems 5.1.1, 5.2.1 and 5.3.2
5.7.1 The I = H relations
Proof of Theorem 5.1.1. We’ll just show the calculation for Equation (5.1.1), with ‘merging’
vertices; the calculation for Equation (5.1.2) is identical. It’s an easy induction, beginning
by calculating dGT of both sides, using the calculations from §4.3.
= (−1)c + (−1)n+b+cq−a +
+ (−1)bq−a−b + (−1)b ,
while
= (−1)n+a+c + (−1)n+a+b+cq−a +
+ (−1)a+bq−a−b + (−1)a+b
= (−1)n(a−1)(−1)n+a+c + (−1)na(−1)n+a+b+cq−a +
+ (−1)na(−1)a+bq−a−b + (−1)na(−1)a+b ,
which differs by exactly an overall factor of (−1)(n+1)a.
For the base cases of the induction, we need consider the possibility that a, b or
c is 0 or a + b + c = n. It’s actually more convenient to consider one of the equivalent
relations in Equation (5.1.3), where the corresponding base cases are a, b, c or d = 0. Then
I claim
= (−1)(n+1)a (5.7.1)
in Symn (and similarly in the other 3 cases when instead b, c or d is zero), and so the
difference is automatically in the kernel of Rep. Each of the four variations of Equation
(5.7.1) follows from the ‘degeneration’ relations in Symn, given in Equation (2.2.10). If
a = 0, we have
= = = (−1)(n+1)0 . (5.7.2)
The middle equality here uses ‘tag cancellation’ from Equation (2.2.8). The other cases are
similar, but use ‘tag flipping’ from Equation (2.2.9), producing the desired signs.
5.7.2 The square-switch relations
Proof of Theorem 5.2.1. This is quite straightforward, by induction. We’ll just do the n +
Σa− Σb ≥ 0 case; the other follows immediately by rotation, by §2.4.1.
The base cases are easy; since either each bi ≥ ai, or APna,b = AQna,b = 0, when
we’re at n = 0 the only interesting case is a = b = (0, 0). Then SS0(0,0),(0,0) is simply the
span of P0
(0,0),(0,0);0
(0,0),(0,0);0
; this element is actually zero in Sym0, by Lemma 2.4.1,
and so automatically in the kernel of Rep.
For the inductive step, we simply show that for l = max b, . . . ,min a + n the
component of
Pna,b;l −
min b
m=max a
n+Σa−Σb
m+ l − Σb
Qna,b;m
(5.7.3)
in HomSymn−1(∅,L(a+ a′, b+ b′)) lies in SS
a+a′,b+b′ for each a
′, b′. The expression in Equa-
tion (5.7.3) has one matrix for each subset of the four boundary points. The only nonzero
matrix entries correspond to those subsets for which there is a reduction path with end-
points coinciding with the subset. Thus, numbering the four boundary points 1, 2, 3 and 4,
we have dGT = dGT ∅+dGT {1,2}+dGT {1,4}+dGT {3,4}+dGT {3,2}+dGT ∂ . In the notation
of §4.4,
dGT {1,2}|AP = dGT (1,1),(0,1), dGT {1,2}|AQ = dGT (0,0),(−1,0),
dGT {1,4}|AP = dGT (1,0),(0,0), dGT {1,4}|AQ = dGT (1,0),(0,0),
dGT {3,4}|AP = dGT (1,1),(1,0), dGT {3,4}|AQ = dGT (0,0),(0,−1),
dGT {3,2}|AP = dGT (0,1),(0,0), dGT {3,2}|AQ = dGT (0,1),(0,0).
We now compute all of the components of Equation (5.7.3). First, use Equations (4.4.2) and
(4.4.3) to write down
dGT {1,2}
Pna,b;l
= (−1)Σaq−n−Σa+ΣbPn−1
a+(1,1),b+(0,1);l
dGT {1,2}
Qna,b;l
= (−1)Σaq−n−Σa+ΣbQn−1
a+(0,0),b+(−1,0);l
= (−1)Σaq−n−Σa+ΣbQn−1
a+(1,1),b+(0,1);l+1
dGT {1,4}
Pna,b;l
= q−n−a1+b2Pn−1
a+(1,0),b+(0,0);l
dGT {1,4}
Qna,b;l
= q−n−a1+b2Qn−1
a+(1,0),b+(0,0);l
dGT {3,4}
Pna,b;l
= (−1)Σaq−n−Σa+ΣbPn−1
a+(1,1),b+(1,0);l
dGT {3,4}
Qna,b;l
= (−1)Σaq−n−Σa+ΣbQn−1
a+(0,0),b+(0,−1);l
= (−1)Σaq−n−Σa+ΣbQn−1
a+(1,1),b+(1,0);l+1
dGT {3,2}
Pna,b;l
= q−a1+b1Pn−1
a+(0,1),b+(0,0);l
dGT {3,2}
Qna,b;l
= q−a1+b1Qn−1
a+(0,1),b+(0,0);l
and notice that the coefficients appearing are in each case independent of whether we’re
acting on AP or AQ, and moreover that the coefficients are independent of the internal
flow labels l. We can use this to show that dGT {i,j}
Pna,b;l −
∑min b
m=max a
n+Σa−Σb
m+l−Σb
Qna,b;m
lies in SSn−1 for each of the four pairs {i, j}. For example,
dGT {1,2}
Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l − Σb
Qna,b;m
= (−1)Σaq−n−Σa+Σb×
a+(1,1),b+(0,1);l
min b
m=max a
n+Σa− Σb
m+ l − Σb
a+(1,1),b+(0,1);m+1
= (−1)Σaq−n−Σa+Σb×
a+(1,1),b+(0,1);l
min b
m=max a
n− 1 + Σ(a+ (1, 1)) − Σ(b+ (0, 1))
m+ 1 + l − Σ(b+ (0, 1))
a+(1,1),b+(0,1);m+1
which becomes, upon reindexing the summation,
(−1)Σaq−n−Σa+Σb×
a+(1,1),b+(0,1);l
min b+1
m=max a+1
n− 1 + Σ(a+ (1, 1)) − Σ(b+ (0, 1))
m+ l − Σ(b+ (0, 1))
a+(1,1),b+(0,1);m
Now, making use of the fact that if min b+1 > min (b+ (0, 1)), the last term vanishes since
a+(1,1),b+(0,1);min b+1
= 0, we can rewrite the summation limits and obtain
dGT {1,2}
Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l −Σb
Qna,b;m
= (−1)Σaq−n−Σa+Σb ×
a+(1,1),b+(0,1);l
min (b+(0,1))
m=max (a+(1,1))+1
n− 1 + Σ(a+ (1, 1)) − Σ(b+ (0, 1))
m+ l − Σ(b+ (0, 1))
a+(1,1),b+(0,1);m
The parenthesised expression is exactly an element of the spanning set for SSn−1
a+(1,1),b+(0,1)
It remains to show that dGT ∅ and dGT ∂ carry SSn into SSn−1. Both are similar; here’s the
calculation establishing the first.
dGT ∅
Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l − Σb
Qna,b;m
= q−a1
qlPn−1a,b;l + q
l−n−Σa+ΣbPn−1a,b;l−1 −
min b
m=max a
q−mqΣb
n+Σa− Σb
m+ l −Σb
Qn−1a,b;m
which, using the q-binomial identity
+ qt−s
, we can rewrite as
= q−a1
qlPn−1a,b;l −
min b
m=max a
n− 1 + Σa− Σb
m+ l − Σb
Qn−1a,b;m+
ql−n−Σa+ΣbPn−1
a,b;l−1 −
min b
m=max a
ql−n−Σa+Σb
n− 1 + Σa− Σb
m+ l − 1− Σb
a,b;m
which is in SSn−1a,b .
5.7.3 The Kekulé relations
We now prove Theorem 5.3.2, describing the relations amongst polygonal diagrams. We
begin with a slight reformulation of the goal; it turns out that the orthogonal complement
APR⊥ of the relations in the dual space AP∗ is easier to describe.
Lemma 5.7.1. The orthogonal complement APRna,b⊥ in APna,b∗ is spanned by the elements
a,b;j∗
k∗=Σb
n+Σa− Σb
k∗ − Σb
(Pna,b;k∗+j∗)∗
for j∗ = −
b, . . . ,−
Proof. First note that APRna,b ⊂ APna,b is an (n+Σa−Σb)-dimensional subspace of APna,b;
the spanning set we’ve described for it really is linearly independent in Symn. It’s a sub-
space of a (min a−max b+ n + 1)-dimensional vector space, so we expect the orthogonal
complement to be (1 +
a)-dimensional, as claimed in the lemma. We thus need
only check that each e
a,b;j∗
annihilates each d
a,b;j
, for j = Σb, . . . ,Σa + n − 1. Thus we
calculate
j∗ (d
j ) =
min(−
a+1,j∗−j+n+Σa)
k=max(−
b,Σb+j∗−j)
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
n+Σa−Σb
j − j∗ + k − Σb
and observe that the limits of the summation are actually irrelevant; outside the limits at
least one of the three quantum binomial coefficients is zero.
Lemma A.1.1 in the appendix §A.1 shows that this q-binomial sum is zero.
Corollary 5.7.2. The orthogonal complement AQRna,b⊥ in AQna,b∗ is spanned by the elements
a,b;j∗ =
k∗=Σa
Σb−Σa− n
k∗ − Σa
Qna,b;j∗+k∗⊥
for j∗ = −
a, . . . ,−
Proof. This follows immediately from the previous lemma, by rotation, as in §2.4.1.
The proof of Theorem 5.3.2 now proceeds inductively. We’ll assume that
kerRep∩APn−1
= APRn−1
for all flows a, b. We’ll look at dGT ∅ first; we know that anything in kerRep∩APna,b must
lie inside dGT −1
(APRn−1a,b ), and we’ll discover that in fact dGT
(APRn−1a,b ) = APR
a,b, in
Lemma 5.7.3. Subsequently, we need to check, in Lemma 5.7.4, that dGT a′,b′(APRna,b) ⊂
APRn−1
a+a′,b+b′
for each allowed pair of boundary flow reduction patterns a′, b′. Modulo
these lemmas, that proves the first part of Theorem 5.3.2 (at least the statements about
APR; Lemma 5.7.5 deals with the statements about AQR). Later, Lemma 5.7.8 deals with
the other parts of Theorem 5.3.2.
Explicit formulas for dGT ∅ and dGT a′,b′ are easy to come by, from §4.4:
dGT ∅
Pna,b;l
= ql−a1
Pn−1a,b;l + q
Σb−Σa−nPn−1a,b;l−1
(5.7.4)
dGT a′,b′(Pna,b;l) = κn,a,b,a′,b′q−l(m(a
′,b′)−1)Pn−1a+a′,b+b′;l (5.7.5)
where here κn,a,b,a′,b′ is some constant not depending on l, which for now we don’t need to
keep track of, but can be reconstructed from Equation (4.4.2), and m(a′, b′) is the number
of components of the reduction path indexed by a′, b′, which is just Σa′ −Σb′.
It’s worth noting at this point that the P-webs appearing in the right hand sides
of Equations (5.7.4) and (5.7.5) are sometimes zero; the internal flow value might be out of
the allowable range.
In particular, the allowed values of l for Pna,b;l ∈ APna,b are l = max b, . . . ,min a+n,
and so the target of dGT ∅ is one dimension smaller than its source. Moreover, it’s easy to
see from Equation (5.7.4) that the kernel of dGT ∅ is exactly one dimensional. Life is a little
more complicated for dGT a′,b′ ; if b′i = 1 for any i where bi = max b, then
dGT a′,b′
Pna,b;max b
and if a′i = 1 for every i where ai = min a, then
dGT a′,b′
Pna,b;mina+n
and otherwise dGT a′,b′ is actually nonzero on each P-web. Thus the kernel of dGT a′,b′ may
be 0, 1 or 2 dimensional.
Lemma 5.7.3. The only candidates for the kernel of the representation functor on APna,b are in
APRna,b, since
dGT −1
(APRn−1a,b ) = APR
Proof. We’ve already noticed that the kernel of dGT ∅ is 1-dimensional, so we actually only
need to check that dGT ∅
APRna,b
⊂ APRn−1
, or, equivalently, that
dGT ∗∅(APR
) ⊂ APRna,b⊥.
Easily, we obtain
dGT ∗∅(P
a,b;l∗
) = ql
Pna,b;l∗∗ + qΣb−Σa−n+1Pna,b;l∗+1∗
simply by taking duals in Equation (4.4.4). Then
dGT ∗∅(e
P,n−1
a,b;j∗ ) = dGT
n−1+Σa
k∗=Σb
n− 1 + Σa− Σb
k∗ − Σb
(Pn−1a,b;k∗+j∗)
= q−a1
n−1+Σa
k∗=Σb
n− 1 + Σa− Σb
k∗ − Σb
Pna,b;k∗+j∗∗ + qΣb−Σa−n+1Pna,b;k∗+j∗+1∗
and by reindexing the summation of the second terms, we can rewrite this as
dGT ∗∅(e
P,n−1
a,b;j∗ ) = q
j∗−a1+Σb
k∗=Σb
n− 1 + Σa− Σb
k∗ − Σb
∗−n−Σa
n− 1 + Σa− Σb
k∗ − 1− Σb
Pna,b;k∗+j∗∗
∗−a1+Σb
k∗=Σb
n+Σa− Σb
k∗ − Σb
Pna,b;k∗+j∗∗
∗−a1+Σbe
a,b;j∗
tidily completing the proof.
Remark. By rotation, we can also claim that
dGT −1∂ (AQR
a,b ) = AQR
and that if you’re in the kernel of Rep, and in AQna,b, you must also be in AQRna,b.
Lemma 5.7.4. The candidate relations APRna,b really are in the kernel of the representation functor,
since
dGT a′,b′(APRna,b) ⊂ APRn−1a+a′,b+b′
for each pair a′, b′ (other than the pairs a′ = b′ =
0 and a′ = b′ =
1 ; the previous Lemma dealt
with that matrix entry of dGT ).
Proof. For this we need the quantum Vandermonde identity [39],
= qyz
q−(x+y)i
z − i
Throughout, we’ll write m = Σa′ − Σb′ for the number of components of the reduction
path. We want to show
dGT ∗a′,b′(e
P,n−1
a+a′,b+b′;j∗
) = κn,a+a′,b+b′,a′,b′×
n+Σa+m−1
k∗=Σb
∗+k∗+Σb′)(m−1)
n− Σa− Σb+m− 1
k∗ − Σb
Pna,b;k∗+j∗+Σb∗ (5.7.6)
is equal to
a,b,j∗+Σb+i =
k∗=Σb
n+Σa− Σb
k∗ − Σb
Pna,b;k∗+j∗+i+Σb′
∗ ∈ APRna,b⊥
for some coefficients Xi. Looking at the coefficient of Pna,b;k∗+j∗+Σb′
n in Equation (5.7.6)
and applying the quantum Vandermonde identity with x = n + Σa− Σb, y = m − 1, and
z = k∗ − Σb, we obtain
κn,a+a′,b+b′,a′,b′q
−(j∗+k∗+Σb′)(m−1)q(m−1)(k
∗−Σb)×
q−(n+Σa−Σb+m−1)
n+Σa− Σb
k∗ − Σb− i
Choosing Xi = κn,a+a′,b+b′,a′,b′q
(m−1)(−j∗−Σb−Σb′)q−(n+Σa−Σb+m−1)i, (which, note, is inde-
pendent of k∗, the index of the term we’re looking at) this is exactly the coefficient of
Pna,b;k∗+j∗+Σb′
i=0 Xie
a,b,j∗+Σb+i
Lemma 5.7.5. The intersection of kerRep and AQ is exactly AQR.
Proof. Again, this is just by rotation, see §2.4.1.
The next two Lemmas, and the subsequent proof of Lemma 5.3.3, are kinda hairy.
Hold on tight!
Lemma 5.7.6. For any n-hexagonal flows (a, b) of length k ≥ 3 such that APna,b 6= 0 one of the
following must hold:
1. The flows (a, b) are also (n− 1)-hexagonal.
2. The flows (a+
1 , b) are (n− 1)-hexagonal.
3. There is some (a′, b′) ∈ ({0, 1}k)2, so (a + a′, b + b′) is (n − 1)-hexagonal and moreover
dGT a′,b′ maps APna,b faithfully into APn−1a+a′,b+b′ .
4. n ≤ 3.
Proof. For (a, b) not to be (n− 1)-hexagonal, at least k − 5 edge labels must be (n− 1), and
for (a+
1 , b) not to be (n− 1)-hexagonal, at least k− 5 edge labels must be 1. Already, this
establishes the lemma for k > 10 and n > 2 (and it’s trivially true for n ≤ 2).
We thus need to deal with k = 6, 8 or 10. If k = 10, one of the first two alternatives
hold, unless L(a, b) is some permutation of (1, 1, 1, 1, 1, n − 1, n− 1, n− 1, n− 1, n− 1). It’s
easy enough to see that no permutation is possible, given the condition that the incoming
edges and outgoing edges have the same sum. For k = 6 or 8, we can be sure that both 1
and n − 1 ∈ L(a, b). Notice that this implies APRna,b = AQRna,b = 0, and both APna,b and
AQna,b are at most 2-dimensional, with APna,b spanned by Pna,b;max b and Pna,b;min a+n and
AQna,b spanned by Qna,b;max a and Qna,b;min b. Let’s now assume that we never see (not even
‘cyclically’) as contiguous sequences of boundary edges labels either
((n− 1,−), (∗ ≤ 1,+), (∗,−), (∗ ≤ 1,+), (∗,−), . . . , (∗,−), (∗ ≤ 1,+), (n − 1,−)) (5.7.7a)
((n− 1,+), (∗ ≤ 1,−), (∗,+), (∗ ≤ 1,−), (∗,+), . . . , (∗,+), (∗ ≤ 1,−), (n − 1,+)). (5.7.7b)
We can then prescribe a reduction path on the P-polygons, which corresponds to a pair
(a′, b′) establishing the third alternative above. Consider the reduction path which starts
at each incoming edge labeled n− 1, and ends at the next (heading counterclockwise) out-
going edge with a label greater than 1, and which additionally ends at each outgoing edge
labeled n−1, having started at the previous incoming edge with a label greater than 1. That
this forms a valid reduction path (i.e. there are no overlaps between the components de-
scribed) follows immediately from the assumption of the previous paragraph, that certain
sequences do not appear. The pair (a′, b′) corresponding (via the correspondence discussed
in §4.4) to this reduction path then satisfies the first condition of the third alternative above,
that (a+ a′, b+ b′) be (n− 1)-hexagonal. However, it doesn’t obviously satisfy the second
condition. The only way that dGT a′,b′ might not be faithful on APna,b is if it kills Pna,b;max b
or Pna,b;mina+n (remember max b = min a+ n− 1 or min a+ n, since 0 < dimAP
a,b ≤ 2). To
kill Pna,b;max b, the reduction path we’ve described would have to traverse an internal edge
labeled 0. However, it turns out there’s ‘no room’ for this. Consider some component of
the reduction path starting at an incoming edge labeled n− 1. It then looks like:
for some 0 ≤ l ≤ n, 1 < κ ≤ n, λi = 0 or 1 and 0 ≤ µi ≤ n. If any internal 0 edge gets
reduced, there must be some i so that the internal edge li also gets reduced. However,
l1 = l + n − 1 − λ1 ≥ l + n − 2, and generally li = l + n − 1 +
j=1(−λj + µi) − λi ≥
l + n+ ♯ (µi ≥ 1)− i− 1. Thus for li to be 0, we must have
m ≥ i ≥ n− 1 + ♯ (µi ≥ 1) . (5.7.8)
Now, if k = 6, none of the external edges can be zero, since (a, b) is n-hexagonal. Thus
♯ (µi ≥ 1) = m, so as long as n ≥ 2, the inequality of Equation (5.7.8) cannot be satisfied.
If, on the other hand k = 8, we can only say ♯ (µi ≥ 1) ≥ m − 1 (since any non-adjacent
pair of external edges being zero makes it impossible to satisfy the ‘alternating’ part of the
condition for being n-hexagon), so m ≥ n− 1 +m− 1, or n ≤ 2. The argument preventing
dGT a′,b′ from killing Pna,b;min a+n is much the same.
Finally, we need to show that the appearance of one of the ‘forbidden sequences’
from Equation (5.7.7) forces n ≤ 3, so that the last alternative holds. Suppose that, for
some m ≥ 0, ((n− 1,−), (λ1,+), (κ1,−), . . . , (κm,−), (λm+1,+), (n− 1,−) with λi = 0 or 1
appears as a contiguous subsequence of L(a, b). The internal edge labels on either side of
this subsequence then differ by 2n − 2 +
i=1 κi −
i=1 λi. We know this quantity must
be no more than n, since APna,b 6= 0, i.e., there is some internal flow value l so Pna,b;l 6= 0.
However, n−2+
i=1 κi−
i=1 λi ≥ n−2+♯ (κi 6= 0)−♯ (λi = 1) ≥ n+♯ (κi 6= 0)−m−3.
Thus we must have n ≤ m + 3 − ♯ (κi 6= 0). If such a subsequence appears with m = 0,
this immediately forces n ≤ 3. If such a subsequence appears with m > 0, either n ≤ 3 or
♯ (κi 6= 0) < m, so there’s at least one κi = 0. This certainly isn’t possible for k = 3; (a, b)
couldn’t be n-hexagonal.
We’re now left with a fairly finite list of cases to check for the k = 4 case; for
each one, even in the presence of a ‘forbidden sequence’ we’ll explicitly describe an ad
hoc reduction path, so that the corresponding (a′, b′) pair satisfies the third alternative. We
need to consider both m = 1 and m = 2; there’s no room for m ≥ 3 if k = 4. For m = 1, the
boundary edges (up to a rotation) are
((n − 1,−), (λ1,+), (0,−), (λ2,+), (n − 1,−), (µ1,+), (µ2,−), (µ3,+))
with λi = 0 or 1 and some µi. In order for this to be n-hexagon, we can’t have µ1 = n,
µ2 = 0, µ3 = n or both λi = 0. Subject then to the condition that incoming and outgoing
labels have the same sum, we must have one of
µ2 = 2 λ1 = 1 λ2 = 1 µ1 = n− 1 µ3 = n− 1,
µ2 = 1 λ1 = 1 λ2 = 1 µ1 = n− 2 µ3 = n− 1,
µ2 = 1 λ1 = 1 λ2 = 1 µ1 = n− 1 µ3 = n− 2,
µ2 = 1 λ1 = 0 λ2 = 1 µ1 = n− 1 µ3 = n− 1,
µ2 = 1 λ1 = 1 λ2 = 0 µ1 = n− 1 µ3 = n− 1.
For the first three, however, there’s an (a′, b′) pair:
((1, 0, 1, 1), (0, 0, 0, 0)),
((1, 1, 1, 0), (0, 1, 0, 0)),
((1, 0, 1, 1), (0, 0, 0, 1)),
and for the last two, we find that APna,b = 0. For m = 2, the boundary edges are
((n− 1,−), (λ1,+), (κ1,−), (λ2,+), (κ2,−), (λ3,+), (n − 1,−), (µ,+))
with λi = 0 or 1 and at least one of the κi = 0. If both κi = 0, or if µ = n, we can’t achieve
n-hexagon-ness. But then the sum of the incoming labels is 2n−2+κ1+κ2 ≥ 2n−1, while
the sum of the outgoing edges is λ1 +λ2 +λ3 +µ ≤ n+2. If 2n− 1 ≤ n+2, n ≤ 3 anyway,
and we’re done.
The argument for the forbidden sequence with opposite orientations is almost
identical.
Lemma 5.7.7. The intersection APRna,b ∩ spanA
Pna,b;max b,Pna,b;max b+1, . . . ,Pna,b;Σb−Pa
zero.
Proof. Say x =
i=0 xiPna,b;max b+i, write imax for the greatest i so xi 6= 0. Define j∗ =
imax − Σb, and note that since max b ≤ imax ≤ Σb−
a we must have −
b ≤ j∗ ≤ −
Then apply e
a,b;j∗ , as defined in Lemma 5.7.1 to x, obtaining:
a,b;j∗(x) = ximax 6= 0,
so x /∈ APRna,b.
We now give the proof of an earlier lemma, as we’re about to need it.
Proof of Lemma 5.3.3. We can just prove the ∞-circumference case. Moreover, if we’re as-
suming (a, b) has ∞-circumference 2k, we can further assume that a and b each actually
have length k. (Otherwise, there’s some i so bi = ai, or bi = ai+1; in that case, ‘snip
out’ those entries from a and b.) Now, pick some imin so aimin = min a, and then pick as
imax the first value of i ≥ imin (in the cyclic sense) so that bimax = max b. If imax = imin,
we’re done;
i 6=imax
bi − ai ≥ k − 1. Otherwise, we’ll show how to modify a
and b, preserving the value of
a, to make imax − imin (again, measured cyclically)
smaller. Specifically, for each imin ≤ i < imax, increase bi by bimax − bimax−1, and for each
imax < i ≤ imax, increase ai by the same. The resulting sequence is still valid, in the sense
that bi ≥ ai, ai+1 for each i. Clearly, this doesn’t change any of max b, min a, Σb or Σa, but
it reduces the minimal value of imax − imin by one.
We the previous three monstrosities tamed, we can finish of Theorem 5.3.2.
Lemma 5.7.8. If (a, b) is a pair of flow labels of length at least 3, and all the external edge labels
in L(a, b) are between 1 and n − 1 inclusive, the intersection of kerRep and APna,b + AQna,b is
APRna,b +AQRna,b.
Proof. If AQna,b 6= 0 but APna,b = 0, first apply a rotation, as described in §2.4.1. We need to
deal with the four alternatives of the previous lemma.
Supposen−1 /∈ L(a, b), so we’re in the first alternative. Then after applying dGT ∅,
we’re in a situation where 5.3.2 holds in full; the kernel of Rep is just APRn−1+AQRn−1.
(The argument below works also if 1 /∈ L(a, b) and we’re in the second alternative, mutatis
mutandis, exchanging the roles of dGT ∅ and dGT ∂ .) Thus suppose X ∈ APn +AQn is in
kerRep. Then dGT ∅ (X) = X∅;P +X∅;Q, for some X∅;P ∈ APRn−1 and X∅;Q ∈ AQRn−1.
Pick some XP ∈ dGT −1∅ (X∅;P), which we know by Lemma 5.7.3 must also be an element
of APRn. Now dGT ∅ (X −XP) = X∅;Q ∈ AQRn−1, so X −XP must lie in AQn. We want
to show it’s actually in AQRn. We know (via Lemma 5.7.4), that X − XP ∈ kerRep, so
dGT ∂(X−XP) ∈ AQRn−1, by the remark following Lemma 5.7.3. Thus X−XP ∈ AQRn.
The decomposition X = XP + (X −XP) establishes the desired result.
Suppose that there’s some (a′, b′) with dGT a′,b′ |APn
faithful and (a + a′, b + b′)
(n−1)-hexagonal. Now, if 1, n−1 /∈ L(a, b), as in the last lemma we know APna,b is spanned
by Pna,b;max b and Pna,b;mina+n and AQ
a,b is spanned by Qna,b;max a and Qna,b;min b (remember
that in each case the pairs of elements might actually coincide). Thus
dGT a′,b′
Pna,b;max b
= (−1)b′·a+rotl(a)qmax b(Σb′−Σa′+1)×
× qrotl(a′)·b−b′·a−a1−na′1Pn−1a+a′,b+b′;max b 6= 0,
dGT a′,b′
Pna,b;min a+n
= (−1)b′·a+rotl(a)q(min a+n)(Σb′−Σa′+1)×
× qrotl(a′)·b−b′·a−a1−na′1Pn−1
a+a′,b+b′;mina+n
6= 0,
dGT a′,b′
Qna,b;max a
= (−1)b′·a+rotl(a)qmax a(Σa′−Σb′−
× qΣb−nΣb′+b′·rotl(a)−a′·b−a1−na′1Qn−1
a+a′,b+b′;max a
dGT a′,b′
Qna,b;min b
= (−1)b′·a+rotl(a)qmin b(Σa′−Σb′−
× qΣb−nΣb′+b′·rotl(a)−a′·b−a1−na′1Qn−1
a+a′,b+b′;min b
Thus suppose x1Pna,b;max b + x2Pna,b;min a+n + y1Qna,b;max a + y2Qna,b;min b is in kerRepn. Then
x′1Pn−1a+a′,b+b′;max b + x
2Pn−1a+a′,b+b′;mina+n + y
1Qn−1a+a′,b+b′;max a + y
2Qn−1a+a′,b+b′;min b
is in kerRepn−1, where x
1 is a nonzero multiple of x1, and x
2 is a nonzero multiple of x2,
by the formulas above. Since (a+a′, b+b′) is (n−1)-hexagonal, we can inductively assume
kerRepn−1 = APRn−1a+a′,b+b′ + AQR
a+a′,b+b′
, so x′1Pn−1a+a′,b+b′;max b + x′2P
a+a′,b+b′;mina+n
APRn−1
a+a′,b+b′
. However, by Lemma 5.7.7, along with the knowledge that the relations in
APRn−1
a+a′,b+b′
have ‘breadth’ at least 3, as per Lemma 5.3.3, this implies x′1 = x
2 = 0, and
thus x1 = x2 = 0. Now we know y1Qna,b;max a + y2Qna,b;min b is in kerRepn, which when
restricted to AQna,b is just AQRna,b, by Lemma 5.7.5. Again by Lemma 5.7.7, y1 = y2 = 0.
The final alternative is n ≤ 3, where the lemma is obvious from the known com-
plete descriptions of kerRep2 and kerRep3.
5.7.4 More about squares
To prove Theorem 5.4.1, that SS = APR ⊕ SS ′, we need to establish the following three
lemmas. (Trivially, APR ∩ SS ′ = 0.) We’ll in fact only deal with the n+Σa− Σb > 0 case
of Theorem 5.4.1; the n + Σa − Σb = 0 is trivial, and, as usual, the n + Σa − Σb < 0 case
follows by rotation.
Lemma 5.7.9. APR ⊂ SS .
Proof. In the quotient (AP +AQ)/SS ,
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
Pna,b;j+k =
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
min b
m=max a
n+Σa− Σb
m+ j + k − Σb
Qna,b;m.
The coefficient of Qna,b;m in this is
min b
m=max a
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
n+Σa− Σb
m+ j + k − Σb
Not only does this vanish, but each term indexed by a particular value of m vanishes
separately: replacing m with −j∗, this is precisely the q-binomial identity of Lemma A.1.1.
Lemma 5.7.10. SS ′ ⊂ SS.
Proof. We need to show that each element of the spanning set of SS ′ presented in Theorem
5.4.1 is in SS . In SS ′/(SS ∩ SS ′),
Qna,b;m −
n+mina
l=n+Σa−m
(−1)m+l+n+Σa
m+ l − 1− Σb
m+ l − n− Σa
Pna,b;l =
= Qna,b;m −
n+mina
l=n+Σa−m
(−1)m+l+n+Σa
m+ l − 1− Σb
m+ l − n− Σa
min b
m′=max a
n+Σa− Σb
m+ l − Σb
Qna,b;m′ .
The coefficient of Qna,b;m′ here is
δmm′ − (−1)m+n+Σa
n+mina
l=n+Σa−min b
(−1)l
m+ l − 1− Σb
m+ l − n− Σa
n+Σa− Σb
m′ + l − Σb
That this is zero follows from the q-binomial identity proved in §A.1 as Lemma A.1.2.
Lemma 5.7.11. SS ⊂ SS ′ +APR.
Proof. We take an element of the spanning set described for SS, considered as an element
of (AP +AQ)/SS ′. Using the relations from SS ′, we write this as an element of AP/SS ′,
and check that this element is annihilated by each element of the spanning set for APR⊥
described in Lemma 5.7.1. Thus,
Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l − Σb
Qna,b;m =
= Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l − Σb
n+min a
l′=n+Σa−m
(−1)m+l′+n+Σa
m+ l′ − 1−Σb
m+ l′ − n− Σa
Pna,b;l′ .
Applying
∑n+Σa
k∗=Σb
n+Σa−Σb
k∗−Σb
(Pna,b;k∗+j∗)∗ to this we get
n+Σa− Σb
k∗ − Σb
δl,k∗+j∗−
min b
m=max a
n+mina
l′=n+Σa−m
k∗=Σb
(−1)m+l′+n+Σa
n+Σa− Σb
m+ l − Σb
m+ l′ − 1− Σb
m+ l′ − n− Σa
n+Σa− Σb
k∗ − Σb
δl′,k∗+j∗ .
First noticing that the lower limit for l′ is redundant, a slight variation Lemma A.1.2 lets
us evaluate the sum over m, obtaining
n+Σa− Σb
k∗ − Σb
δl,k∗+j∗ −
n+mina
l′=−∞
k∗=Σb
δl,l′
n+Σa− Σb
k∗ − Σb
δl′,k∗+j∗
n+Σa− Σb
k∗ − Σb
δl,k∗+j∗ −
k∗=Σb
n+Σa− Σb
k∗ − Σb
δl,k∗+j∗
n+Σa− Σb
k∗ − Σb
δl,k∗+j∗ −
n+Σa− Σb
k∗ − Σb
δl,k∗+j∗
Chapter 6
Relationships with previous work
6.1 The Temperley-Lieb category
The usual Temperley-Lieb category (equivalently, the Kauffman bracket skein module) has
unoriented strands, and a ‘loop value’ of − [2]q. The category for Uq(sl2) described here has
oriented strands, and a ‘loop value’ of [2]q, along with orientation reversing ‘tags’, which
themselves can be flipped for a sign. Nevertheless, the categories are equivalent, as de-
scribed below (in parallel with a description of the relationship between my category for
Uq(sl4) and Kim’s previous conjectured one) in §6.4. The modification of Khovanov homol-
ogy described by myself and Kevin Walker in [26], producing a fully functorial invariant, is
based on a categorification of the quantum group skein module, rather than the Kauffman
skein module.
6.2 Kuperberg’s spider for Uq(sl3)
Kuperberg’s work on Uq(sl3) is stated in the language of spiders. These are simply (strict)
pivotal categories, with an alternative set of operations emphasised; instead of composi-
tion and tensor product, ‘stitch’ and ‘join’. Further, in a spider the isomorphisms such as
Hom(a, b⊗ c) ∼= Hom(a⊗ c∗, b) are replaced with identifications.In this and the following
sections, I’ll somewhat freely mix vocabularies.
In his work, the only edges that appear are edges labelled by 1, and there are
no tags. As previously pointed out, the only trivalent vertices in my construction at n =
3 involve three edges labelled by 1 (here I’m not counting the flow vertices; remember
they’re secretly defined in terms of the original vertices and tags). This means that in any
diagram, edges labelled by 2 only appear either at the boundary, connected to a tag, or in
the middle of an internal edge, with tags on either sides. Since tags cancel without signs
in the n = 3 theory, we can ignore all the internal tags.
There’s thus an easy equivalence between Kuperberg’s n = 3 pivotal category
(defined implicitly by his spider) and mine. (In fact, this equivalence holds before or after
quotienting out by the appropriate relations.) In one direction, to his category, we send
(1,±) → (1,±) and (2,±) → (1,∓) at the level of objects, and ‘chop off’ any external 2-
edges, and their associated tag, at the level of morphisms.1 The functor the other direction
is just the ‘inclusion’. Of the two composition functors, the one on his category is actually
the identity; the one on mine is naturally isomorphic to the identity, via the tensor natural
transformation defined by
φ(1,+) = φ(1,−) =
φ(2,+) = φ(2,−) =
6.3 Kim’s proposed spider for Uq(sl4)
In the final chapter of his Ph.D. thesis, Kim [21] conjectured relations for the Uq(sl4) spider.
These relations agree exactly with mine at n = 4. He discovered his relations by calculating
the dimensions of some small Uq(sl4) invariant spaces. As soon as it’s possible to write
down more diagrams with a given boundary than the dimension of the corresponding
space in the representation theory, there must be linear combinations of these diagrams
in the kernel. Consistency conditions coming from composing with fixed diagrams which
reduce the size of the boundary enabled him to pin down all the coefficients, although, as
with my work, he’s unable to show that he’s found generators for the kernel.
6.4 Tags and orientations
In this section I’ll describe an equivalence between my categories and the categories pre-
viously described for n = 2, in §6.1 and for n = 4, in §6.3. The description will also
encompass the equivalence described for n = 3 in §6.2; I’m doing these separately because
further complications arise when n is even.
The usual spider for Uq(sl2), the Temperley-Lieb category, has unoriented edges.
Similarly, Kim’s proposed spider for Uq(sl4) does not specify orientations on the ‘thick’
edges, that is, those edges labelled by 2. Moreover, as in Kuperberg’s Uq(sl3) spider, only
a subset of the edge labels I use appear; he only has edges labelled 1 and 2.
Nevertheless, those spiders are equivalent to the spiders described here. First, the
issue of the edge label 3 being disallowed is treated exactly as above in §6.2; we notice that
at n = 4, there are no vertices involving edges labelled 3, so we can remove any internal
3-edges by cancelling tags, and, at the cost of keeping track of a natural isomorphism,
remove external 3-edges too. There’s a second issue, however, caused by the unoriented
edges. The category equivalences we define will have to add and remove orientation data,
and tags.
1This picturesque description needs a patch for the identity 2-edge; there we create a pair of tags first, so
we have something to chop off at either end.
Thus we define two functors, ι, which decorates spider diagrams with ‘complete
orientation data’, and π, which forgets orientations and tags (entirely for n = 2, and only
on the 2 edges for n = 4). The forgetful functor also ‘chops off’ any 3 edges (in the n = 4
case), just like the functor in §6.2. The decorating functor ι simply fixes an up-to-isotopy
representative of a diagram, and orients each unoriented edge up the page, placing tags at
critical points of unoriented edges as follows:
It’s well defined because opposite tags cancel. The composition π ◦ ι is clearly the identity.
The other composition ι ◦π isn’t quite, but is natural isomorphic to the identity, via the tag
maps.
6.5 Murakami, Ohtsuki and Yamada’s trivalent graph invariant
In [27], Murakami, Ohtsuki and Yamada (MOY, hereafter) define an invariant of closed
knotted trivalent graphs, which includes as a special case the HOMFLYPT polynomial.2
Their graphs carry oriented edge labels, and the trivalent vertices are exactly as my ‘flow
vertices’. (They don’t have ‘tags’.) They don’t make any explicit connection with Uq(sln)
representation theory, although this is certainly their motivation. In fact, they say:
We also note that our graph invariant may be obtained (not checked yet) by
direct computations of the universal R-matrix. But the advantage of our defi-
nition is that it does not require any knowledge of quantum groups nor repre-
sentation theory.
One of their closed graphs can be interpreted in my diagrammatic categorySymn;
pushing it over into the representation theory RepUq(sln), it must then evaluate to a num-
ber. Presumably, this number must be a multiple of their evaluation (modulo replacing
their q with my q2), with the coefficient depending on the vertices appearing in the dia-
gram, but not their connectivity. Thus given a suitable closed trivalent graph D, the two
evaluations would be related via
〈D〉MOY = λ(D)Repn (D) . (6.5.1)
Knowing this evaluation coefficient λ(D) explicitly would be nice. You might approach it
by either ‘localising’ the MOY formulation3, or deriving a recursive version of the MOY
2There’s also a ‘cheat sheet’, containing a terse summary of their construction, at http://katlas.math.
toronto.edu/drorbn/index.php?title=Image:The_MOY_Invariant_-_Feb_2006.jpg.
3If you’re interested in trying, perhaps ask Dror Bar-Natan or myself, although we don’t have that much to
say; the localising step is easy enough.
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:The_MOY_Invariant_-_Feb_2006.jpg
http://katlas.math.toronto.edu/drorbn/index.php?title=Image:The_MOY_Invariant_-_Feb_2006.jpg
evaluation function, writing the evaluation of a graph for n in terms of the evaluation
of slightly modified graphs for n − 1. In particular, there’s an evaluation function in my
category, obtained by branching all the way down to n = 0, which is of almost the same
form as their evaluation function. It’s a sum over multiple reduction paths, such that each
edge is traversed by as many reductions paths as its label. Moreover, each reduction path
comes with an index, indicating which step of the branching process it is ‘applied’ to the
diagram at. This index corresponds exactly to the labels in N in MOY, after a linear change
of variable. Writing down the details of this should produce a formula for λ(D).
Modulo the translation described in the previous paragraph, their (unproved)
Proposition A.10 is presumably equivalent to the n+Σa− Σb ≥ 0 case of Theorem 5.2.1.
6.6 Jeong and Kim on Uq(sln)
In [11], Jeong and Kim independently discovered a result analogous to both cases of Theo-
rem 5.2.1, using a dimension counting argument to show that there must be such relations,
and finding coefficients by gluing on other small webs. (Of course, they published first,
and have priority on that theorem.) They never describe an explicit map from trivalent
webs to the representation theory of Uq(sln), however; they posit relations for loops and
bigons, and I = H relations, which differ from ours up to signs, and use these to show
that the relations of Theorem 5.2.1 must hold in order to get the dimensions of spaces of
diagrams right.
They later make a conjecture which says (in my language) that even for 2k-gons
with k ≥ 3, each P-2k-gon can be written in terms of Q-2k-gons and smaller polygons. My
results show this conjecture is false.
6.7 Other work
Sikora, in [34], defines an invariant of oriented n-valent braided ribbon graphs. His graphs
do not have labels on the edges, and allow braidings of edges. The invariant is defined by
some local relations, including some normalisations, a ‘traditional skein relation’ for the
braiding,
and a relation expressing a pair of n-valent vertices as a linear combination of braids
Easily, every closed graph evaluates to a number: the second relation above lets you re-
move all vertices, resulting in a linear combination of links, which can be evaluated via the
first relation. He further explains the connection with Murakami, Ohtsuki and Yamada’s
work, giving a formula for their evaluation in terms of his invariant.
Although his work does not expressly use the language of a category of diagrams,
it’s straightforward to make the translation. He implicitly defines a braided tensor cate-
gory, with objects generated by a single object, the unlabeled strand, and morphisms gen-
erated by caps, cups, crossings, and the n-valent vertices.
This category, with no more relations than he explicitly gives, ought to be equiv-
alent to the full subcategory of RepUq(sln) generated by tensor powers of the standard
representation. Note that this isn’t the same as the category FundRepUq(sln) used here;
it has even fewer objects, although again its Karoubi envelope is the entire representation
category. However, I don’t think that this equivalence is obvious, at least with the cur-
rently available results. Certainly he proves that there is a functor to this representation
category (by explicitly construct Uq(sln) equivariant tensors for the braiding, coming from
the R-matrix, and for the n-valent vertices). Even though he additionally proves that there
are no nontrivial quotients of his invariant, this does not prove that the functor to the rep-
resentation theory is faithful. Essentially, there’s no reason why ‘open’ diagrams, such as
appear in the Hom spaces of the category, shouldn’t have further relations amongst them.
Any way of closing up such a relation would have to result in a linear combination of
closed diagrams which evaluated to zero simply using the initially specified relations. Al-
ternatively, we could think of this as a question about‘nondegeneracy’: there’s a pairing
on diagrams, giving by gluing together pairs of diagrams with the same, but oppositely
oriented, boundaries. It’s C(q)-valued, since every closed diagram can be evaluated, but it
may be degenerate. That is, there might be elements of the kernel of this pairing which do
not follow from any of the local relations. See [25, §3.3] for a discussion of a similar issue
in sl3 Khovanov homology.
Perhaps modulo some normalisation issues, one can write down functors be-
tween his category and mine, at least before imposing relations. In one direction, send
an edge labeled k in my category to k parallel ribbons in Sikora’s category, and send each
vertex, with edges labeled a, b and c to either the incoming or outgoing n-valent vertex in
Sikora’s category. In the other direction, send a ribbon to an edge labeled 1, an n-valent
vertex to a tree of trivalent vertices, with n leaves labeled 1 (up to a sign it doesn’t mat-
ter, by the I = H relations, which tree we use), and a crossing to the appropriate linear
combination of the two irreducible diagrams with boundary ((1,+), (1,+), (1,−), (1,−)).
This suggests an obvious question: are the elements of the kernel of the repre-
sentation functor I’ve described generated by Sikora’s relations? They need not be, if the
diagrammatic pairing on Sikora’s category is degenerate.
Yokota’s work in [40] is along similar lines as Sikora’s, but directed towards giv-
ing an explicit construction of quantum SU(n) invariants of 3-manifolds.
Chapter 7
Future directions
Sad though it is to say, I’ve barely scratched the surface here. There are a number
of obvious future directions.
• Prove Conjecture 5.5; that the whole kernel of Rep is generated by the relations I’ve
described via tensor product and composition.
• Try to find a ‘confluent’ set of relations, according to the definition of [35]. There’s
no obvious basis of diagrams modulo the relations I’ve given, for n ≥ 4; this is
essentially just saying that the relations I’ve presented are not confluent.
• Find an evaluation algorithm; that is, describe how to use the relations given here to
evaluate any closed web. The authors of [11] claim to do so, using only the I = H
and ‘square-switch’ relations, although I have to admit not being able to follow their
proof.
• Complete the discussion from §6.5, explicitly giving the relationship between the
evaluation of a closed diagram in my category, and its evaluation in the MOY theory.
• In fact, the first Kekulé relation, appearing in Equation (5.3.1) at n = 4, follows from
the ‘square switch’ and ‘bigon’ relations, as follows. First, applying Equation (5.6.6)
across the middle of a hexagon, we see
and then, using the I = H relation to switch two of the edges in the first term, and
Equation (5.6.5) to resolve each of the squares in the second term,
q + [2]q + [2]q +
Next, we resolve the remaining squares using Equation (5.6.6), obtaining
 + + +
q + [2]q + [2]q +
and then resolve the newly created squares using Equation (5.6.5) again, and resolve
the newly created bigons using Equation (5.6.2)
1 + [3]q − [2]
q + 1
= + − .
Is there any evidence that this continues to happen? Perhaps all the Kekulé relations
follow in a similar manner from ‘square-switch’ relations?
• Try to do the same thing for the other simple Lie algebras. Each Lie algebra has a set
of fundamental representations, corresponding to the nodes of its Dynkin diagram.
You should first look for intertwiners amongst these fundamental representations,
until you’re sure (probably by a variation of the Schur-Weyl duality proof given here)
that they generate the entire representation theory. After that, describe some rela-
tions, and perhaps prove that you have all of them. I have a quite useful computer
program for discovering relations, which is general enough to attempt this problem
for any simple Lie algebra. On the other hand, the methods of proof used here, based
on the multiplicity free branching rule for sln, will need considerable modification.
For the notion of Gel‘fand-Tsetlin basis in the orthogonal and symplectic groups, see
[36].
• Describe the representation theory of the restricted quantum group at a root unity in
terms of diagrams; in particular find a diagrammatic expression for the idempotent
in ⊗aV maa<n which picks out the irrep of high weight
amaλa, for each high weight.
Kim solved this problem for Uq(sl3) in his Ph.D. thesis [22]. It would be nice to start
simply by describing the ‘lowest root of unity’ quotients, in which only the funda-
mental representations survive. After this, one might learn how to use these cate-
gories of trivalent graphs to give a discussion, parallel to that in [3], of the modular
tensor categories appearing at roots of unity.
• Categorify everything in sight! Bar-Natan’s work [1] on local Khovanov homology
provides a categorification of the Uq(sl2) theory. Khovanov’s work [18] on a foam
model for sl3 link homology categorifies the Uq(sl3) theory, although it’s only made
explicitly local in later work by Ari Nieh and myself [25]. Finding an alternative to
the matrix factorisation method [19, 20] of categorifying the Uq(sln) knot invariants,
based explicitly on a categorification of the Uq(sln) spiders, is a very tempting, and
perhaps achievable, goal!
If you have answers to any of these problems, I’d love to hear about them; if you have
partial answers, or even just enthusiasm for the problems, I’d love to work on them with
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Appendix A
Appendices
A.1 Boring q-binomial identities
In this section we’ll prove some q-binomial identities needed in various proofs in the body
of the thesis. The identities are sufficiently complicated that I’ve been unable to find com-
binatorial arguments for them. Instead, we’ll make use of the Zeilberger algorithm, de-
scribed in [31]. In particular, we’ll use the Mathematica‘ implementation in [30, 33], and
the implementation of the q-analogue of the Zeilberger algorithm described in [29, 32].
If you’re unhappy with the prospect of these identities being proved with com-
puter help, you should read Donald Knuth’s foreword in [31]
Science is what we understand well enough to explain to a computer. Art is ev-
erything else we do. During the past several years an important part of math-
ematics has been transformed from an Art to a Science: No longer do we need
to get a brilliant insight in order to evaluate sums of binomial coefficients, and
many similar formulas that arise frequently in practice; we can now follow a
mechanical procedure and discover the answers quite systematically.
For each use made of the the q-Zeilberger algorithm, I’ve included a Mathematica
notebook showing the calculation. These are available in the /code/ subdirectory, after
you’ve downloaded the source of this thesis from the arXiv.
Unfortunately, the conventions I’ve used for q-binomials don’t quite agree with
those used in the implementation of the q-Zeilberger algorithm. They are related via
qk(k−n)
]qZeil
, although in my uses of their algorithms, I’ve replaced q2 with q throughout;
this apparently produces cleaner looking results.
Lemma A.1.1. For Σb ≤ j ≤ Σa+ n− 1 and −
b ≤ j∗ ≤ −
min(−
a+1,j∗−j+n+Σa)
k=max(−
b,Σb+j∗−j)
(−1)j+k
j + k −max b
j −Σb
min a+ n− j − k
Σa+ n− 1− j
n+Σa− Σb
j − j∗ + k − Σb
= 0. (A.1.1)
Proof. Writing Σn for the sum above (suppressing the dependence on Σa,Σb,min a,max b, j
and j∗), the q-Zeilberger algorithm reports (see /code/triple-identity.nb/) that
a+Σb−j+j∗−1)
(1− q(2+2n+2min a−2max b))
1− q(2Σa+2
b−2j+2j∗+2n)
as long as
n+Σa− Σb 6= 0 and n+Σa+
b− j + j∗ 6= 0. (A.1.2)
In the case that n+Σa−Σb = 0, the top of the third q-binomial in Equation (A.1.1) is zero,
so the sum is automatically zero unless k = Σb+ j∗ − j, in which case the sum collapses to
(−1)Σb+j∗
Σb+ j∗
j − Σb
a− j∗
Σb− 1− j
However, since j ≥ Σb, the bottom of the second q-binomial here is less than zero, so the
whole expression vanishes.
In the case that n+Σa+
b−j+j∗ = 0, we can replace n everywhere in Equation
(A.1.1), obtaining
(−1)j+k
j + k −max b
j − Σb
b− j∗ − k
b− j∗ − 1
b+ j − j∗
j − j∗ − k − Σb
Again, the second q-binomial vanishes, since j∗ ≥ −
Now that we’re sure the identity holds when either of the inequalities of Equation
(A.1.2) are broken, we can easily establish the identity when they’re not; if it holds for some
value of n, it must hold for n + 1 (notice that the restriction on j in the hypothesis of the
lemma doesn’t depend on n). Amusingly, the easiest starting case for the induction is not
n = 0, but n sufficiently large and negative, even though this makes no sense in the original
representation theoretic context! In particular, at n ≤ −Σa+ Σb− 1, the third binomial in
Equation (A.1.1) is always zero, so the sum vanishes.
For the next lemma, we’ll need q-Pochhammer symbols,
(a; q)k =
j=0(1− aqj) if k > 0,
1 if k = 0,
j=1(1− aq−j)−1 if k < 0.
In particular, note that (q; q)
k<0 = 0, and (q; q)k>0 6= 0.
Lemma A.1.2. For any n,m,m′ ∈ Z, and a = (a1, a2), b = (b1, b2) ∈ Z2, with n+Σa−Σb > 0
δmm′ − (−1)m+n+Σa
n+mina
l=n+Σa−min b
(−1)l
m+ l − 1− Σb
m+ l − n− Σa
n+Σa−Σb
m′ + l − Σb
Proof. First, notice that the expression is invariant under the transformation adding some
integer to each of m,m′, l, a1, a2, b1, b2. We’ll take advantage of this to assume Σb + 2m −
1,Σb+m+m′−1, 2Σb+2m−1−n−Σa, min a+m+n, min a+m′+n, and Σb−maxa+m
are all positive, and −2Σb−m−m′ + n− Σa and max a− Σb−m′ − 1 are both negative.
Again, writing Σn for the left hand side of the above expression, the q-Zeilberger
algorithm reports (see /code/SS-identity.nb/) that as long as m 6= m′,
Σn = (−1)•q•
(q; q)Σb+2m−1
(qm − qm′) (q; q)Σb+m+m′−1 (q; q)2Σb+2m−1−n−Σa (q; q)−2Σb−m−m′+n−Σa
+ (−1)•q•
(q; q)mina+m+n
(qm − qm′) (q; q)mina+m′+n (q; q)Σb−max a+m (q; q)max a−Σb−m′−1
This is exactly zero, using the inequalities described in the previous paragraph, and the
definition of the q-Pochhammer symbol.
When m = m′, the q-Zeilberger algorithm reports that
Σn = (−1)•q•
(qn+Σa−Σb − 1)
(qmin a+m+n − 1) (q; q)Σb−max a+m (q; q)−Σb+max a−m
+Σn−1. (A.1.3)
The first term is zero, since −Σb+max a−m < 0, so (q; q)−1−Σb+max a−m = 0.
Thus we just need to finish off the case m = m′, n + Σa− Σb = 1, where the left
hand side of Equation (A.1.2) reduces to
δmm′ − (−1)m+Σb+1
(−1)l
m+ l − 1− Σb
m+ l − 1− Σb
m′ + l − Σb
There are then two terms in the summation, l = Σb−m′ and l = Σb−m′ +1, so we obtain
δmm′ − (−1)m−m
m−m′ − 1
m−m′ − 1
− (−1)m−m′
If m = m′, the first q-binomial vanishes, but the second is 1, while if m > m′, both q-
binomials are equal to 1, and cancel, and if m < m′, both q-binomials vanish.
A.2 The Uq(sln) spider cheat sheet.
A.2.1 The diagrammatic category, §2.2.
Generators.
, , ,
for each a = 1, . . . , n− 1,
, , , and
for each a = 1, . . . , n− 1, and
for 0 ≤ a, b, c ≤ n, with a+ b+ c = n.
Relations: planar isotopy.
Relations: (anti-)symmetric duality.
= (−1)(n+1)a = (−1)(n+1)a
Relations: degeneration.
A.2.2 The map from diagrams to representation theory, §3.6.
 = Rep
= Rep
= (−1)(n+1)a Rep
= (−1)(n+1)a
= Rep
= Rep
A.2.3 The diagrammatic Gel‘fand-Tsetlin functor, §4.
Pairings and copairings.
dGT ′
= + dGT ′
= qn−a + q−a
dGT ′
= + dGT ′
= qa−n + qa ,
(4.1.2)
Identifications with duals.
dGT ′
= + (−1)aq−a
dGT ′
= (−1)n+a + q−a
dGT ′
= qa + (−1)n+a
dGT ′
= (−1)aqa + . (4.1.3)
Curls.
= qn−a + q−a dGT
= qa−n + qa
Trivalent vertices.
dGT ′
 = (−1)cqb+c + (−1)aqc + (−1)b ,
dGT ′
 = (−1)c + (−1)aq−a + (−1)bq−a−b
 = (−1)a
 + (−1)n+b + q−a
 = (−1)a
+ (−1)n+bqb +
= (−1)a
+ (−1)n+bqb +
= (−1)a
qa + + (−1)n+bqa−n
Larger webs.
 = +
+ q−a + +
+ q−a + (−1)b+cqb (4.3.1)
= qb +
+ + +
+ qb + (−1)a+cqb+c−n (4.3.2)
P’s and Q’s, §2.4.
dGT a′,b′(Pna,b;l) = (−1)b
′·(a+rotl a)ql(Σb
′−Σa′+1)qrotl a
′·b−b′·a−a1−na
1Pn−1
a+a′,b+b′;l
dGT a′,b′(Qna,b;l) = (−1)b
′·(a+rotl a)ql(Σa
′−Σb′−
+1)qΣb+nΣb
′+b′·rotl a−a′·b−a1−na
1Qn−1a+a′,b+b′;l
dGT ∅(Pna,b;l) = dGT −→0 ,−→0 (P
a,b;l) + dGT −→1 ,−→1 (P
a,b;l)
= ql−a1
Pn−1a,b;l + q
−n−Σa+ΣbPn−1a,b;l−1
dGT ∅(Qna,b;l) = dGT −→0 ,−→0 (Q
a,b;l)
= ql−a1qΣb−
2 Qn−1a,b;l
dGT ∂(Pna,b;l) = dGT −→1 ,−→0 (P
a,b;l)
= ql−a1qΣb−
−nPn−1
1 ,b;l
dGT ∂(Qna,b;l) = dGT −→0 ,−→−1(Q
a,b;l) + dGT −→1 ,−→0 (Q
a,b;l)
= ql−a1
qΣb−Σa−
2 Qn−1
1 ,b;l+1
+ q−nQn−1
1 ,b;l
A.2.4 Conjectural complete set of relations, §5.
The I = H relations
= (−1)(n+1)a ,
= (−1)(n+1)a .
The ‘square-switching’ relations
SSna,b =
spanA
Pna,b;l −
min b
m=max a
n+Σa− Σb
m+ l − Σb
Qna,b;m
}mina+n
l=max b
if n+Σa− Σb ≥ 0
spanA
Qna,b;l −
n+mina
m=max b
Σb− n− Σa
m+ l − Σa− n
Pna,b;m
}min b
l=maxa
if n+Σa− Σb ≤ 0
The Kekulé relations
APRna,b = spanA
a,b;j =
(−1)j+k
j + k −max b
j − Σb
min a+ n− j − k
Σa+ n− 1− j
Pna,b;j+k
Σa+n−1
(A.2.1)
AQRna,b = spanA
a,b;j =
(−1)j+k
j + k −max a
j − Σa
min b− j − k
Σb− n( |∂|
− 1)− 1− j
Qna,b;j+k
Σb−n(
−1)−1
(A.2.2)
with orthogonal complements
APRna,b⊥ = spanA
a,b;j∗ =
k∗=Σb
n+Σa− Σb
k∗ − Σb
Pna,b;j∗+k∗⊥
AQRna,b⊥ = spanA
a,b;j∗
k∗=Σa
Σb− Σa− n
k∗ − Σa
×Qna,b;j∗+k∗⊥
(A.2.3)
	Introduction
	Summary
	The Temperley-Lieb algebras
	Kuperberg's spiders
	The `diagrammatic' category Symn
	Pivotal categories
	Quotients of a free tensor category
	Flow vertices
	Polygonal webs
	Just enough representation theory
	The Lie algebra sln
	The quantum groups Uq(sln)
	Representations
	Strictifying Rep Uq(sln)
	Generators for FundRepUq(sln)
	The representation functor
	The diagrammatic Gel`fand-Tsetlin functor
	Definition on generators
	Descent to the quotient
	Calculations on small webs
	A path model, and polygons.
	Describing the kernel
	The I=H relations
	The square-switch relations
	The Kekulé relations
	More about squares
	A conjecture
	Examples: Uq(sln), for n=2,3,4 and 5.
	Proofs of Theorems 5.1.1, 5.2.1 and 5.3.2
	Relationships with previous work
	The Temperley-Lieb category
	Kuperberg's spider for Uq(sl3)
	Kim's proposed spider for Uq(sl4)
	Tags and orientations
	Murakami, Ohtsuki and Yamada's trivalent graph invariant
	Jeong and Kim on Uq(sln)
	Other work
	Future directions
	Bibliography
	Appendices
	Boring q-binomial identities
	The Uq(sln) spider cheat sheet.
