REPRESENTATIONS OF THE
ORIENTED BRAUER
CATEGORY
by
ANDREW REYNOLDS
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2015
DISSERTATION APPROVAL PAGE
Student: Andrew Reynolds
Title: Representations of the Oriented Brauer Category
This dissertation has been accepted and approved in partial fulfillment of the requirements for the
Doctor of Philosophy degree in the Department of Mathematics by:
Jonathan Brudan Chair
Alexander Kleshchev Core Member
Victor Ostrik Core Member
Marcin Bownik Core Member
Dejing Dou Institutional Representative
and
Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate School.
Degree awarded June 2015.
ii
c©2015, Andrew Reynolds.
iii
DISSERTATION ABSTRACT
Andrew Reynolds
Doctor of Philosophy
Department of Mathematics
June 2015
Title: Representations of the Oriented Brauer Category
We study the representations of a certain specialization OB(δ) of the oriented Brauer cate-
gory in arbitrary characteristic p. We exhibit a triangular decomposition of OB(δ), which we use to
show its irreducible representations are labelled by the set of all p-regular bipartitions. We then ex-
plain how its locally finite dimensional representations can be used to categorify the tensor product
V (−$m′)⊗V ($m) of an integrable lowest weight and highest weight representation of the Lie alge-
bra slk. This is an example of a slight generalization of the notion of tensor product categorification
in the sense of Losev and Webster and is the main result of this paper. We combine this result with
the work of Davidson to describe the crystal structure on the set of irreducible representations. We
use the crystal to compute the decomposition numbers of standard modules as well as the charac-
ters of simple modules assuming p = 0. We give another proof of the classification of irreducible
modules over the walled Brauer algebra. We use this classification to prove that the irreducible
OB(δ)-modules are infinite dimensional unless δ = 0, in which case they are all infinite dimensional
except for the irreducible module labelled by the empty bipartition, which is one dimensional.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Andrew Scott Reynolds
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon
University of California, Los Angeles
DEGREES AWARDED:
Doctor of Philosophy, University of Oregon, 2015
Master of Science, University of Oregon, 2014
Bachelor of Science, University of California, Los Angeles, 2010
AREAS OF SPECIAL INTEREST:
Combinatorial representation theory and categorification
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, 2010 - 2015
GRANTS, AWARDS AND HONORS:
Johnson Award, Department of Mathematics, University of Oregon, Eugene, 2014.
Sherwood Award, Department of Mathematics, University of California, Los Angeles, 2010.
PUBLICATIONS:
J. Brundan, J. Comes, D. Nash and A. Reynolds, A basis theorem for the affine
oriented Brauer category and its cyclotomic quotients, to appear in Quantum Top.;
arXiv:1404.6574.
v
ACKNOWLEDGMENTS
I wish to thank my advisor, Jon Brundan, for his directness and endless patience. His careful
and thorough instruction over the years has been invaluable.
I thank the members of my Ph.D. committee, who have never accepted less than my best
work. I also thank the many influential professors I had before beginning graduate school, notably
Andrea Colbaugh and Curtis Paul, who convinced me to pursue mathematics further.
I cannot give enough thanks to my wife, Kaeli. I have relied on her support throughout my
graduate schoold experience, and could not have succeeded without her.
vi
DEDICATION
To Ron Clayborn (1955 - 2014). Rock on!
vii
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II.1. The Symmetric Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II.2. Locally Unital Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
II.3. The Lie Algebra slk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
II.4. Quotient Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II.5. Standard Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
III. THE ALGEBRA OB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
III.1. Oriented Brauer Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
III.2. Triangular Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
III.3. Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
III.4. OB0 and the Symmetric Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
III.5. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
III.6. Classification of Simple Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
IV. BRANCHING RULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
IV.1. Functors E,F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
IV.2. Analogues for OB0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
IV.3. Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
V. TENSOR PRODUCT CATEGORIFICATION . . . . . . . . . . . . . . . . . . . . . . . . 40
V.1. Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
V.2. Projective Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
V.3. Categorical Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
V.4. Crystal Graph Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
VI. APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
VI.1. Decomposition Numbers in Characteristic 0 . . . . . . . . . . . . . . . . . . . . . 50
VI.2. Characters of Simple Modules in Characterstic 0 . . . . . . . . . . . . . . . . . . 55
VI.3. Representations of the Walled Brauer Algebra . . . . . . . . . . . . . . . . . . . 58
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
viii
LIST OF FIGURES
Figure Page
1. Branching graph when p = m = m′ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2. Crystal graph when m = m′ = 0, p = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3. Illustration of k-value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. Well-oriented composite diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5. Markers of the composition factors of a standard module. . . . . . . . . . . . . . . . . . . 55
6. Decomposition numbers of a standard module. . . . . . . . . . . . . . . . . . . . . . . . . 56
7. Branching graph (computation of character of standard module). . . . . . . . . . . . . . 58
ix
CHAPTER I
INTRODUCTION
Let k be an algebraically closed field of characteristic p ≥ 0. The oriented Brauer category
OB is the free k-linear symmetric monoidal category generated by a single object ↑ and its dual ↓.
Its objects are finite sequences of the symbols ↑, ↓, including the empty sequence ∅, which is the
unit object. The set of such sequences is denoted 〈↓, ↑〉. If a, b ∈ 〈↓, ↑〉, then the space HomOB(a, b)
has a basis consisting of oriented Brauer diagrams with bubbles of type a → b. Such a diagram is
obtained by drawing the sequence a below b and pairing the vertices ↑, ↓ by drawing strings in the
space between a, b. The strings are allowed to cross, and may connect any two vertices as long as
they induce an orientation on the string. For example, a pair ↑, ↑ may be connected by a string if
and only if one belongs to a and the other to b; and a pair ↑, ↓ may be connected if and only if
they both belong to a or both belong to b. We shall refer to strings which pair two vertices from
the bottom row caps and those which pair two vertices from the top row cups. All other strings
will be called vertical strings. Additionally, there may be some number of closed (oriented) curves
(called bubbles) in the space between a, b. Two such diagrams D1, D2 are equivalent if they have
the same number of bubbles and the remaining strings partition the vertices in a and b in the same
way. That is, the apparent topological structure of oriented Brauer diagrams is irrelevant; they
are a convenient way of visually representing combinatorial information (see section III.1 for their
combinatorial definition). For example, the following two oriented Brauer diagrams with bubbles of
type ↑↓↓→↑↓↓ are equivalent:
=
Given two such diagrams D1, D2, their composition D1 ◦D2 is given by vertically stacking
D1 on top of D2 to get another such diagram with bubbles. The identity morphism of an object a
1
is simply a row of parallel vertical strings, orientations determined by a.
The k-linear category OB is actually linear over the polynomial algebra over k by letting
the polynomial generator act by adding a bubble. By fixing δ ∈ k and specializing the polynomial
generator at δ we obtain a k-linear monoidal category OB(δ). This amounts to introducing a relation
to the presentation of OB identifying a bubble with δ ·∅, where ∅ denotes the empty diagram, which
is the identity morphism of the identity object. So HomOB(δ)(a, b) has basis consisting of oriented
Brauer diagrams of type a→ b without bubbles, and composition is performed by vertically stacking
diagrams and removing any bubbles formed, multiplying by δ for each bubble removed.
The algebra EndOB(δ)(↓r↑s) is isomorphic to the walled Brauer algebra Br,s(δ), which was
introduced independently by Turaev [T] and Koike [Ko] in the late 1980s, motivated in part by a
Schur-Weyl duality between Br,s(δ) and GLm(C) arising from mutually commuting actions on the
“mixed” tensor space V ⊗r ⊗W⊗s, where V is the natural representation of GLm(C) and W = V ∗.
The walled Brauer algebra is spanned by walled Brauer diagrams which are obtained by drawing
two rows of r + s vertices, one above the other, and drawing strings between pairs of vertices. We
imagine a wall separating the first r vertices and the last s vertices in both rows. We require that
the endpoints of each string are either on the same row of vertices and opposite sides of the wall,
or else they are on opposite rows and the same side of the wall. So forgetting orientations defines a
linear isomorphism EndOB(δ)(↓r↑s)→ Br,s(δ). The multiplication in Br,s(δ) is such that this map is
an algebra isomorphism. It is worth noting that the Karoubi envelope of OB(δ) is Deligne’s tensor
category Rep(GLδ) (see [CW]).
The main goal of this paper is to show how OB(δ) can be used to categorify a tensor product
of representations of the Lie algebra slk. The Lie algebra slk is defined as the Kac Moody Lie algebra
associated to the graph with vertices k and an edge between i and i + 1 for each i ∈ k. Writing
δ = m −m′ for m,m′ ∈ k we will ultimately be able to categorify V (−$m′) ⊗ V ($m), the tensor
product of an integrable lowest weight representation and an integrable highest weight representation
of lowest (resp. highest) weight −$m′ (resp. $m), where $i is the ith fundamental dominant weight
of slk (see section II.3).
For convenience, we often replace OB(δ) with a locally unital algebra OB(δ) whose represen-
tations are equivalent to those of OB(δ), and we suppress δ in the notation, writing OB for OB(δ).
Briefly, a locally unital algebra A is a nonunital algebra with a distinguished collection of mutually
orthogonal idempotents {1i : i ∈ I} satisfying A =
⊕
i,j∈I 1iA1j . It is locally finite dimensional
if each 1iA1j is finite dimensional. In our situation I = 〈↓, ↑〉, 1a is the identity morphism of the
2
object a, and
1aOB1b = HomOB(δ)(b, a),
which is finite dimensional. The multiplication making OB =
⊕
a,b∈〈↓,↑〉 1aOB1b into a locally finite
dimensional locally unital algebra is induced by composition inOB(δ). By a locally finite dimensional
module over a locally unital algebra A, we mean an A-module V satisfying V =
⊕
i∈I 1iV and
dim 1iV <∞ for i ∈ I. The category of such modules is denoted A -mod.
Let K =
⊕
a∈〈↓,↑〉 k ·1a. The first important observation made in this thesis is that OB has a
triangular decomposition OB = OB−⊗KOB0⊗KOB+ (see section III.2), where OB+ (resp. OB−)
is the subalgebra spanned by all diagrams with no cups (resp. caps) and no crossings among vertical
strings. The role of the Cartan subalgebra is played by OB0, which is the subalgebra spanned by
diagrams with no cups or caps. We observe in section III.4 that OB0 is Morita equivalent to
⊕
r,s≥0
kSr ⊗ kSs. (I.0.0.1)
Thus the simple modules of OB0 are parametrized by the set Λ of p-regular bipartitions, that is,
pairs of partitions which have no p rows of the same length (or simply all pairs of partitions if
p = 0). We use this triangular decomposition to define the standard modules by analogy with the
Verma modules. Explicitly, we use a natural projection OB0 ⊗K OB+ → OB0 to inflate the action
of OB0 on its projective indecomposable modules Y (λ), λ ∈ Λ, to an action of OB0 ⊗K OB+, and
then induce to an action of OB to obtain the standard module ∆(λ). This construction (inflation
followed by induction) defines the standardization functor ∆ : OB0 -mod→ OB -mod.
Mimicking standard arguments from Lie theory, we show in section III.6 that the standard
modules have unique irreducible quotients, giving a complete set of inequivalent irreducible OB-
modules {L(λ) : λ ∈ Λ}. In section III.3 we define a preorder on the set Λ. This preorder is
essentially a version of the “inverse dominance order” of Losev and Webster (see Definition 3.2 of
[LW]). It should be viewed as the appropriate analog of the Bruhat order from Lie theory. We prove
the following theorems in sections IV.3 and V.2, respectively.
Theorem. If L(λ) arises as a subquotient of ∆(µ) then λ ≤ µ.
Theorem. The projective cover P (λ) of L(λ) surjects onto ∆(λ) with kernel having a finite filtration
with sections ∆(µ) with µ > λ.
This essentially means that the category OB -mod is locally standardly stratified, which is a
mild weakening of the notion of a standardly stratified category introduced by Losev and Webster in
3
[LW] as part of the structure required for a tensor product categorification. The remaining data of
a tensor product categorification is a categorical action in the sense of Rouquier (see [R]). Roughly,
this means that for each i ∈ k there are biadjoint endofunctors Ei, Fi of OB -mod which induce
an action of slk on the split Grothendieck group of the category of finitely generated projective
OB-modules. Additionally, these functors need to be equipped with some endomorphisms (natural
transformations) satisfying certain relations as prescribed by Rouquier.
The notion of a categorical action of a Kac-Moody algebra g was first defined in a paper
of Chuang and Rouquier, [CR] in the case g = sl2. The general case was defined in [R] and also
in the work of Khovanov and Lauda, [KL1]-[KL3]. This new direction was first motivated by the
observation that many categories occurring in representation theory, such as the representations of
symmetric groups, of Hecke algebras, of the general linear groups or of Lie algebras of type A have
endofunctors that on the level of the Grothendieck group give actions of Kac-Moody Lie algebras of
type A. These ideas led Chuang and Rouquier first to the definition of categorical actions of type
A algebras, which they then generalized to Kac-Moody algebras of arbitrary type.
To construct the categorical slk-action on OB -mod we begin by defining endofunctors E,F
of OB -mod in section IV.1 as tensoring with certain bimodules. The functors E,F are analogous to
the induction and restriction functors for the symmetric group. In fact, using obvious notation for
induction and restriction between products of symmetric groups (see I.0.0.1), we define endofuctors
of OB0 -mod
E↑ =
⊕
r,s≥0
resr,s+1r,s E
↓ =
⊕
r,s≥0
indr+1,sr,s
F ↑ =
⊕
r,s≥0
indr,s+1r,s F
↓ =
⊕
r,s≥0
resr+1,sr,s
and we show in Theorem IV.2.1 that we have short exact sequences of functors
0→ ∆ ◦ E↑ → E ◦∆→ ∆ ◦ E↓ → 0
0→ ∆ ◦ F ↓ → F ◦∆→ ∆ ◦ F ↑ → 0.
To split the functors E,F into direct sums of Ei, Fi, we next construct certain endomor-
phisms of the bimodules defining E,F , and define Ei, Fi to be the generalized i-eigenspace of E,F ,
respectively. These endomorphisms come naturally from the affine oriented Brauer category AOB,
which is obtained from OB by adjoining an additional monoidal generator, an endomorphism of ↑,
4
along with an extra relation. If we depict this morphism diagrammatically as a dot on a string
oriented upward, then morphisms in AOB can be unambiguously be represented as dotted oriented
Brauer diagrams, i.e. oriented Brauer diagrams with some nonnegative number of dots on each
segment of each string (including bubbles). In particular, we view OB as a subcategory of AOB.
The extra relation imposed in AOB is depicted diagrammatically as
• = • +
This relation comes from the degenerate affine Hecke algebra Hn. If Sn is represented by
permutation diagrams and the polynomial generator xi is represented as a dot on string i, then it
is easy to see that EndAOB(↑n) is isomorphic to the degenerate affine Hecke algebra Hn. For any
choice of m ∈ k, there is a full functor AOB → OB which restricts to the identity functor on the
subcategory OB and sends a dot on an ↑-string on the leftmost boundary to m (Theorem III.1.4).
Note that with the identifications EndAOB(↑n) = Hn and EndOB(↑n) = kSn, a dot on the ith string
is sent to m + Li, where Li is the i
th Jucys-Murphy element. We can use this functor to interpret
dotted diagrams as morphisms in OB.
The Jucys-Murphy elements split the induction and restriction functors F ↓, E↓, E↑, F ↑ into
generalized eigenspaces:
E↑ =
⊕
i∈k
E↑i E
↓ =
⊕
i∈k
E↓i
F ↑ =
⊕
i∈k
F ↑i F
↓ =
⊕
i∈k
F ↓i .
It follows from [Groj] that the functors F ↓i , E
↓
i , E
↑
i , F
↑
i give a categorical sl
↓
k⊕sl↑k-action on OB0 -mod
under the assignments
e↓i 7→ E↓i f↓i 7→ F ↓i
e↑i 7→ E↑i f↑i 7→ F ↑i
(see Theorem V.3.1).
As a vector space, the bimodule defining the functor F is
⊕
a∈〈↓,↑〉OB1a↑. The endomor-
phism of this bimodule which we use to split F is given on OB1a↑ by multiplying by a dot on the
bottom right ↑. The endomorphism for E is defined similarly, using a dot on a ↓ string.
5
Theorem. The above short exact sequences split:
0→ ∆ ◦ E↑i → Ei ◦∆→ ∆ ◦ E↓i → 0
0→ ∆ ◦ F ↓i → Fi ◦∆→ ∆ ◦ F ↑i → 0.
We use these split sequences to compute the formal characters of the standardized Specht
modules ∆˜(λ), that is, the modules obtained by applying the standardization functor ∆ to the Specht
modules for OB0. The coefficients are the numbers of paths of various types in the branching graph.
The vertices of the branching graph are all bipartitions and there is an edge between bipartitions λ
and µ whenever µ = (µ↓, µ↑) is obtained by adding a box to λ = (λ↓, λ↑). If the box is added to row
i and column j of λ↑, then the edge is colored m+ j − i, read mod p. If the box is added to row i
and column j of λ↓, then the edge is colored m′ + i− j, mod p (see section IV.3).
We now state our main theorem (Theorem V.3.2).
Theorem. The endofunctors Ei, Fi of OB -mod together with certain natural transformations de-
fined in detail in section IV.2 define a categorical slk-action. This action is compatible with the
locally stratified structure on OB -mod and makes OB -mod into a (generalized) tensor product cat-
egorification of V (−$m′)⊗ V ($m) in the sense of [LW].
This theorem allows us to apply the main result of Davidson [D] which implies that EiL(λ)
is either zero or else its head and socle are both isomorphic to some simple module parameterized by
some bipartition e˜iλ. This result of Davidson is a generalization of a result of Chuang and Rouquier
[CR], which in turn extended ideas of Grojnowski, Vazirani and Kleshchev (see [K]). Letting L(e˜iλ)
denote the zero module in the event that EiL(λ) = 0, we then have head(EiL(λ)) = L(e˜iλ) for all
i ∈ k and λ ∈ Λ. A similar statement holds for FiL(λ): head(FiL(λ)) = L(f˜iλ). The main results
of this paper and [D] also enable us to compute the bipartitions e˜iλ, f˜iλ explicitly, which we do in
section V.4. The graph whose vertices are Λ with an edge colored i between λ and f˜iλ whenever
f˜iλ 6= 0 is called the crystal graph. It is the Kashiwara tensor product of the crystals associated to
the slk-modules V (−$m′), V ($m), which are known (see section 11.1 of [K]). In characteristic zero
we are able to use the crystal to describe the actions of Ei, Fi on the projective covers of the simple
modules, which we require for our proof of the following theorem (see section VI.1).
Theorem. If p = 0 the composition multiplicities of standard modules and the characters of the
simple modules can be computed by explicit combinatorics as described in chapter VI of the present
thesis.
6
As a final application we recover the classification of irreducible Br,s(δ)-modules, which was
first proved by A. Cox, M. De Visscher, S. Doty and P. Martin, (see [CDDM]). We then use this
classification to show that L(λ) is (globally) finite dimensional if and only if δ = 0 and λ = (∅,∅),
in which case it is one dimensional.
Organization of thesis
In Chapter II we recall some classical theory and introduce some slight modifications of
standard notions. We recite some facts about the representation theory of the symmetric group,
which is central to our approach. Next we define locally unital algebras and develop some theory
including versions of Schur’s Lemma and the Krull-Schmidt theorem. We also define the Lie algebra
slk, which is the Kac-Moody algebra underlying the categorical action we present here. We then
recall some of the classical theory of quotient functors and then discuss a weakened version of
standard stratification (see [LW]).
Chapter III introduces our main object of study, the oriented Brauer category. We describe
its triangular decomposition, which enables us to use some techniques from Lie theory. We show in
section III.4 that its Cartan subalgebra is Morita equivalent to the direct sum of all group algebras
of products of two symmetric groups. After defining a duality functor on a certain category of
representations of the oriented Brauer category, we prove the classification of its simple modules.
The endofunctors E,F of OB -mod leading to the categorical slk-action are defined in chap-
ter IV. We also define the endofunctors E↑, E↓, F ↑, F ↓ of OB0 -mod and explain their relation to
E,F , namely the short exact sequences mentioned above. We use this relation to compute the formal
characters of the standardized Specht modules in terms of the branching graph, which is defined in
section IV.3.
In chapter V we prove our main theorem. First we show that OB -mod is standardly
stratified in sections V.1 and V.2. We then explain the compatibility of this stratified structure with
the categorical slk-action in section V.3. The description of the crystal graph is given in section V.4.
In the final chapter of this thesis we use the above crystal to determine the composition
multiplicities of the standard modules and the characters of the simple modules, assuming p = 0.
Finally, we deduce the known classification of simple Br,s(δ)-modules due to Cox et al. (see [CDDM])
and use it to prove L(λ) is (globally) finite dimensional if and only if δ = 0 and λ = (∅,∅), in which
case it is one dimensional.
7
CHAPTER II
PRELIMINARIES
In this chapter we gather some basic facts which we shall need throughout this paper. In
section II.1 we recall some classical results about the representation theory of the symmetric group.
In section II.2 we define the notion of a locally unital algebra and develop some basic theory, similar
to that of unital algebras. Next we introduce the Lie algebra slk and its weights in section II.3. We
recall some basic facts about quotient functors and idempotent truncations in section II.4. Finally,
we define locally stratified categories in section II.5.
II.1. The Symmetric Group
Let Sn denote the symmetric group on n letters and kSn -mod the category of finite dimen-
sional kSn-modules. A central character is an algebra homomorphism χ : Z(kSn)→ k. The central
characters split kSn -mod into blocks:
kSn -mod =
⊕
χ
kSn -mod[χ],
where kSn -mod[χ] consists of those Sn-modules V such that (z − χ(z))NV = 0 for N  0 (see
section 1.1 in [K]).
We identify a partition with its Young diagram as usual. We use the English convention so
that the top row represents the first part of the partition. For p > 0, a partition λ is called p-regular
if for any k > 0 we have #{j : λj = k} < p. In terms of the Young diagram, this means that no
p rows have the same length. By definition, all partitions are 0-regular. Let Pp(n) be the set of
p-regular partitions of n.
Given a partition λ of n, we define the content of a box in row i and column j to be
j − i (mod p). The content of λ is the tuple cont(λ) = (γi)i∈Z·1k , where γi is the number of boxes of
content i.
For any partition λ of n there is an explicit construction (see [J]) of a finite dimensional
8
kSn-module S(λ), the corresponding Specht module. Assuming λ is p-regular, it is known that S(λ)
has irreducible head, which we denote by D(λ).
Theorem II.1.1. The modules {D(λ) : λ ∈ Pp(n)} form a complete set of inequivalent irreducible
kSn-modules. Moreover, for λ, µ ∈ Pp(n) we have:
(i) D(λ) is self-dual;
(ii) modules D(λ) and D(µ) belong to the same block of kSn if and only if cont(λ) = cont(µ).
Proof. This is part of Theorem 11.2.1 in [K].
Since kSn is a symmetric algebra, the projective cover of D(λ) is isomorphic to its injective
envelope, the corresponding Young module Y (λ). Actually Young modules are defined for any
partition, but shall only need those associated to p-regular partitions which happen to be projective
covers and injective envelopes as previously mentioned.
Define the kth Jucys-Murphy element Lk ∈ kSn by
Lk :=
∑
1≤m<k
(m, k).
Embedding kSn−1 ⊂ kSn with respect to the first n− 1 letters, it is easy to see that Ln commutes
with kSn−1. In particular, the Jucys-Murphy elements commute with each other. Now let i =
(i1, . . . , in) ∈ kn and let V be a finite dimensional kSn-module. We let Vi denote its simultaneous
generalized eigenspace of L1, . . . , Ln corresponding to the tuple of eigenvalues i. That is,
Vi = {v ∈ V : (Lk − ik)Nv = 0 for k = 1, . . . , n, and N  0}.
Now define the formal character of a kSn-module to be the following element of the free
Z-module on basis {ei : i ∈ k}.
chV :=
∑
i∈kn
(
dimVi
)
ei.
Lemma II.1.2. The characters of the irreducible kSn-modules are Z-linearly independent.
Proof. This is Lemma 11.2.5 in [K].
II.2. Locally Unital Algebras
By a locally unital k-algebra we mean a non-unital associative k-algebra A containing a
distinguished collection of mutually orthogonal idempotents {1i : i ∈ I}, for some index set I, such
9
that
A =
⊕
i,j∈I
1iA1j . (II.2.0.1)
We can build a locally unital k-algebra A out of a small k-linear category A as follows. Set
A =
⊕
a,b∈ob(A)
HomA(a, b) (II.2.0.2)
and define multiplication of two composable morphisms to be their composition, and let the product
of two non-composable morphisms be zero. It is easy to see that this makes A into a locally unital
k-algebra, with the identity morphisms in A serving as the system of idempotents. For example, we
have 1aA1b = HomA(b, a) so that (II.2.0.1) is satisfied.
A homomorphism of locally unital algebras is an algebra homomorphism which sends dis-
tinguished idempotents to distinguished idempotents. This corresponds to the notion of a k-linear
functor A → B if A,B are built from A,B as above.
By a module over a locally unital k-algebra A =
⊕
i,j∈I 1iA1j , we will always mean a locally
unital module, that is, a module V such that
V =
⊕
i∈I
1iV. (II.2.0.3)
We denote by A -Mod the category of all such modules. The nonzero spaces 1iV are called weight
spaces, and the nonzero elements of 1iV are called weight vectors. We note that if A is built out of a
small k-linear category A as above then A -Mod is equivalent to the category of representations of A,
ie. k-linear functors A → k -Vec to the category of k-vector spaces. A module V is said to be locally
finite dimensional if each 1iV is finite dimensional. The category of all locally finite dimensional
A-modules will be denoted A -mod. A locally unital algebra A is called locally finite dimensional if
each 1iA1j is finite dimensional.
Suppose A is a locally unital algebra, and V is a finitely generated A-module with generators
v1, . . . , vn. Since A is locally unital, 1ivj is nonzero for only finitely many choices of i, j, and vj is
the sum of the nonzero 1ivj . So we may assume v1, . . . , vn are weight vectors. Let V0 denote their
k-linear span in V . Then V0 is a homogeneous generating subspace, that is, a subspace V0 satisfying
V0 =
⊕
i∈I 1iV0 which generates V . We have shown that V has a finite dimensional homogeneous
generating subspace.
Proposition II.2.1. If A is locally finite dimensional, then every finitely generated A-module is
10
locally finite dimensional.
Proof. Choose a finite dimensional homogeneous generating subspace V0 of V . Then V is a quotient
of the finitely generated projective module
P =
⊕
i∈I
A1⊕ dim 1iV0i . (II.2.0.4)
In particular, 1jV is the image of 1jP =
⊕
i∈I 1jA1
⊕ dim 1iV0
i . Now 1jA1
⊕ dim 1iV0
i is nonzero for
only finitely many i, and these summands are all finite dimensional as A is locally finite dimensional
and V0 is finite dimensional. So 1jV is the image of the finite dimensional vector space 1jP , which
proves the proposition.
We have the following version of Schur’s lemma.
Lemma II.2.2. If A is locally finite dimensional, and L a simple A-module, then EndA(L) = k.
Proof. Let f ∈ EndA(L). For each i ∈ I, f restricts to a linear endomorphism fi of 1iL, which is
finite dimensional by Proposition II.2.1. Fix any i ∈ I. As k is algebraically closed, fi has some
eigenvalue λ. Then ker(f − λ · Id) 6= 0 implies that f = λ · Id.
Proposition II.2.3. Assume A is any locally unital algebra. Let V be a finitely generated A-
module and W a locally finite dimensional A-module. Then HomA(V,W ) is finite dimensional. In
particular, if A is locally finite dimensional, then EndA(V ) is finite dimensional.
Proof. Choose a finite dimensional homogeneous generating subspace V0 of V . There is a finite
subset J ⊂ I such that V0 =
⊕
i∈J 1iV0. Any f ∈ HomA(V,W ) is determined by its restriction to
V0. So we have
dim HomA(V,W ) ≤
∑
i∈J
dim Homk(1iV0, 1iW ) =
∑
i∈J
dim 1iV0 · dim 1iW <∞.
Corollary II.2.4. If A is locally finite dimensional, then every finitely generated A-module V has a
direct sum decomposition into finitely many indecomposable modules. This decomposition is essen-
tially unique in the sense of the Krull-Schmidt theorem.
Proof. If V is indecomposable, then EndA(V ) is a finite dimensional algebra in which 1 is a primitive
idempotent, hence a local ring. This observation proves uniqueness in the usual way (see Theorem
7.5 of [La]).
11
To prove existence, suppose there is some finitely generated module which is not a finite
direct sum of indecomposables. Choose such a module V , which has a finite dimensional homoge-
neous generating subspace V0 satisfying V0 =
⊕
i∈J 1iV for some finite subset J ⊂ I. Choose V
with the dimension of such V0 minimal. Then as V is not indecomposable, we have V = V1 ⊕ V2
for some nonzero submodules V1, V2. It is clear that each Vk is generated by
⊕
i∈J 1iVk, which has
smaller dimension than V0. By minimality, we must have that V1, V2 are each a finite direct sum of
indecomposables. Therefore so is V .
Proposition II.2.5. If A is locally finite dimensional, then every simple A-module L has a projective
cover.
Proof. We know L is a quotient of some finitely generated projective module (see (II.2.0.4)). Take an
indecomposable summand P mapping onto L. Call this map pi. Then to see that P is a projective
cover of L, suppose Q ⊂ P is a submodule which maps onto L. Then projectivity gives a map
g : P → Q such that pi ◦ f ◦ g = pi, where f : Q → P is inclusion. Then 1 − f ◦ g is a non-unit
in EndA(P ), which is a local ring as P is indecomposable. Therefore f ◦ g is an isomorphism. In
particular f is surjective, so Q = P . So P is a projective cover of L.
Let L be a simple A-module. For V ∈ A -Mod we define the composition multiplicity of L
in V as usual by
[V : L] = sup #{i : Vi+1/Vi ∼= L} (II.2.0.5)
the supremum being taken over all filtrations by submodules 0 = V0 ⊂ · · · ⊂ Vn = V . If [V : L] 6= 0,
we call L a composition factor of V , even if V does not have a composition series.
Proposition II.2.6. If V is locally finite dimensional, then [V : L] is finite for any simple L.
Moreover, if P is a projective cover of L, then we have
[V : L] = dim HomA(P, V )
Proof. Choose i ∈ I so that 1iL 6= 0. Suppose n sections of a given filtration of V are isomorphic
to L. Then n ≤ dim 1iV . Hence [V : L] ≤ dim 1iV < ∞. In particular, the supremum sup #{i :
Vi+1/Vi ∼= L} is attained by some finite filtration of V . It remains to show that dim HomA(P, V )
is equal to the number of sections in this filtration which are isomorphic to L. This follows from
exactness of HomA(P, ?).
12
Right modules and bimodules are defined for locally unital algebras in the obvious way. If
(A, I), (B, J) are locally unital algebras and V is a (B,A)-bimodule, then V⊗A? is a well-defined
functor A -Mod→ B -Mod. Moreover, if V is locally finite dimensional as a B-module, then V⊗A?
is a well-defined functor A -mod → B -mod. However, if I is infinite then HomB(V, ?) is not a
well-defined functor B -Mod → A -Mod. This is because HomB(V,W ) need not be a locally unital
module if I is infinite:
HomB(V,W ) = HomB
(⊕
i∈I
V 1i,W
)
=
∏
i∈I
HomB(V 1i,W ) (II.2.0.6)
Instead, the right adjoint of V⊗A? is
⊕
i∈I HomB(V 1i, ?). In particular, if B ⊂ A is a subalgebra
with I = J , then induction A⊗B? is left adjoint to restriction
⊕
j∈J HomA(A1j , ?) ∼= A⊗A?, which
is left adjoint to coinduction
⊕
i∈I HomB(A1i, ?).
II.3. The Lie Algebra slk
Make k into a graph by connecting i and i+ 1 for each i ∈ k. The result is a disjoint union
of graphs of type A∞ if p = 0 or A˜p−1 if p > 0. We let slk denote the Kac-Moody Lie algebra
associated to this graph (see chapter 1 of [Kac]). So
slk =
⊕
ŝlp
where ŝl0 = sl∞ and the sum is over the cosets of Z · 1k in k.
Alternatively, slk can be described as follows. First assume p > 0 and choose a set k˜ of
Z · 1k-coset representatives in k. Let h be the vector space with basis {α∨i : i ∈ k} ∪ {di : i ∈ k˜}.
Define the weight lattice
P = {λ ∈ h∗ : λ(α∨i ) ∈ Z, for all i ∈ k}. (II.3.0.7)
Inside P there are elements {$i : i ∈ k} ∪ {δi : i ∈ k˜} which are dual to the chosen basis of h. That
is,
$i(α
∨
j ) = δi,j $i(dj) = 0
δi(α
∨
j ) = 0 δi(dj) = δi,j ,
where δi,j is the Kronecker delta. We define the simple roots αi = 2$i − $i−1 − $i+1 and set
cij = αi(α
∨
j ).
13
Then slk is generated by {ei, fi}i∈k and h subject only to the following relations:
[h, h′] = 0 [ei, fi] = δi,jα∨i
[h, ei] = αi(h)ei [h, fi] = −αi(h)fi
(ad ei)
1−cij (ej) = 0 (ad fi)1−cij (fj) = 0
for all h, h′ ∈ h and i ∈ k.
In characteristic zero slk can be described as the set of finitely supported, traceless matrices
(aij)i,j∈k with aij = 0 unless i, j lie in the same coset of Z · 1k. Explicitly, aij 6= 0 for only finitely
many choices of i, j, each such pair necessarily lying in the same coset of Z · 1k, and
∑
i aii = 0.
The Lie bracket is given by the commutator of matrices. Let h be the Cartan subalgebra of diagonal
matrices in slk. Let ei denote that matrix with a 1 in row i and column i+ 1 and zeros elsewhere.
Let fi be the transpose of ei and set α
∨
i = [ei, fi]. Define the weight lattice P as in (II.3.0.7). Inside
P we have the diagonal coordinate function εi defined by εi(diag(aj)) = ai. Set αi = εi−εi+1. Note
that the infinite sum $i :=
∑
i−j∈Z≥0·1k
εj can be interpreted as an element of h
∗. That is, $i is the
function h → k sending diag(ai) to
∑
i−j∈Z≥0·1k
aj , which is a finite sum as matrices in h have finite
support.
In any characteristic, the weight lattice P is partially ordered by dominance. Explicitly,
given x, y ∈ P , we define x ≤ y if y − x ∈⊕i∈k Z≥0 · αi.
II.4. Quotient Functors
Let A be an abelian category. A full subcategory C is called a Serre subcategory (sometimes
called a thick subcategory) if for every short exact sequence
0→M ′ →M →M ′′ → 0
in A, M is an object in C if and only if M ′,M ′′ are objects in C (see chapter 3 of [G]). Given an
abelian category A and a Serre subcategory C, we can form the quotient category A/C as follows.
• The objects of A/C are the same as those of A.
• HomA/C(M,N) = lim−→
M ′,N ′
HomA(M ′, N/N ′),
where the limit runs through all subobjects M ′ ⊂M , N ′ ⊂ N such that M/M ′ and N ′ are objects
of C. We define a functor pi : A → A/C by pi(M) = M for any object M , and given a morphism
14
f : M → N , pi(f) is the image of f in the inductive limit lim
−→
M ′,N ′
HomA(M ′, N/N ′). We call pi a
quotient functor, or the canonical functor A → A/C.
Proposition II.4.1. If C is a Serre subcategory of A, then the category A/C is abelian and the
canonical functor A → A/C is an exact additive functor.
Proof. This is Lemme 1 and Proposition 1 of [G] combined. See section 1 of chapter 3 of [G] for
their proofs.
The above quotient has the following universal property.
Corollary II.4.2. Let A,D be abelian categories. If C is a Serre subcategory of A and G : A → D is
an exact functor with G(M) = 0 for any object M of C, then there is a unique functor H : A/C → D
such that G = H ◦ pi. Moreover, the functor H is exact.
Proof. The first statement is Corollaire 2 of [G]. Exactness of H follows from Corollaire 3 of [G].
See section 1 of chapter 3 of [G] for their proofs.
We conclude this section by giving an important example of a quotient functor, namely
idempotent truncation (see section V.1).
Let A be a locally finite dimensional locally unital algebra, and let e ∈ A be a nonzero
idempotent. Then e lies in the span of finitely many of the 1iA1j , which implies that the unital
algebra eAe lies in the same (finite dimensional) subspace. We recall the functor f : A -mod →
eAe -mod from section 6.2 of [Gr]. If V ∈ A -mod, then the subspace eV is an eAe-module, and we
define f(V ) = eV . If θ : V → V ′ is a morphism in A -mod, then f(θ) is the restriction of f to eV .
It is easy to see that f is exact.
Theorem II.4.3. The functor f is the quotient functor by the Serre subcategory consisting of
modules V with eV = 0.
Proof. First note that the functor h = Ae⊗eAe? is isomorphic to a right inverse to f . Let D be an
abelian category and G : A -mod → D an exact functor satisfying G(V ) = 0 whenever eV = 0. If
H : eAe -mod → D satisfies G = H ◦ f , then composing on the right with h gives H = G ◦ h. On
the other hand, if we define H = G ◦ h, then we have G = H ◦ f as follows. The product map
m : Ae⊗eAe eV → V fits into an exact sequence
0→ K → Ae⊗eAe eV m→ V → C → 0
15
where K = ker f and C = coker f . Since the restriction of m to eAe ⊗eAe eV is an isomorphism,
we have f(K) = f(C) = 0 and therefore G(K) = G(C) = 0. Applying the exact functor G to the
above exact sequence now shows that G is isomorphic to G ◦ h ◦ f = H ◦ f .
Theorem II.4.4. Suppose {Vλ : λ ∈ Λ} is a full set of irreducible modules in A -mod, indexed by a
set Λ, and let Λ′ = {λ ∈ Λ : eVλ 6= 0}. Then {eVλ : λ ∈ Λ′} is a full set of irreducible modules in
eAe -mod.
Proof. This is Theorem 6.2g of [Gr]. The proof given in section 6.2 of [Gr] is valid for any locally
unital algebra, providing we interpret (1− e)V as the kernel of the linear endomorphism of V given
by multiplication by e.
II.5. Standard Stratification
We need to relax the finiteness conditions slightly in the definition of a standardly stratified
category given in [LW]. Specifically, let C denote an abelian category which is equivalent to A -mod
for some locally finite dimensional locally unital k-algebra A. Let Λ be a preordered set with a fixed
bijection λ 7→ L(λ) to the isomorphism classes of simple objects in C. Fix a projective cover P (λ)
for L(λ) in C, which exists by Proposition II.2.5.
Let Ξ be the poset induced by Λ. That is, Ξ is the quotient of Λ by the equivalence relation
which identifies λ, µ whenever λ ≤ µ and λ ≥ µ. Given ξ ∈ Ξ, let C≤ξ (resp. C<ξ) be the full
subcategory of C consisting of all modules whose composition factors L(λ) satisfy [λ] ≤ ξ (resp.
[λ] < ξ). Note that C<ξ is a Serre subcategory of C≤ξ (see section II.4). Set Cξ := C≤ξ/C<ξ. For
λ ∈ ξ, let Lξ(λ) denote the image of L(λ) in Cξ. Let Pξ(λ) denote its projective cover in Cξ. Let piξ
denote the (exact) quotient functor C≤ξ → Cξ (see section II.4). By the general theory piξ has a left
adjoint. We fix one and denote it henceforth by ∆ξ.
We call the category C as above a locally stratified category if ∆ξ is exact and, setting
∆(λ) = ∆[λ](P[λ](λ)), there is an epimorphism P (λ)→ ∆(λ) whose kernel admits a finite filtration
by objects ∆(µ) with µ > λ. Said differently, we define a standard filtration or ∆-flag of a module
to be a finite filtration with with sections isomorphic to standard modules. Then P (λ) is required
to have a standard filtration with ∆(λ) at the top, and other sections isomorphic to ∆(µ) for µ > λ.
Given a locally stratified category C, we define the associated graded category to be gr C =⊕
ξ∈Ξ Cξ. We call ∆ =
⊕
∆ξ : gr C → C the standardization functor. The objects ∆(λ) are called
the standard objects and the objects ∆(λ) = ∆[λ](L[λ](λ)) are called the proper standard objects.
We remark that more classical notion of a highest weight category of [CPS] is a special case
16
of this notion of local stratification. If each Cξ is equivalent to the category of finite dimensional
k-vector spaces, then C is called a locally highest weight category. This is the case if the preorder on
Λ is actually a partial order. If in addition, the set Λ is finite, then the subcategory of C consisting
of objects of finite length is a highest weight category in the sense of [CPS].
17
CHAPTER III
THE ALGEBRA OB
In this chapter we meet the oriented Brauer categories and their locally unital algebra
counterparts. Their definitions are given in section III.1. We describe its triangular decomposition
in section III.2. The poset Λ parameterizing the simple OB-modules is defined in section III.3. We
then show in section III.4 that OB0 and
⊕
r,s≥0 kSr ⊗ kSs are Morita equivalent. In section III.5
we define a duality functor on OB -mod. We then give the classification of simple OB-modules in
section III.6.
III.1. Oriented Brauer Categories
Let 〈↓, ↑〉 denote the set of all words in the alphabet {↑, ↓}, including the empty word ∅.
Given two words a = a1 . . . an, b = b1 . . . bm ∈ 〈↓, ↑〉, an oriented Brauer diagram of type a→ b is a
diagrammatic representation of a bijection
{i : ai =↑} ∪ {i : bi =↓} → {i : ai =↓} ∪ {i : bi =↑}.
We draw such a diagram by aligning the sequences a, b in two rows, b above a, and drawing consis-
tently oriented strands between a and b connecting pairs of letters prescribed by the bijection. For
example,
is a diagram of type ↓↑↓↑↑↓↓↓↑↓↑→↓↑↑↓↓. Two diagrams are equivalent if they are of the same
type and represent the same bijection. The strands connecting a vertex in a to one in b are called
vertical strands and all other strands are called horizontal. The horizontal strands which connect
two vertices in a are called caps, while those which connect two vertices in b are called cups. Given
18
a diagram D of type a → b we define the source of D to be s(D) = a and the target of D to be
t(D) = b. We also define D′ to be the diagram obtained from D by switching the orientation of
every strand.
Given a, b, c ∈ 〈↓, ↑〉, we may stack an oriented Brauer diagram of type b→ c on top of one
of type a→ b to obtain an oriented Brauer diagram of type a→ c along with some number of loops
made up of strands which were connected only to vertices in b, which we call bubbles. Two oriented
Brauer diagrams with bubbles are equivalent if they have the same number (possibly zero) of bubbles
(regardless of orientation), and the underlying oriented Brauer diagrams obtained by ignoring the
bubbles are equivalent in the earlier sense. For example, if we stack the above diagram on top of
we get
The oriented Brauer category OB
Now we can define the oriented Brauer category OB to be the k-linear category with objects
〈↓, ↑〉 and morphisms HomOB(a, b) consisting of all formal k-linear combinations of equivalence
classes of oriented Brauer diagrams with bubbles of type a → b. The composition D1 ◦ D2 of
diagrams with bubbles is given by stacking D1 on top of D2. This is clearly associative. There
is also a tensor product making OB into a strict monoidal category. If D1, D2 are diagrams, then
D1 ⊗D2 is obtained by horizontally stacking D1 to the left of D2. We will often omit ⊗ from the
notation.
As a k-linear monoidal category, OB is generated by objects ↑, ↓ and the morphisms c :
∅→↑↓, d :↓↑→ ∅ and s :↑↑→↑↑ given by
c = d = s =
which satisfy the following relations.
19
(↑ d) ◦ (c ↑) =↑ (III.1.0.1)
(d ↓) ◦ (↓ c) =↓ (III.1.0.2)
(↑ s) ◦ (s ↑) ◦ (↑ s) = (s ↑) ◦ (↑ s) ◦ (s ↑) (III.1.0.3)
s2 =↑↑ (III.1.0.4)
(d ↑↓) ◦ (↓ s ↓) ◦ (↓↑ c) is invertible. (III.1.0.5)
We denote the inverse of the diagram in III.1.0.5 by t.
Theorem III.1.1. As a k-linear monoidal category, OB is generated by the objects ↑, ↓ and mor-
phisms c, d, s, t subject only to the relations III.1.0.1-III.1.0.5, where t is the inverse referred to in
relation III.1.0.5.
Proof. See section 3.1 of [BCNR] for the proof of this theorem. Ultimately it is a consequence of a
more general result of Turaev for ribbon categories.
Let δ ∈ k. We shall be studying the representation theory of the category OB(δ) obtained
from OB by imposing the additional relation d ◦ t ◦ c = δ, which is represented diagrammatically as
= δ. (III.1.0.6)
Since diagrams in OB are viewed up to equivalence, this relation says that a bubble is equal to
δ · ∅ in OB(δ). So in OB(δ), composition D1 ◦ D2 of two diagrams is defined by stacking D1 on
top of D2, then removing all bubbles and multiplying by the scalar δ
n, where n is the number
of bubbles removed. Then HomOB(δ)(a, b) has a basis consisting of equivalence classes of oriented
Brauer diagrams of type a→ b with no bubbles.
The affine oriented Brauer category AOB
The affine oriented Brauer category AOB is the monoidal category generated by objects
↑, ↓ and morphisms c : ∅ →↑↓, d :↓↑→ ∅, s :↑↑→↑↑, and x :↑→↑ subject to (III.1.0.1)-(III.1.0.5)
plus one extra relation
(↑ x) ◦ s = s ◦ (x ↑)+ ↑↑ . (III.1.0.7)
Note that by Theorem III.1.1 there is a functor OB → AOB sending the generators of OB
20
to those of AOB with the same name. Hence we can interpret any oriented Brauer diagram with
bubbles as a morphism in AOB. Let us now represent the new generator x by the diagram
x = •
We define a dotted oriented Brauer diagram with bubbles to be an oriented Brauer diagram with
bubbles, such that each segment is decorated in addition with some non-negative number of dots,
where a segment means a connected component of the diagram obtained when all crossings are
deleted. The following is a typical example of a dotted oriented Brauer diagram with bubbles.
•
•••
•
Two dotted oriented Brauer diagrams with bubbles are equivalent if one can be obtained
from the other by continuously deforming strands through other strands and crossings, and also by
sliding dots along strands without pulling them past any crossings.
Any dotted oriented Brauer diagram with bubbles is equivalent to one that is a vertical
composition of diagrams of the form acb, adb, asb, axb and atb for various a, b ∈ 〈↓, ↑〉. Hence it can
be interpreted as a morphism in AOB. Moreover, the resulting morphism is well defined independent
of the choices made, and it depends only on the equivalence class of the original diagram. For
example, the following diagram x′ represents the morphism (d ↓) ◦ (↓ x ↓) ◦ (↓ c) ∈ EndAOB(↓):
x′ = •
Also, we can represent the relation (III.1.0.7) as the first of the following two diagrammatic relations;
the second follows from the first by composing with s on top and bottom:
• = • + , • =
• +
These local relations explain how to move dots past crossings in any diagram, introducing an “error
term” with fewer dots.
A dotted oriented Brauer diagram with bubbles is normally ordered if it is equivalent to a
tensor (horizontal) product of diagrams b1 . . . bnD, where b1, . . . , bn are each a clockwise, crossing free
21
bubble with some nonnegative number of dots and D is a dotted oriented Brauer diagram without
bubbles, with all dots on outward-pointing boundary segments, i.e. segments which intersect the
boundary at a point that is directed out of the picture. For example, of the two diagrams below,
the one on the left is not normally ordered, but the diagram on the right is.
•
•••
•
••
••
•
(III.1.0.8)
Theorem III.1.2. For a, b ∈ 〈↓, ↑〉, the space HomAOB(a, b) has basis given by equivalence classes
of normally ordered dotted oriented Brauer diagrams with bubbles of type a→ b.
Proof. This is Theorem 1.2 of [BCNR] and is proved in section 5.4 of that paper.
The cyclotomic oriented Brauer category COB
SupposeM is a monoidal category. A right tensor ideal I ofM is the data of a submodule
I(a, b) ⊆ HomM(a, b) for each pair of objects a, b in M, such that for all objects a, b, c, d we have
h◦g◦f ∈ I(a, d) whenever f ∈ HomM(a, b), g ∈ I(b, c), h ∈ HomM(c, d), and g⊗1c ∈ I(a⊗c, b⊗c)
whenever g ∈ I(a, b). The quotient M/I ofM by right tensor ideal I is the category with the same
objects as M and morphisms given by HomM/I(a, b) := HomM(a, b)/I(a, b).
Let ` ≥ 1 be a fixed level and f(u) ∈ k[u] be a monic polynomial of degree ` in the auxiliary
variable u. The cyclotomic oriented Brauer category COB is the quotient of AOB by the right tensor
ideal generated by f(x) ∈ EndAOB(↑). Since COB has the same objects as AOB while its morphism
spaces are quotients of those in AOB, any morphism in AOB can be viewed as one in COB.
Theorem III.1.3. For a, b ∈ 〈↓, ↑〉, the space HomCOB(a, b) has basis given by equivalence classes
of normally ordered dotted oriented Brauer diagrams with bubbles of type a → b, subject to the
additional constraint that each segment is decorated by at most (`− 1) dots.
Proof. This is Theorem 1.5 of [BCNR]. It is proved is sections 5.1-3 of that paper.
Theorem III.1.4. Suppose that f(u) = u − m ∈ k[u] is monic of degree one. Then the functor
OB → COB defined as the composite first of the functor OB → AOB then the quotient functor
AOB → COB is an isomorphism.
Proof. This is Theorem 3.3 of [BCNR].
22
The above theorem implies in particular that the functor OB → AOB is faithful. It also
shows that there is a functor AOB → OB which restricts to identity on OB and sends xa to m1↑a
for any a ∈ 〈↓, ↑〉. Using this functor we can interpret dotted diagrams as morphisms in OB, which
we shall do from now on. For example the relation
• = • − δ
shows that the functor AOB → OB sends x′a to m′1↓a, which is how we interpret a dot on a
downward-oriented string on the left edge of a diagram. Lemma III.1.5 below explains how to
interpret a dot found anywhere in a diagram. We stress that from now on all diagrams refer to
morphisms in OB; this will not be ambiguous as we shall not need the category AOB again.
Define ci1a, di1a, si, xi1a, ti1a, s
′
i1a, x
′
i1a (also written 1bci, 1bdi, 1bsi, 1bxi, 1bti, 1bs
′
i, 1bx
′
i, re-
spectively) to be the following morphisms in OB, for all i, a, b for which they make sense. Here, i
labels the ith string from the left. The object on the bottom of each diagram is a, and the one on
top is b (for different choices of a, b).
ci1a = 1bci = . . . . . .
i i+ 1
di1a = 1bdi = . . . . . .
i i+ 1
si1a = 1bsi = . . . . . .
i i+ 1
xi1a = 1bxi = . . . . . .
i
•
ti1a = 1bti = . . . . . .
i i+ 1
s′i1a = 1bs
′
i = . . . . . .
i i+ 1
x′i1a = 1bx
′
i = . . . . . .
i
•
If D is a diagram and D = 1aD1b, then we allow ourselves to write compositions of D with the
above morphisms without writing the 1a, 1b. For example, if ai =↓ and ai+1 =↑, then we can define
t−1i 1a = disi+1ci+21a and if bi =↑ and b =↓, then 1bt−1i = 1bdisi+1ci+2. Note that the symbols
ci, di, si, xi, etc. are ambiguous until an object is specified by multiplying by some well-defined
morphism in OB.
23
To express xi1a↑, x′i1a↓ in terms of undotted oriented Brauer diagrams we have some more
notation to introduce. For a ∈ 〈↓, ↑〉, 1 ≤ i < j ≤ `(a), define (i j)1a to be whichever one of the
following two morphisms a→ a matches the orientations of the vertices ai, aj .
. . . . . .
i j
if ai = aj − . . . . . .
i j
if ai 6= aj
Lemma III.1.5.
If ai =↑ then xi1a = m1a +
∑
1≤j<i
(j i)1a and
if ai =↓ then x′i1a = m′1a −
∑
1≤j<i
(j i)1a.
Proof. These two statements are proved by induction. The induction step follows from the relations
xi+11a = sixisi1a + (i i+ 1)1a if ai = ai+1 =↑
x′i+11a = s
′
ix
′
is
′
i1a + (i i+ 1)1a if ai = ai+1 =↓
xi+11a = tixit
−1
i 1a + (i i+ 1)1a if ai =↓, ai+1 =↑
x′i+11a = t
−1
i x
′
iti1a + (i i+ 1)1a if ai =↑, ai+1 =↓
The base case is easy. We already know x11a = m1a. Now x
′
11a = m
′1a follows from the
relation
• = • − δ
III.2. Triangular Decomposition
We apply the construction of a locally unital algebra starting from the small k-linear category
OB(δ) to obtain the algebra OB(δ) (see section II.2). Since we shall never consider the algebra built
out of OB, we can unambiguously abbreviate OB(δ) to OB. We note that then OB is locally finite
dimensional.
We now describe a triangular decomposition for OB analogous to that of the universal
enveloping algebra of a semisimple Lie algebra (see sections 0.5 and 1.1 of [H]). Let K denote the
span of the distinguished idempotents: K =
⊕
a∈〈↓,↑〉 k·1a. Let OB+ (resp. OB−) denote the span of
24
all diagrams with no cups (resp. caps) and no crossings among vertical strands. Also let OB0 denote
the span of all diagrams with no cups or caps. Then OB+, OB−, OB0 are subalgebras containing K.
Observe that if V is a right K-module and W a left K-module, then V ⊗KW ∼=
⊕
a∈〈↓,↑〉 V 1a⊗k1aW .
Proposition III.2.1. The product map OB−⊗KOB0⊗KOB+ → OB is an isomorphism of vector
spaces.
Proof. Surjectivity is clear as OB− ⊗K OB0 ⊗K OB+ =
⊕
a,b∈〈↓,↑〉OB
−1a ⊗k 1aOB01b ⊗k 1bOB+.
Any element of the kernel of the product map can be written as
∑
D cDD, summing over all D =
D−⊗D0⊗D+ with D−, D0, D+ in the respective standard bases for OB−, OB0, OB+. We then have∑
D cDD
+D0D− = 0. Now the nonzero D+D0D− are all distinct diagrams, so linear independence
of diagrams shows cD = 0 whenever D 6= 0, so injectivity is proved.
Define OB] (resp. OB[) to be the subalgebra spanned by all diagrams with no cups (resp.
caps). So OB], OB[ are the images of OB0 ⊗K OB+, OB− ⊗K OB0 under the product map. The
algebras OB], OB[ are graded by the number of caps or cups appearing in a diagram, respectively.
So OB] =
⊕
d≥0OB
][d], where OB][d] is the span of all diagrams in OB] which have exactly
d caps. Similarly OB+ =
⊕
d≥0OB
+[d]. Also OB[ =
⊕
d≤0OB
[[d], where OB[[d] is the span
of all diagrams in OB[ which have exactly −d cups, and similarly define OB−[d]. Observe that
OB0 = OB][0] is a quotient of OB] by the two-sided ideal
⊕
d≥1OB
][d]. Pulling back through this
quotient gives an exact functor OB0 -mod → OB] -mod. To simplify the notation, we shall simply
view OB0-modules as OB]-modules in this way without further comment. Composing this with
OB⊗OB]? defines a functor ∆ : OB0 -mod → OB -mod. Later in this thesis, we will show that
OB -mod is locally standardly stratified in the sense of section II.5 with gr OB -mod = OB0 -mod
and standardization functor ∆; see Proposition V.1.2 below. We therefore call ∆ the standardization
functor. Since OB = OB− ⊗K OB], it is easy to see that as a right OB]-module, OB is isomorphic
to a direct sum of copies of 1aOB
], for various a’s. Hence OB is a projective right OB] module, so
that ∆ is an exact functor.
If V is an OB0-module, then
∆V = OB− ⊗K OB] ⊗OB] V = OB− ⊗K V =
⊕
a
OB−1a ⊗k 1aV.
Let Ba be a basis for 1aV . Then ∆V has a basis consisting of D⊗ v with D a diagram with no caps
and no crossings among vertical strands, and v ∈ Bs(d).
25
III.3. Weights
A bipartition is a pair of partitions. Recall the definition of a p-regular partition from section
II.1. A bipartition is p-regular if both its constituent partitions are p-regular. Let Λ denote the set
of all bipartitions and Λ the set of all p-regular bipartitions. Let Λr,s be the set of all bipartitions
λ = (λ↓, λ↑) such that λ↓ ` r and λ↑ ` s, and let Λr,s = Λ ∩ Λr,s. For such a bipartition we let
|λ| = r+ s. We define a preorder on Λ (hence on Λ) as follows. Recall that m,m′ ∈ k are fixed and
δ = m−m′. Given a box in row i and column j, the ↑-content of this box is cont↑() = m+ j − i,
and its ↓-content is cont↓() = m′ + i − j. The content of a box in either constituent partition of
a bipartition λ = (λ↓, λ↑) refers to its ↓-content if it belongs to λ↓ and its ↑-content if it belongs to
λ↑.
Recall the weight lattice P , which is partially ordered by dominance (see section II.3). Now
define the ↑-content of a bipartition λ = (λ↓, λ↑) to be
cont↑(λ) =
∑
∈λ↑
αcont↑() ∈ P.
We also define the ↓-content of a bipartition to be
cont↓(λ) =
∑
∈λ↓
αcont↓().
Next we define the content of λ = (λ↓, λ↑) to be
cont(λ) = cont↑(λ)− cont↓(λ).
Now given two bipartitions, λ, µ, we define λ ≤ µ if cont(λ) = cont(µ) and cont↑(λ) ≥ cont↑(µ), or
equivalently, if cont(λ) = cont(µ) and cont↓(λ) ≥ cont↓(µ).
Note that this preorder depends on δ, but not on m,m′ individually. This preorder induces
a partial order on the set Ξ of equivalence classes of Λ under the equivalence relation given by λ ∼ µ
if λ ≤ µ and µ ≤ λ. So λ ∼ µ if and only if (cont↓(λ), cont↑(λ)) = (cont↓(µ), cont↑(µ)). Note that
in characteristic zero, the preorder on Λ = Λ is already a partial order, ie. Ξ = Λ. Also note that if
δ /∈ Z · 1k and λ ≤ µ, then cont↑(λ) = cont↑(µ) and cont↓(λ) = cont↓(µ). That is, the partial order
on Ξ is trivial if δ /∈ Z · 1k. In any case, we see that λ ≤ µ implies |λ| ≥ |µ|, and therefore Λ is upper
finite. We remark that the above partial order is a special case of the “inverse dominance order”
mentioned in Definition 3.2 of [LW].
26
Example III.3.1. The bipartitions (∅,∅) and (∅, ) are not comparable because the first has con-
tent equal to 0 while the second has content equal to αm. However, we do have (∅,∅) > ( , )
assuming δ = 0.
Example III.3.2. If p = 2, then ( , ) ∼ ( , ) because they both have ↓-content equal to
2αm′ + 2αm′+1 and ↑-content equal to αm. So these two 2-regular bipartitions are identified in Ξ.
In a similar fashion one can see that Λ 6= Ξ whenever p > 0.
III.4. OB0 and the Symmetric Groups
For a ∈ 〈↓, ↑〉, define `↓(a) = #{i : ai =↓}, `↑(a) = #{i : ai =↑}, and `(a) = `↑(a) + `↓(a).
Let 〈↓, ↑〉r,s = {a ∈ 〈↓, ↑〉 : `↓(a) = r and `↑(a) = s}. For a, b ∈ 〈↓, ↑〉r,s let aσb denote the unique
OB-diagram b → a in OB0 such that no strings of the same orientation (up or down) cross. For
convenience, we shall write aσr,s for aσ(↓r↑s) and r,sσa for (↓r↑s)σa. For example,
(↑↓↓↑↓↑↓)σ4,3 =
Note that we evidently have cσbbσa = cσa and aσa = 1a. Let r, s ≥ 0. We have an algebra
isomorphism
Mat(r+sr )
(kSr ⊗ kSs)→
⊕
a,b∈〈↓,↑〉r,s
1aOB
01b,
∑
a,b∈〈↓,↑〉r,s
τa,bea,b 7→
∑
a,b∈〈↓,↑〉r,s
aσr,s τa,b r,sσb,
where the rows and columns of matrices are indexed by 〈↓, ↑〉r,s, and ea,b is the corresponding matrix
unit. This implies that
OB0 =
⊕
r,s≥0
 ⊕
a,b∈〈↓,↑〉r,s
1aOB
01b
 is Morita equivalent to S := ⊕
r,s≥0
kSr ⊗ kSs.
For λ = (λ↓, λ↑) ∈ Λ, the outer tensor product of Specht modules S(λ↓)  S(λ↑) is an
S-module with simple head whose projective cover coincides with its injective hull (see section
II.1). Transporting S(λ↓) S(λ↑) through the Morita equivalence gives us the Specht module S(λ).
We denote its irreducible head by D(λ). The projective cover (and injective hull) of D(λ) shall
be denoted Y (λ). Define the standard modules ∆(λ) = ∆Y (λ), and the proper standard modules
∆(λ) = ∆D(λ). Write ∆˜(λ) for the standardized Specht modules ∆S(λ).
27
III.5. Duality
Given a diagram D ∈ 1aOB1b, let τ(D) be the diagram in 1bOB1a which represents the
inverse of the bijection represented by D (see section III.1). That is, τ(D) is obtained by flipping the
diagram D vertically, then reversing all orientations. For example, τ(d) = c′ (recall the definition
of the diagram D′ from section III.1). Then τ extends to an anti-involution of OB which fixes
K, preserves OB0, and swaps OB+ with OB−. Given an OB-module V , we give an OB-module
structure to the restricted dual
V ∗ =
⊕
a∈〈↓,↑〉
Homk(1aV,k) (III.5.0.9)
via the formula
(D · f)(v) = f(τ(D)v), D ∈ OB, f ∈ V ∗, v ∈ V. (III.5.0.10)
We have an exact contravariant involution V 7→ V ∗ of OB -mod. Since τ preserves OB0 we similarly
get a duality functor on OB0 -mod. Passing through the Morita equivalence described in section
III.4, we obtain the usual duality functor for the symmetric groups.
III.6. Classification of Simple Modules
Theorem III.6.1. Let λ = (λ↓, λ↑) ∈ Λ. Then ∆(λ) is an indecomposable module which has a
unique maximal submodule. Let L(λ) denote its unique irreducible quotient. Then {L(λ) : λ ∈ Λ} is
a complete set of inequivalent irreducible OB-modules.
Proof. Let r = |λ↓|, s = |λ↑|, and let a ∈ 〈↓, ↑〉r,s. Then
1a∆(λ) = 1aOB
− ⊗K D(λ) = 1a ⊗k D(λ),
which generates ∆(λ). This shows that any proper submodule of ∆(λ) must lie in the subspace
⊕
a∈〈↓,↑〉r+t,s+t
t≥1
1a∆(λ) ( ∆(λ),
and hence so does the sum of all proper submodules, proving the first statement.
Let L be an irreducible OB-module. Choose a ∈ 〈↓, ↑〉 of minimal length such that 1aL 6= 0.
Let V be the OB0-submodule generated by 1aL. Note that 1bV = 1bOB
01aL = 0 unless a, b ∈ 〈↓, ↑
〉r,s for some r, s ≥ 0. Let v, w ∈ V be nonzero. There is some f ∈ 1t(w)OB1t(v) with fv = w. Since
caps act as zero on V , f can be chosen in the span of diagrams having no caps. But then we must
28
have f ∈ OB0 as 1t(w)fv = w 6= 0.
Hence we have V = OB01aL ∼= D(λ) for some λ ∈ Λr,s and therefore ∆(λ) ∼= OB ⊗OB] V ,
where V is viewed as an OB]-module as usual. Then
HomOB(∆(λ), L) = HomOB](V,L) = HomOB0(V,L) 6= 0, (III.6.0.11)
where the last equality follows from the fact that the image of any OB0-homomorphism V → L lies
in V , and caps act as zero on V . Then there is a surjective module homomorphism ∆(λ) → L, so
that L ∼= L(λ).
It is worth emphasizing a fact which was deduced in the proof of this theorem. If V is an
OB-module, let n be the minimal length of a ∈ 〈↓, ↑〉 such that 1aV 6= 0. Then we call
⊕
`(a)=n
1aV
the shortest word space of V . It is a submodule of the restriction of V to OB0. The fact we wish to
emphasize is that the shortest word space of L(λ) is isomorphic to D(λ), as an OB0-module.
We deduce that
L(λ)∗ ∼= L(λ) (III.6.0.12)
by comparing shortest word spaces as OB0-modules, using the fact that simple OB0-modules are
self dual.
29
CHAPTER IV
BRANCHING RULES
In this chapter we construct the advertised categorical action. We begin by defining functors
E,F in section IV.1. Then in section IV.2 we construct certain short exact sequences of functors
which are central to many of our arguments. For example, in section IV.3 we use our short exact
sequences to compute the formal characters of the modules ∆˜(λ). These characters are described in
terms of the branching graph, which is defined in the same section.
IV.1. Functors E,F
Recall that our main theorem will be that locally finite dimenstional OB-modules categorify
a tensor product of representations of slk. In this section we begin gathering the data of this
categorification.
Define OB-bimodules OB↑, OB↓, ↑OB, ↓OB as follows.
1aOB↑1b = 1aOB1b↑ 1aOB↓1b = 1aOB1b↓
1a (↑OB) 1b = 1a↑OB1b 1a (↓OB) 1b = 1a↓OB1b
The left (resp. right) actions on OB↑, OB↓ (resp. ↑OB, ↓OB) are just the usual multipli-
cation. The remaining actions are defined as follows. For f ∈ OB and
x ∈ OB↑, x · f = x ◦ (f ↑)
x ∈ OB↓, x · f = x ◦ (f ↓)
x ∈ ↑OB, f · x = (f ↑) ◦ x
30
x ∈ ↓OB, f · x = (f ↓) ◦ x
Proposition IV.1.1. OB↓ ∼= ↑OB and OB↑ ∼= ↓OB.
Proof. We have a linear isomorphism ϕ : OB↓ → ↑OB,
D 7→ D
. . .
which is a bimodule homomorphism as follows:
ϕ(D1 ◦D2) =
D2
D1
. . .
= D1 · ϕ(D2), and
ϕ(D1 ·D2) =
D1
D2
. . .
=
D1
D2
. . . = ϕ(D1) ◦D2.
The other isomorphism is obtained from ϕ by reversing the orientation of the added string.
We now have two functors OB -mod→ OB -mod:
E = OB↓⊗OB? ∼= ↑OB⊗OB?
F = OB↑⊗OB? ∼= ↓OB⊗OB?
The functors E,F are exact because they are biadjoint. For example, the counit of the
adjunction (E,F ) is induced by the bimodule homorphism OB↓⊗OBOB↑ → OB given on OB↓⊗OB
OB↑1a = OBa↑↓ by right multiplication by ac. The unit of this adjunction is induced by the bimodule
homomorphism OB → OB↑ ⊗OB OB↓ given on OB1a by right multiplication by ad, identifying
OB↑ ⊗OB OB↓1a = OB1a↓↑. The unit and counit of the adjunction (F,E) are defined similarly,
replacing c, d with c′, d′.
IV.2. Analogues for OB0
We similarly define some endofunctors of OB0 -mod. Define OB0-bimodules OB0↑ , OB
0
↓ ,
↑OB0, ↓OB0 by replacing “OB” by “OB0” everywhere in the definitions of OB↑, OB↓, ↑OB, ↓OB,
31
respectively. Now we define functors by tensoring with these bimodules:
E↑ = ↑OB0⊗OB0? E↓ = OB0↓⊗OB0?
F ↑ = OB0↑⊗OB0? F ↓ = ↓OB0⊗OB0?
These functors correspond, under the Morita equivalence OB0 ∼= S discussed in section III.4, to
induction and restriction in the two tensor factors. Specifically, recall that kSr is embedded in kSr+1
with respect to the first r letters. This induces embeddings kSr,s ⊂ kSr+1,s and kSr,s ⊂ kSr,s+1,
where kSr,s is shorthand for kSr ⊗ kSs. These last two embeddings give induction and restriction
functors indr+1,sr,s , res
r+1,s
r,s and ind
r,s+1
r,s , res
r,s+1
r,s . Then the Morita equivalence identifies
E↑ =
⊕
r,s≥0
resr,s+1r,s E
↓ =
⊕
r,s≥0
indr+1,sr,s
F ↑ =
⊕
r,s≥0
indr,s+1r,s F
↓ =
⊕
r,s≥0
resr+1,sr,s
Theorem IV.2.1. There exist short exact sequences of functors OB0 -mod→ OB -mod
0→ ∆ ◦ E↑ → E ◦∆→ ∆ ◦ E↓ → 0
0→ ∆ ◦ F ↓ → F ◦∆→ ∆ ◦ F ↑ → 0
Proof. The functors appearing in the short exact sequences are given by tensoring by certain bimod-
ules. So to get these short exact sequences, it suffices to find short exact sequences of (OB,OB0)-
bimodules of the form
E : 0→ OB ⊗OB] ↑OB0 ϕE−→ ↑OB ⊗OB] OB0 ψE−→ OB ⊗OB] OB0↓ → 0 (IV.2.0.1)
F : 0→ OB ⊗OB] ↓OB0 ϕF−→ ↓OB ⊗OB] OB0 ψF−→ OB ⊗OB] OB0↑ → 0. (IV.2.0.2)
To define the above homomorphisms, we write pure tensors D1⊗D2 in the above bimodules
by drawing D1 over D2, with the tensor sign separating them. Then the above maps can be depicted
32
as follows:
ϕE

D1
D2
⊗
 =
D1
D2
⊗ ψE

D1
D2
⊗
 =
D1
D2
⊗
. . .
(IV.2.0.3)
ϕF

D1
D2
⊗
 =
D1
D2
⊗ ψF

D1
D2
⊗
 =
D1
D2
⊗
. . .
(IV.2.0.4)
It is easy to see that these maps are well defined and make IV.2.0.1 and IV.2.0.2 into chain
complexes of bimodules. We check that IV.2.0.1 is exact. Exactness of IV.2.0.2 is proved similarly.
To see that ψE is surjective, let D1 ∈ OB, D2 ∈ OB0↓ be diagrams. Then
D1
⊗
D2
=
D1
⊗
D2
=
D1
⊗
D2
=
d1
⊗
d2
. . .
. . . = ψE

d1
⊗
d2
. . .

where the string shown in diagram D2 is the one connected to its bottom-right vertex, d2 is obtained
from D2 by deleting this string, and d1 is the diagram above the tensor sign in the third frame.
To see that kerψE ⊂ imφE we first define I to be the standard basis of OB− consisting
of diagrams with no caps and no crossings among vertical strings. Now let I1 to be the set of all
diagrams in I whose target object ends in ↑ and this vertex is on a cup. We define I2 to be the
set of all diagrams in I whose target object ends in ↑ and this vertex is on a vertical string. Now
observe that
↑OB ⊗OB] OB0 =
⊕
D∈I1∪I2
D ⊗k 1s(D)OB0.
So any x ∈ kerψE can be written as x =
∑
D∈I1∪I2 D⊗fD, with fD ∈ OB0. Now if D ∈ I1,
and has a cup connected to its top-right vertex and some other vertex i, then let Dˆ denote the
diagram in I obtained from D by replacing the cup with a new vertical string connected to i on
top (its vertex on bottom is determined by the fact that vertical strings are not allowed to cross).
Let τD = 1s(Dˆ)τD1s(D)↓ be the diagram in OB
0 which, when placed beneath Dˆ, connects the “new”
vertex in s(Dˆ) to the bottom-right vertex, and none of the other vertical strings cross each other.
33
For example, if
D =

 then Dˆ =
and the corresponding τ is
Now
0 = ψE(x) =
∑
D∈I1
D
fD
⊗
. . .
=
∑
D∈I1
Dˆ
fD
τD
⊗ =
∑
D0∈I,D∈I1, Dˆ=D0
D0
fD
τD
⊗
Therefore
∑
D∈I1, Dˆ=D0
τD ◦(fD ↓) = 0 for all D0 ∈ I. Now since each diagram appearing in τD ◦(fD ↓)
has its bottom right vertex connected to some vertex on top, determined by D ∈ I1, we have by
linear independence of diagrams that τD ◦ (fD ↓) = 0 for all D0 ∈ I and all D ∈ I1 with Dˆ = D0,
ie. for all D ∈ I1. Then fD = 0 for all D ∈ I1, whence x =
∑
D∈I2
D ⊗ fD ∈ imϕE .
It remains to show that ϕE is injective. Note that ϕE =
⊕
D∈I
ϕ
(D)
E , where ϕ
(D)
E is the
restriction of ϕE to D ⊗k ↑OB0, which is a linear isomorphism onto (D ↑) ⊗k OB0. Then since
D 7→ D ↑ is an injective map I → I1, we have that ϕE is injective.
Recall from theorem III.1.4 that we can interpret dotted diagrams as elements of OB.
It is immediate from the definition of the bimodule OB↑ that the linear endomorphism of OB↑
given on OB1a↑ by right multiplication by ax is a bimodule homomorphism, which we denote x.
This induces an endomorphism of the functor F , denoted X. Since k is algebraically closed and X
preserves the individual spaces 1aV which are finite dimensional, we see that we have a decomposition
F =
⊕
i∈k Fi, where Fi is the generalized i-eigenspace of X acting on F . By adjointness, the
endomorphism X induces an endomorphism X of E which is given on OB1a↓ by right multiplication
by ax′. The functor E splits into the direct sum of Ei, defined as the generalized i-eigenspace of
x′ acting on E. It should be noted that using the isomorphism from proposition IV.1.1 we can
alternatively describe X ∈ EndF as left multplication on ↓OB by a dot on a “down” strand, and
34
X ∈ EndE as left multiplication on ↑OB by a dot on an “up” strand. Note that E2 = (E ↑OB)⊗OB?
and 1aE ↑OB = 1a↑↑OB. So we also have an endomorphism T of E2 induced by the bimodule
endomorphism given on 1a↑↑OB by left multiplication by the diagram as.
To similarly refine the functors E↑, F ↑, E↓, F↓, we introduce the Jucys-Murphy elements:
z↑1a↑ =
∑
1≤j<`(a)
aj=↑
(j i)1a↑ z↓1a↓ =
∑
1≤j<`(a)
aj=↓
(j i)1a↓
It is easy to show that (D ↑)z↑1a↑ = z↑1b↑(D ↑) and (D ↑)z↑1a↑ = z↑1b↑(D ↑) for D ∈
1bOB
01a. Therefore the linear endomorphism of ↑OB0 (resp. OB0↑) given on 1a↑OB
0 (resp. OB01a↑)
by left (resp. right) multiplication by m1a↑ + z↑1a↑ is an OB0-bimodule homomorphism. Call this
endomorphism x↑. Similarly, we have an OB0-bimodule endomorphism of ↓OB0 (resp. OB0↓)
given on 1a↓OB0 (resp. OB01a↓) by left (resp. right) multiplication by m′1a↓ − z↓1a↓. Call this
endomorphism x↓.
Since x↓, x↑ preserve the spaces 1aV , we have decompositions
E↑ =
⊕
i∈k
E↑i E
↓ =
⊕
i∈k
E↓i
F ↑ =
⊕
i∈k
F ↑i F
↓ =
⊕
i∈k
F ↓i
where the subscript i means to take the generalized i-eigenspace of x↑ or x↓ as appropriate. For
example, since E↑ = ↑OB0⊗OB0?, x↑ induces an endomorphism of E↑, and E↑i is the corresponding
generalized i-eigenspace.
Corollary IV.2.2. There exist short exact sequences of functors OB0 -mod→ OB -mod
0→ ∆ ◦ E↑i → Ei ◦∆→ ∆ ◦ E↓i → 0
0→ ∆ ◦ F ↓i → Fi ◦∆→ ∆ ◦ F ↑i → 0
Proof. First note that each of the functors Ei, Fi, E
↑
i , F
↑
i , E
↓
i , F
↓
i can be alternately described as
tensoring with its value on the appropriate regular module. For example, if ↑OB0i denotes the
generalized i-eigenspace of x↑ acting on ↑OB0 (that is, ↑OB0i = E
↑
i OB
0), then E↑i = ↑OB
0
i⊗OB0?.
Indeed, by the definition of x↑ it is clear that E↑i ⊃ ↑OB0i⊗OB0?. Then taking the direct sum over
35
i ∈ k produces an equality. Therefore none of the inclusions E↑i ⊃ ↑OB0i⊗OB0? can be proper.
Now it suffices to verify that the short exact sequences of bimodules in the proof of theorem
IV.2.1 intertwine the homomorphism x with the analogues for OB0. The homomorphisms x, x′
obviously preserve the images of OB ⊗OB] ↓OB0 and OB ⊗OB] ↑OB0.
Let us show that the short exact sequence for E intertwines x′ with the analogues for OB0.
The proof for the short exact sequence for F is similar. Let D1 ∈ 1aOB,D2 ∈ ↑OB0 be diagrams
with (D1 ↑)D2 6= 0. Then
xϕE(D1⊗D2) = x`(a)+1(D1 ↑)⊗D2 = (D1 ↑)x`(a)+1⊗D2 = ϕE(D1⊗x`(a)+1D2) = ϕEx↑(D1⊗D2).
Now let D1 ∈ 1a↑OB1b, D2 ∈ 1bOB01c be diagrams. Then
ψEx(D1 ⊗D2) =
D1
D2
⊗
. . . •
=
D1
D2
⊗
. . .
•
= x↓ψE(D1 ⊗D2) (IV.2.0.5)
Note that by Lemma III.1.5 the diagram beneath the tensor symbol in the third frame above repre-
sents
(m′1b↓ − z↓1b↓) ◦ (D2 ↓) = (D2 ↓) ◦ (m′1c↓ − z↓1c↓).
IV.3. Characters
Let V ∈ OB -mod and a ∈ 〈↓, ↑〉, n = `(a). For each i = 1, . . . , n let xi be the endomorphism
of 1aV given by left multiplication by xi1a if ai =↑ or x′i1a if ai =↓. Suppose n ≥ 1. Then in the finite
dimensional commutative subalgebra of 1aOB1a generated by x1, . . . , xn, there exist idempotents
{1a;i : i ∈ kn} such that 1a;i acts on any module as projection onto the simultaneous generalized
i-eigenspace of x1, . . . , xn. Let 1∅;∅ = 1∅. We define the formal character of an OB-module V to
be the formal sum
chV =
∑
a,i
(dim 1a;iV )e
ai (IV.3.0.6)
where ai is the sequence of subscripted symbols obtained from a by labelling aj with the subscript
ij . Note that since our interpretation of xi1a in OB depends on our choice of m,m
′, so does chV .
Let Supp(V ) be the set of all (a, i) such that 1a;iV 6= 0.
36
Proposition IV.3.1. The characters of irreducible OB-modules are linearly independent.
Proof. Suppose
∑
λ∈Λ aλch L(λ) = 0 is a nontrivial relation. Choose λ0, with |λ0| minimal, such
that aλ0 6= 0. If λ0 ` (r, s) define a linear map T by Teai = 0 for a 6=↓r↑s, and Te↓
r
i ↑sj =
e(i1−m,...,ir−m,m
′−j1,...,m′−js). Suppose aλT ch L(λ) 6= 0. Then
∑
i(dim 1↓r↑s;iL(λ))e
i 6= 0 so λ `
(r − t, s − t) for some t ≥ 0. But aλ 6= 0 implies |λ| ≥ |λ0|, which yields t = 0, so that λ ` (r, s).
Then if D(λ) is the the irreducible kSr⊗kSs-module labeled by λ, then we have T ch L(λ) = ch D(λ),
where ch D(λ) denotes the usual character of modules for the symmetric group (see section II.1),
because 1↓r↑s;iL(λ) = 1↓r↑s;iD(λ). We then have
∑
λ`(r,s) aλch D(λ) = 0 which implies aλ = 0
whenever λ ` (r, s) (see Lemma II.1.2). In particular, aλ0 = 0.
We can describe the character of ∆˜(λ) combinatorially. We define the branching graph B
by taking Λ for the set of vertices. There is an edge between λ and µ whenever µ is obtained from
λ by adding a single box to either of the constituent partitions. We color the edge with the content
of the box added. For example, the part of the branching graph involving only bipartitions of size
3 or less is shown in Figure 1, in the case p = m = m′ = 0.
(∅,∅)
( ,∅) (∅, )
(
,∅
)
( ,∅) ( , )
(
∅,
)
(∅, )
(
,∅
) (
,∅
)
( ,∅)
(
,
)
( , )
(
,
)
( , )
(
∅,
) (
∅,
)
(∅, )
0 0
1 −1 0 0 −1 1
2 −1
0 1
−2
0 1
−1 −1
1 0
−2
1 0
−1 2
Figure 1: Branching graph when p = m = m′ = 0.
By a path in B we mean a finite sequence of vertices λ0λ1 . . . λn with each λi, λi+1 connected
by an edge. We require λ0 = (∅,∅). The type of a path is ai = (a1)i1 . . . (an)in where ij is the color
of the jth edge traversed in the path and aj ∈ {↓, ↑} is determined as follows. If λj+1 is obtained
from λj by adding a box to λ
↑
j or removing a box from λ
↓
j then aj =↑. Otherwise aj =↓.
37
Proposition IV.3.2.
ch ∆˜(λ) =
∑
p
etype(p)
where the sum is over all paths to λ in B.
Proof. We show 1a;i∆˜(λ) has dimension equal to the number of paths in B to λ of type ai by induction
on `(a). We have dim 1∅;∅∆˜(λ) = dim 1∅∆˜(λ), which is 0 if λ 6= (∅,∅) and 1 if λ = (∅,∅). In
both cases this equals the number of paths to λ of type ∅.
The induction step follows from the identities 1a↑;iiV = 1a;iEiV and 1a↓,iiV = 1a,iFiV . For
example, since Ei∆˜(λ) has a filtration with sections isomorphic to ∆˜(µ), where µ is obtained from
λ either by adding a box of ↓-content i to λ↓ or removing a box of ↑-content i from λ↑, each possible
such µ appearing exactly once, it follows that dim 1a↑;ii∆˜(λ) =
∑
µ dim 1a;i∆˜(µ) summing over all
µ obtained from λ by removing a box of ↓-content i from λ↓ or adding a box of ↑-content i to λ↑.
The case involving Fi∆˜(λ) is similar.
Example IV.3.3. We compute the terms eai in ch ∆˜( ,∅) with `(a) ≤ 3. There is only one
path to ( ,∅) of length ≤ 2, which contributes a term e↓0 . There are six paths to ( ,∅) of length
3. Two of these begin by departing from (∅,∅), and then immediately returning to (∅,∅) before
proceeding to ( ,∅). These two paths contribute e↓0↑0↓0 + e↑0↓0↓0 . Another path to ( ,∅) of length
3 is (∅,∅) → (∅, ) → ( , ) → ( ,∅), which contributes the term e↑0↓0↓0 . The remaining three
paths to ( ,∅) of length 3 all begin by following the edge (∅,∅)→ (∅, ), then visiting a bipartition
of size 3 before returning to ( ,∅). These paths contribute e↓0↓1↑1 + e↓0↓−1↑−1 + e↓0↑0↓0 . So the
terms eai in ch ∆˜( ,∅) with `(a) ≤ 3 are: e↓0 + 2e↓0↑0↓0 + 2e↑0↓0↓0 + e↓0↓1↑1 + e↓0↓−1↑−1 .
Corollary IV.3.4. If [∆˜(λ) : L(µ)] 6= 0, then there is a path in B to µ of length |µ|, and a path to
λ of the same type.
Proof. Choose a ∈ 〈↓, ↑〉|µ↓|,|µ↑| so 1aL(µ) 6= 0. Then we can choose i so that (a, i) ∈ SuppL(µ) ⊂
∆˜(µ). Then proposition IV.3.2 guarantees a path to µ of type ai. But also (a, i) ∈ SuppL(µ) ⊂
Supp ∆˜(λ), so there is a path to λ of type ai.
Theorem IV.3.5. If [∆˜(λ) : L(µ)] 6= 0, then µ ≤ λ.
Proof. From the corollary we have a path to µ of length |µ| and a path to λ of the same type. We
will prove that the existence of such a pair of paths implies µ ≤ λ by induction on |µ|. If |µ| = 0,
then λ = µ = (∅,∅) so µ ≤ λ is trivial. Now assume |µ| ≥ 1. Delete the last edge in each path
to obtain paths to µ′ and λ′, each of length |µ′|. By induction, µ′ ≤ λ′, ie. cont(λ′) = cont(µ′)
38
and cont↑(µ′) ≥ cont↑(λ′). Since the type of the deleted edge in each path is the same, we have
cont(µ) = cont(λ).
Suppose the type of deleted edge is ↑i. Then cont↑(µ) = cont↑(µ′)+αi. Now if λ is obtained
from λ′ by adding a box to λ′↑ then cont↑(λ) = cont↑(λ′) + αi so that cont↑(µ) − cont↑(λ) =
cont↑(µ′)− cont↑(λ′) ≥ 0. On the other hand, if λ is obtained from λ′ by removing a box from λ′↓,
then cont↑(λ) = cont↑(λ′) so that cont↑(µ)− cont↑(λ) = cont↑(µ′) + αi − cont↑(λ′) ≥ αi > 0.
The case when the type of the last edge is ↓i is treated similarly.
39
CHAPTER V
TENSOR PRODUCT CATEGORIFICATION
In this chapter we assemble the pieces of our main theorem (Theorem V.3.2). In section
V.1 we show that we have the data of a locally stratified structure on OB -mod, namely that our
standardization functor ∆ is the left adjoint of a quotient functor (see section II.5). We go on to
show in section V.2 that the axioms of a locally stratified structure are met. In section V.3 we show
that the functors Ei, Fi define a categorical slk-action on OB -mod and state our main theorem
(Theorem V.3.2). Then in section V.4 we describe the crystal of OB, which is a consequence of
Theorem V.3.2 and the main result of [D].
V.1. Standardization
The poset Ξ parameterizes the blocks of the category of finite dimensional OB0-modules.
For ξ ∈ Ξ, the block OB0 -mod[ξ] consists of finite dimensional OB0-modules whose composition
factors are D(λ), with λ ∈ ξ. Denote by ∆ξ the restriction of ∆ to OB0 -mod[ξ]. Then we have
seen that the image of ∆ξ lies in OB -mod≤ξ, which is the full subcategory of OB -mod consisting
of modules V with [V : L(λ)] = 0 unless λ ∈ ξ′ for some ξ′ ∈ Ξ satisfying ξ′ ≤ ξ.
We’ve defined ∆ξ as inclusion OB
0 -mod[ξ] ↪→ OB0 -mod, followed by the inflation
OB0 -mod → OB] -mod, followed by induction to OB. Therefore, ∆ξ is left adjoint to the functor
piξ defined as the composite prξ ◦R ◦ resOBOB] , where prξ is projection to the block OB0 -mod[ξ], and
R is the right adjoint to the inflation functor. We have RV = {v ∈ V : OB][1]v = 0}.
For V ∈ OB -mod≤ξ, let 1ξ =
∑
a∈〈↓,↑〉r,s 1a, where the representatives of ξ are bipartitions
of (r, s). Then clearly piξV ⊂ 1ξV and 1ξV ⊂ R ◦ resOBOB] V . So piξV = 1ξV once we can see that
1ξV ∈ OB0 -mod[ξ]. This is so because the composition factors of V are L(λ) for [λ] ≤ ξ, and for
such λ we have 1ξL(λ) = 0 unless λ ∈ ξ, in which case 1ξL(λ) = D(λ) as we’ve noted before. In
fact, we now see that piξV is the shortest word space of V whenever V ∈ OB -mod≤ξ and piξV 6= 0.
Now it is clear that piξ is exact and commutes with the duality. Therefore it has a right
adjoint ∇ξ, which is obtained from ∆ξ by composing with duality on the left and right. The
40
result is that ∇ξ is given by inclusion OB0 -mod[ξ] ↪→ OB0 -mod, followed by the inflation functor
OB0 -mod → OB[ -mod (defined by letting cups act as zero), followed by the coinduction to OB.
Explicitly, ∇ξ =
⊕
a∈〈↓,↑〉HomOB[(OB1a, ?). Here, and forever, we view any OB
0-module as an
OB[-module via the inflation functor as we do with inflation to OB]. We set ∇ = ⊕ξ∈Ξ∇ξ and
call this the costandardization functor. We define the costandard modules ∇(λ) = ∇Y (λ) and the
proper costandard modules ∇(λ) = ∇D(λ). Write ∇˜(λ) for ∇S(λ).
Proposition V.1.1. We have ∆(λ)∗ ∼= ∇(λ) and ∆(λ)∗ ∼= ∇(λ). In particular,
[∇(µ) : L(λ)] = [∆(µ) : L(λ)] .
Proof. The stated isomorphisms follow from the fact that D(λ) and Y (λ) are self-dual. Now since
L(λ) is self-dual, the statement about composition multiplicities follows.
Proposition V.1.2. OB -modξ = OB
0 -mod[ξ] and piξ is the quotient functor from section II.5.
Proof. This is a special case of Theorem II.4.3 in light of the above description of piξ.
V.2. Projective Modules
Given V ∈ OB -mod a standard filtration or ∆-flag of V is a finite filtration
0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V
with Vi/Vi−1 ∼= ∆(λi) for λi ∈ Λ. Let (V : ∆(λ)) denote the multiplicity of ∆(λ) as a section of such
a filtration of V . Theorem V.2.2 below proves that this is independent of the standard filtration
chosen. We begin with a lemma.
Lemma V.2.1. For any λ, µ ∈ Λ, we have
dim ExtiOB(∆(µ),∇(λ)) = δi,0δλ,µ.
41
Proof. Using the triangular decomposition of OB, we can see that resOB
OB[
◦∆ = OB[⊗OB0?
ExtiOB(∆(µ),∇(λ)) = ExtiOB
(
∆(µ),
⊕
a
HomOB[(OB1a, D(λ))
)
= ExtiOB[(OB
[ ⊗OB0 Y (µ), D(λ))
= ExtiOB0(Y (µ), D(λ)).
Theorem V.2.2. Let V ∈ OB -mod have a standard filtration. Then for any λ ∈ Λ, we have
(V : ∆(λ)) = dim HomOB(V,∇(λ)). (V.2.0.1)
Proof. Induct on the length of the filtration. If V = ∆(µ), then both sides of V.2.0.1 are equal
to δλ,µ by the lemma. For the induction step, if ∆(µ) is at the top of a standard filtration of V ,
apply HomOB(?,∇(λ)) to 0→ W → V → ∆(µ)→ 0. By Lemma V.2.1 the first Ext1 term is zero.
Therefore by induction, we have
dim HomOB(V,∇(λ)) = dim HomOB(∆(µ),∇(λ)) + dim HomOB(W,∇(λ))
= δλ,µ + (W : ∆(λ))
= (V : ∆(λ)).
Proposition V.2.3. Let V ∈ OB -mod have a ∆-flag: 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = V with
Vi/Vi−1 ∼= ∆(λi) for some λi ∈ Λ, i = 1, . . . , n. Suppose V = V ′ ⊕ V ′′ with V ′, V ′′ ∈ OB -mod.
Then V ′, V ′′ both have ∆-flags.
Proof. If n = 1, then V ∼= ∆(λ1) is indecomposable, so there is nothing to prove. Suppose n > 1.
Choose λ ∈ Λ maximal such that L(λ) appears as a section of some filtration of V . If there are
multiple candidates for λ, choose |λ| minimal.
Now L(λ) is a quotient of some submodule V˜ ⊂ V . Then since D(λ) is a summand of L(λ)
as an OB0-module, we have a nonzero OB0-module homomorphism V˜ → D(λ). By projectivity
of Y (λ), we get a nonzero OB0-module homomorphism Y (λ) → V˜ ↪→ V . In fact, this is an OB]-
module homomorphism, because its image lies in the direct sum of all 1aV with a ∈ 〈↓, ↑〉r,s, where
42
λ ` (r, s). Caps act as zero on this space because 1aV =
∑n
i=1 dim 1a∆(λi) = 0 for any a with
`(a) < |λ|. We therefore have a nonzero induced homomorphism ∆(λ) → V . Then relabeling V ′
and V ′′ if necessary, we have a nonzero homomorphism ϕ ∈ HomOB(∆(λ), V ′).
Choose i minimal so that imϕ ⊂ Vi. Then the composite
∆(λ)
ϕ→ Vi → Vi/Vi−1 ∼= ∆(λi)
is nonzero, which implies λ ≤ λi. By maximality of λ, we then have λ = λi, and this composite
must be an isomorphism. Therefore ϕ is injective and ∆(λ) ∩ Vi−1 = 0. We have V/∆(λ) =
(V ′/∆(λ))⊕ V ′′. If V/∆(λ) has a shorter ∆-flag than V , then induction gives ∆-flags for V ′/∆(λ)
and V ′′, hence for V ′ also.
To see that V/∆(λ) has a shorter ∆-flag than V , observe that the short exact sequence 0→
Vi−1 → Vi → ∆(λ) → 0 splits because ∆(λ) ∩ Vi−1 = 0. Thefore Vi/∆(λ) ∼= Vi−1 so that the short
exact sequence 0 → Vi/∆(λ) → V/∆(λ) → V/Vi → 0 becomes 0 → Vi−1 → V/∆(λ) → V/Vi → 0.
Since Vi−1 has a ∆-flag of length i − 1 and V/Vi has one of length n − i, we see that V/∆(λ) has
one of length n− 1, as needed.
Let λ ∈ Λr,s. We construct the projective cover P (λ) of L(λ) as follows. Recall that the
shortest word space of L(λ) is isomorphic as an OB0-module to D(λ), ie.
⊕
a∈〈↓,↑〉r,s
1aL(λ) ∼= D(λ).
Therefore dim HomOB0(Y (λ), L(λ)) = dim HomOB0(Y (λ), D(λ)) = 1. Consider the projective mod-
ule Pˆ (λ) = OB ⊗OB0 Y (λ). We have dim HomOB(Pˆ (λ), L(λ)) = 1, and so L(λ) is covered by some
unique indecomposable projective summand P (λ).
Proposition V.2.4. P (λ) has a ∆-flag, with ∆(λ) at the top, and all other sections isomorphic to
∆(µ) with λ < µ.
Proof. We shall show that Pˆ (λ) has a ∆-flag, and conclude from Proposition V.2.3 that P (λ) does
also. In particular, the following version of BGG reciprocity follows from V.2.2, II.2.6, and V.1.1:
(P (λ) : ∆(µ)) =
[
∆(µ) : L(λ)
]
(V.2.0.2)
Now Theorem IV.3.5 proves the description of sections of the ∆-flag.
43
To show Pˆ (λ) = OB ⊗OB] OB] ⊗OB0 Y (λ) has a ∆-flag, we define an infinite descending
filtration onOB] by settingOB]i =
⊕
k≥iOB
][k]. It follows thatOB]⊗OB0Y (λ) has a finite filtration
with sections OB][k] ⊗OB0 Y (λ), 0 ≤ k ≤ min{|λ↓|, |λ↑|}, and so Pˆ (λ) has a finite filtration with
sections ∆
(
OB][k]⊗OB0 Y (λ)
)
.
Note that OB][k]⊗OB0 Y (λ) is a summand of the OB0-module OB][k], which is isomorphic
to ⊕
r,s≥0
OB01r,s ⊗k 1r,sOB+[k] ∼=
⊕
r,s≥0
(dim 1r,sOB
+[k])OB01r,s,
which is projective. Hence OB][k] ⊗OB0 Y (λ) is a finite dimensional projective OB0-module, and
therefore has a filtration with sections Y (µ) for various bipartitions µ of (|λ↓| − k, |λ↑| − k). This
finishes the proof.
We have now proved the following theorem.
Theorem V.2.5. The preorder on Λ defined in section III.3 makes OB -mod into a locally stratified
category with standard objects ∆(λ), λ ∈ Λ. If p = 0 then OB -mod is a locally highest weight
category.
V.3. Categorical Action
Let C = OB -mod. We have shown that C is a locally stratified category with gr C isomorphic
to the category of finite dimensional OB0-modules. We also have categorical actions on C and gr C.
The categorical action on gr C is well-known, and the categorical action on C was described in section
IV.1. We now review these categorical actions, and then discuss their compatibility with the local
stratification.
We denote by K0(C) (resp. K0(gr C)) the split Grothendieck group of the category of finitely
generated projective OB-modules (resp. OB0-modules). That is, K0(C) is the free abelian group
on the isomorphism classes of finitely generated projective OB-modules modulo [B] − [A] − [C]
whenever B ∼= A ⊕ C, and similarly with OB0. We let C∆ denote the subcategory of C consisting
of all modules with a ∆-flag. We denote by G0(C∆) its Grothendieck group. That is, G0(C∆) is
the free abelian group on the isomorphism classes of modules in C∆ modulo [B]− [A]− [C] for any
A,B,C ∈ C∆ exhibiting a short exact sequence 0→ A→ B → C → 0. We define [C] = C⊗ZK0(C),
[gr C] = C⊗Z K0(gr C), and [C∆] = C⊗Z G0(C∆).
Theorem V.3.1. The operators E↓i , F
↓
i , E
↑
i , F
↑
i , i ∈ k on [gr C] satisfy the relations of the Chevalley
44
generators of sl↓k ⊕ sl↑k under the assignments
e↓i 7→ E↓i f↓i 7→ F ↓i
e↑i 7→ E↑i f↑i 7→ F ↑i .
Moreover, this module is isomorphic to the tensor product of basic modules V (−$m′)⊗V ($m), and
the weight space decomposition of this module coincides with the decomposition of finite dimensional
OB0-modules into blocks.
Proof. This follows from [Groj] in light of the identifications
E↑i =
⊕
r,s≥0
(i−m)− resr,s+1r,s E↓i =
⊕
r,s≥0
(m′ − i)− indr+1,sr,s
F ↑i =
⊕
r,s≥0
(i−m)− indr,s+1r,s F ↓i =
⊕
r,s≥0
(m′ − i)− resr+1,sr,s
The categorical slk-action on C is a slightly modified version of Rouquier’s definition in [R].
That is, we have an adjoint pair (E,F ) of exact functors C → C and endomorphisms X ∈ EndE,
T ∈ EndE2 (see IV.1). We have E = ⊕i∈kEi and F = ⊕i∈k Fi, where Ei (resp. Fi) is the
generalized i-eigenspace of X acting on E (resp. F ). The functor F is isomorphic to a left adjoint of
E. The action on En of Xi = E
n−iXEi−1 for 1 ≤ i ≤ n and of Ti = En−i−1TEi−1 for 1 ≤ i ≤ n−1
induce an action of the degenerate affine Hecke algebra.
Theorem V.3.1 says that [gr C] = V (−$m′) ⊗ V ($m) as sl↓k ⊕ sl↑k-modules. Pulling this
action back through the diagonal map slk → sl↓k ⊕ sl↑k gives an integrable slk-module. This module
is isomorphic to [C∆] via the standardization functor (see corollary IV.2.2). Then since [C] embeds
into [C∆], we see that [C] is an integrable slk-module. Moreover, this embedding is an isomorphism
[C] ∼= [C∆]. To see this, choose a total order of Λ refining its preorder. Then note that with respect
to the ordered bases {[P (λ)] : λ ∈ Λ} for [C] and {[∆(λ)] : λ ∈ Λ} for [C∆], the embedding is given by
an upper unitriangular matrix (see Proposition V.2.4). Therefore the embedding is an isomorphism.
Note that although we don’t have a decomposition of C into blocks, we do have [C] = ⊕ξ∈Ξ[Cξ].
The local stratification and the categorical action on C are compatible in the following
sense. First, the poset Ξ can be identified with pairs of weights of V (−$m′), V ($m) via the map
[λ] 7→ (−$m′+cont↓ λ,$m−cont↑ λ). This map is clearly surjective, and the remarks in section III.3
45
show that it is a well-defined injection. Transporting the partial order on Ξ through this bijection
induces the inverse dominance order (see Definition 3.2 of [LW]) on the set of pairs of weights of
V (−$m′), V ($m). That is, for two such pairs µ = (µ1, µ2), ν = (ν1, ν2), we have µ ≤ ν if and
only if µ1 ≥ µ2 and µ1 + µ2 = ν1 + ν2. Second, gr C carries a categorical sl↓k ⊕ sl↑k action with
[gr C] ∼= V (−$m′)⊗V ($m) and the weight ξ subcategory of gr C is the quotient Cξ. Lastly, for each
M ∈ Cξ the object Ei∆(M) admits a filtration with successive quotients being ∆(E↓iM),∆(E↑iM),
and similarly with Fi.
We have proved our main theorem, which we now state.
Theorem V.3.2. The endofunctors Ei, Fi of OB -mod define a categorical slk-action. This action
is compatible with the locally stratified structure on OB -mod and categorifies V (−$m′)⊗ V ($m).
V.4. Crystal Graph Structure
We modify the description of the crystal associated to V ($0) from ([K]) to get the crystals
associated to V (−$m′) and V ($m). Fix a partition λ. Label all addable nodes of ↑-content i by
+ and all removable nodes of ↑-content i by −. The ↑i-signature of λ is the sequence of pluses and
minuses obtained by going along the rim of the Young diagram of λ from bottom left to top right
and reading off all the signs. The reduced ↑i-signature of λ is obtained from its ↑i-signature by
successively deleting all neighboring pairs of the form −+. The reduced ↑i-signature is a sequence
of +’s followed by a sequence of −’s.
To define (reduced) ↓i-signature, label all addable nodes of ↓-content i by − and all removable
nodes of ↓-content i by +. The ↓i-signature of λ is the sequence of pluses and minuses obtained
by going along the rim of the Young diagram of λ from top right to bottom left and reading off all
the signs. The reduced ↓i-signature of λ is obtained from its ↓i-signature by successively deleting
all neighboring pairs of the form −+. The reduced ↓i-signature is a sequence of +’s followed by a
sequence of −’s.
We make the set of p-regular partitions into the crystal associated to V ($m) as follows. We
define
ε↑i (λ) = #{−’s in the reduced ↑i -signature of λ}
ϕ↑i (λ) = #{+’s in the reduced ↑i -signature of λ}.
If ε↑i (λ) = 0, set e˜
↑
i λ = 0. Otherwise define e˜
↑
i λ to be the partition obtained by removing the node in
λ corresponding to the leftmost − in its reduced ↑i-signature. If ϕ↑i (λ) = 0, set f˜↑i λ = 0. Otherwise
46
define f˜↑i λ to be the partition obtained by adding the node in λ corresponding to the rightmost +
in its reduced ↑i-signature. Finally, set wt↑(λ) = $m − cont↑ λ.
We make the set of p-regular partitions into the crystal associated to V (−$m′) as follows.
We define
ε↓i (λ) = #{−’s in the reduced ↓i -signature of λ}
ϕ↓i (λ) = #{+’s in the reduced ↓i -signature of λ}.
If ε↓i (λ) = 0, set e˜
↓
i λ = 0. Otherwise define e˜
↓
i λ to be the partition obtained by adding the node in
λ corresponding to the leftmost − in its reduced ↓i-signature. If ϕ↓i (λ) = 0, set f˜↓i λ = 0. Otherwise
define f˜↓i λ to be the partition obtained by removing the node in λ corresponding to the rightmost
+ in its reduced ↓i-signature. Finally, set wt↓(λ) = −$m′ + cont↓ λ.
The crystal associated to V (−$m′)⊗ V ($m) is the Kashiwara tensor product of the above
crystals. Given a bipartition λ = (λ↓, λ↑), its i-signature is obtained by concatenating the ↓i-
signature of λ↓ followed by the ↑i-signature of λ↑. The reduced i-signature of λ is obtained from its
i-signature by successively deleting all neighboring pairs of the form −+. The reduced i-signature
is a sequence of +’s followed by a sequence of −’s.
We make Λ into the crystal associated to V (−$m′)⊗ V ($m) by defining
εi(λ) = #{−’s in the reduced i-signature of λ}
ϕi(λ) = #{+’s in the reduced i-signature of λ}.
If εi(λ) = 0, set e˜iλ = 0. Otherwise define e˜iλ to be the bipartition obtained by adding or removing
the node in λ corresponding to the leftmost − in its reduced ↓i-signature (add the node if it belongs
to λ↓ and remove it if it belongs to λ↑). If ϕi(λ) = 0, set f˜iλ = 0. Otherwise define f˜iλ to be the
partition obtained by adding or removing the node in λ corresponding to the rightmost + in its
reduced ↓i-signature (add the node if it belongs to λ↑ and remove it if it belongs to λ↓). Finally, set
wt(λ) = wt↓(λ↓) + wt↑(λ↑).
Example V.4.1. Let m = m′ = 0, p = 2. Then the 0-signature of
(
,∅
)
is − − −+ (the three
−’s correspond to the three addable nodes in of content 0, and the + corresponds to the addable
node in ∅), and its reduced 0-signature is −−. The node corresponding to the rightmost − in the
reduced signature is the addable node in the first row of . Thus e˜0
(
,∅
)
=
(
,∅
)
.
Theorem V.4.2. Suppose EiL(λ) 6= 0. Then the head and socle of EiL(λ) are both isomorphic to
47
the simple module L(e˜iλ). Moreover, if εi(λ) = 1, then EiL(λ) = L(e˜iλ). The same result holds
with Ei replaced by Fi, e˜i replaced by f˜i, and εi replaced by ϕi.
Proof. This follows from the main result of [D], combined with our main theorem above, which
verifies that the hypotheses of [D] are all satisfied.
We now define the crystal graph, which is a subgraph of the branching graph from section
IV.3. The set of vertices is Λ, and whenever λ ∈ Λ and f˜iλ 6= 0 we connect λ and f˜iλ with an edge
colored i.
Example V.4.3. Let m = m′ = 0, p = 2. The the part of the crystal graph involving bipartitions
of size up to 4 is shown in Figure 2.
(∅,∅)
( ,∅) (∅, )
( ,∅) ( , ) (∅, )
(
,∅
)
( ,∅) ( , ) ( , ) (∅, )
(
∅,
)
(
,∅
)
( ,∅) ( , ) ( , ) ( , ) (∅, )
(
∅,
)
1 0 0 1
1 1
0 1
0
0
1 1
0
0
1 0
Figure 2: Crystal graph when m = m′ = 0, p = 2.
As the above example shows, the crystal graph is not connected in general. For instance,
(∅,∅) is isolated if m = m′ = 0. Also, it is easy to describe the connected component containing
( , ) in the case m = m′ = 0, p = 2. It is a graph of type A∞, with the colors of edges forming the
pattern . . . , 0, 0, 1, 1, 0, 0, 1, 1, . . . . Part of this component is visible in the above example. Starting at
( , ), there are two possible routes. One route passes through (∅, ) and the other through ( ,∅).
With each additional edge, a box is added to the nonempty constituent partition, alternating between
the first and second rows.
In fact the crystal graph has infinitely many connected components if p = 0 and m,m′ ∈
Z · 1k. Presumably this assertion is also true when p > 0, but we will not attempt to prove that
48
here. For the p = 0 case, first define the k-value of a bipartition (λ↓, λ↑) to be the smallest integer
k ≥ −min(m,m′) such that the union of a horizontal strip of height k + m and a vertical strip of
width k + m′ can cover the diagram composed of the Young diagram of λ↓ rotated through 180◦
and adjoined to Young diagram of λ↑ with corner vertices touching. Equivalently it is the smallest
k ≥ −min(m,m′) such that λ↑i + λ↓m+k+2−i ≤ k +m′ for some 1 ≤ i ≤ k +m+ 1. In other words,
λ is a (k +m, k +m′)-cross bipartition in the sense of Comes and Wilson.
Example V.4.4. If m = 0 and m′ = 2, then the k-value of the bipartition
(
,
)
is 2 (See
Figure 3).
Figure 3: Illustration of k-value.
Proposition V.4.5. Let p = 0 and m,m′ ∈ Z · 1k. Any two bipartitions in the same connected
component of the crystal graph have the same k-value. In particular, the crystal graph has infinitely
many connected components.
Proof. Suppose λ↑i +λ
↓
m+k+2−i ≤ k+m′ for some 1 ≤ i ≤ k+m+1 and µ = f˜jλ. If µ↑i +µ↓m+k+2−i >
k +m′ then we must have λ↑i + λ
↓
m+k+2−i = k +m
′ and that f˜j adds a box either to row i of λ↑ or
to row m + k + 2− i of λ↓. In this case, we find that both λ↓, λ↑ have an addable node of content
m − i + 1 + λ↑i . Hence the j-signature of λ is −+, which reduces to ∅. It follows that f˜jλ = 0,
which is a contradiction. This shows that the k-value of µ is less than or equal to that of λ. But
the above argument applies to the situation µ = e˜jλ, implying that the k-value of µ is greater than
or equal to that of λ. Hence they have the same k-value. The last statement follows from the fact
that a bipartition of any k-value exists.
49
CHAPTER VI
APPLICATIONS
This final chapter contains some applications of our results in the case p = 0. We recall
that ∆(λ) = ∆(λ) under this assumption. In section VI.1 we compute the composition multiplicities
of the standard objects explicitly using the combinatorics of arc diagrams. Then in section VI.2
we explain how to compute the characters of the simple modules using the results of section VI.1
together with our computation of the characters of the standard modules in terms of the branching
graph (see section IV.3). Finally, in section VI.3 we give a proof of the known classification of
simple Br,s(δ)-modules due to Cox et al. (see [CDDM]) and use it to prove L(λ) is (globally) finite
dimensional if and only if δ = 0 and λ = (∅,∅), in which case it is one dimensional.
VI.1. Decomposition Numbers in Characteristic 0
We assume for the remainder of this paper that p = 0. In this case we are able to use
the crystal to compute [∆(λ) : L(µ)] = (P (µ) : ∆(λ)). For this it will be convenient to depict a
bipartition by its marker rather than its Young diagram. The notion of a marker is a variation on
the weight diagrams introduced by Brundan and Stroppel. The marker of a bipartition λ = (λ↓, λ↑)
is obtained by decorating the integer points on the y-axis with the symbols , , ,◦ as follows. First
define the sets
I↑(λ) = {λ↑1 +m,λ↑2 +m− 1, λ↑3 +m− 2, . . . }
I↓(λ) = {m′ + 1− λ↓1,m′ + 2− λ↓2,m′ + 3− λ↓3, . . . }
The ith vertex is labelled 
if i lies in I↑ but not I↓
if i lies in I↓ but not I↑
if i lies in both I↑ and I↓
◦ if i lies in neither of I↑ and I↓
50
Observe that
wt↑(λ↑) =
∑
i∈I↑(λ)
εi and wt
↓(λ↓) =
∑
i∈I↓(λ)
εi
so that
wt(λ) =
∑
i∈I↑(λ)∩I↓(λ)
εi −
∑
i∈I↑(λ)c∩I↓(λ)c
εi.
That is, wt(λ) specifies the vertices which are labelled ◦ and .
Suppose λ < µ. Then wt(λ) = wt(µ) and wt↑(λ↑) > wt↑(µ↑). That is, the markers for λ
and µ have ◦ and at all the same vertices and
∑
i∈I↑(λ)
εi −
∑
i∈I↑(µ)
εi > 0
is a finite sum of εi−εj , i < j. In terms of markers, this says that µ is obtained from λ by successively
switching ( ) to ( ) (not just adjacent vertices).
Conversely, if there is a at vertex i and a at vertex j (i < j) in the marker for λ, then
switching the labels of these vertices doesn’t affect the positions of vertices labelled ◦ and , so
wt(λ) is unchanged. But switching the labels adds εi − εj to wt↑(λ↑), which means that switching
the labels produced a bipartition µ > λ: ( ) < ( ). We conclude that λ is maximal if and only if in
its marker every appears above every .
The left arc diagram associated to λ (denoted by λ ) is obtained by drawing non-crossing
rays and arcs in the left half plane incident to some subset of the vertices in the marker in such a
way that
• vertices at the bottom ends of arcs are labelled ;
• vertices at the top ends of arcs are labelled ;
• vertices at the right ends of rays are labelled either by or by in such a way that all rays
labelled appear above all rays labelled ;
• all remaining vertices not at the ends of arcs or rays are labelled either by ◦ or by .
This diagram can be realized by successively drawing arcs connecting pairs of vertices i < j
with vertex i labelled and vertex j labelled and having no unpaired vertices labelled or in
between vertices i and j. Once no more such pairs can be found, draw a ray at all unpaired vertices.
We define the defect of λ to be the number of arcs in its left arc diagram. A bipartition is maximal
if and only if its defect is zero.
51
Given any markers λ, µ with wt(λ) = wt(µ), it makes sense to glue the left arc diagram for
λ onto the marker µ to obtain a composite diagram λµ in which the endpoints of each arc and ray
of the left arc diagram of λ are labelled either or by the marker µ. We say λµ is well-oriented
if
• each arc has exactly one label and one label making it into either a counterclockwise or
clockwise arc;
• all rays labelled are above all rays labelled .
Now we can state the result promised at the beginning of the section.
Theorem VI.1.1. If p = 0 then
[∆(λ) : L(µ)] = (P (µ) : ∆(λ)) =

1 if µλ is well-oriented
0 otherwise
To prove this theorem we will need to know the action of Ei, Fi on ∆(λ) and P (λ) in terms
of markers. Note that I↑(λ) consists exactly of those numbers which appear as the ↑-content of
some node in row i and column λi + 1, i = 1, 2, . . . , and I↓(λ) consists of the ↓-contents of nodes in
row i and column λi, i = 1, 2, . . . . From this observation, we see that if λ
↓ has an addable (resp.
removable) node of ↓-content i then I↓(λ) contains i+ 1 and not i (resp. contains i and not i+ 1).
Similarly, if λ↑ has an addable (resp. removable) node of ↑-content i then I↑(λ) contains i and not
i+ 1 (resp. contains i+ 1 and not i).
The following lemma is a consequence of Corollary IV.2.2.
Lemma VI.1.2. For λ ∈ Λ, i ∈ Z and symbols x, y ∈ {◦, , , }, let λ( yx ) be the marker obtained
from λ by relabelling its ith and (i+ 1)th vertices by x and y, respectively. Then if vλ = [∆(λ)] we
have:
(i) if λ = λ(◦ ), λ(◦ ), λ( ) or λ( ) then eivλ = vµ where µ is obtained from λ by switching the
labels on its ith and (i+ 1)th vertices;
(ii) if λ = λ( ) then eivλ = vµ where µ = λ(
◦ );
(iii) if λ = λ( ) then eivλ = vµ where µ = λ(
◦ );
(iv) if λ = λ(◦ ) then eivλ = vµ + vν where µ = λ( ) and ν = λ( );
(v) in all other situations eivλ = 0.
52
There are also analogous formulae for fivλ, which may be obtained from the above by interchanging
the roles of ◦ and .
Observe that since p = 0, the i-signature of any λ consists of at most two symbols. So
εi(λ) ≤ 2 and ϕi(λ) ≤ 2 for all i. Interpreting L(e˜iλ) (resp. L(f˜iλ)) as zero whenever e˜iλ = 0 (resp.
f˜iλ = 0), Theorem V.4.2 implies
EiL(λ) = L(e˜iλ) whenever εi(λ) 6= 2
FiL(λ) = L(f˜iλ) whenever ϕi(λ) 6= 2.
Lemma VI.1.3. Let the notation be as in Lemma VI.1.2. Then if pλ = [P (λ)] we have:
(i) if λ = λ(◦ ), λ(◦ ), λ( ) or λ( ) then eipλ = pµ where µ is obtained from λ by switching the
labels on its ith and (i+ 1)th vertices;
(ii) if λ = λ( ) then eipλ = pµ where µ = λ(
◦ );
(iii) if λ = λ( ) then eipλ = 2pµ where µ = λ(
◦ );
(iv) if λ = λ(◦ ) then eipλ = pµ where µ = λ( );
(v) if λ = λ( ) and vertex i+ 1 is connected to vertex j > i+ 1 in λ then eipλ = pµ, where µ is
obtained from λ by relabelling vertices i, i+ 1 and j by the symbols ,◦ and , respectively;
(vi) if λ = λ( ) and vertex i is connected to vertex j < i in λ then eipλ = pµ, where µ is obtained
from λ by relabelling vertices j, i and i+ 1 by the symbols , and ◦, respectively;
(vii) in all other situations eipλ = 0.
There are also analogous formulae for fipλ, which may be obtained from the above by interchanging
the roles of ◦ and .
Proof. In fact we shall only need parts (i), (iv) to prove theorem VI.1.1, so we prove these needed
parts only. The remaining part may be proved in a similar fashion to Theorem 3.9 of [BS]. Since
EiP (λ) is a finitely generated projective module, it is a finite direct sum: EiP (λ) =
⊕
µmµP (µ)
where
mµ = dim Hom(EiP (λ), L(µ)) = dim Hom(P (λ), FiL(µ)).
53
If ϕi(µ) ≤ 1, then FiL(µ) = L(f˜iµ) so mµ = 1 if λ = f˜iµ (i.e. if µ = e˜iλ) and mµ = 0 otherwise. If
ϕi(µ) = 2 then µ = µ(
◦ ) so
mµ ≤ dim Hom(P (λ), Fi∆(µ))
= dim Hom(P (λ),∆(µ( ))) + dim Hom(P (λ),∆(µ( ))).
So if mµ 6= 0 then λ ≤ µ( ) and λ ≤ µ( ), which shows that λ does not have a ◦ or at vertices i
or i + 1. We have shown that eipλ = pe˜iλ for any λ which does not have a ◦ or at vertices i or
i+ 1. This proves (i), (iv), and (vii).
Proof of Theorem VI.1.1. We prove by induction on the defect of µ that pµ = [P (µ)] is the sum of
all vλ = [∆(λ)] such that λ is obtained from µ by switching the orientations of some subset of the
arcs in µ , i.e. switching the labels of the endpoints of the arcs.
If the defect of µ is zero, then µ is maximal, and we have P (µ) = ∆(µ) so pµ = vµ as
needed. Suppose µ has positive defect. Find a pair of vertices i < j connected by an arc in µ with
no vertices labelled or in between i and j. If vertex j − 1 is labelled , then P (µ) = Ej−1P (ν),
where ν is obtained by switching the j − 1, j vertices of µ. Or if vertex j − 1 is labelled ◦, then
P (µ) = Fj−1P (ν), where ν is obtained by switching the j − 1, j vertices of µ. We can thus write
P (µ) as a composition of various Ek, Fk applied to P (ν), where ν is obtained from µ by moving the
at vertex j past any ◦’s and ’s onto vertex i + 1. Now P (ν) = FiP (ν(◦ )). Since P (ν(◦ )) has
smaller defect than P (µ), we have that pν( ◦ ) is the sum of all vλ such that λ is obtained from ν(
◦ )
by switching the orientations of some subset of the arcs in its left arc diagram.
Now pµ is obtained by applying the above composition of the various ek, fk to pν . Each vλ
appearing in the expression for pν( ◦ ) satisfies λ = λ(
◦ ). For such λ we have fivλ = vλ( ) + vλ( )
which means pν is the sum of all vλ such that λ is obtained from ν by switching the orientations of
some subset of the arcs in its left arc diagram. Now the effect of applying the composition of the
ek, fk to each of the vλ is to move the label at vertex i+ 1 back up to vertex j past the ◦’s and ’s
from before. This is so because the rules used in selecting the various Ek, Fk apply to the standard
modules as well. That is, if vertex j − 1 in the marker of µ is labelled , then ∆(µ) = Ej−1∆(ν),
where ν is obtained by switching the j − 1, j vertices of µ, and so on.
Example VI.1.4. Let δ = 0. We determine all µ such that L(µ) is a composition factor of ∆(λ),
where λ = ( , ). The marker for λ is (where all vertices above these are and all vertices below
these are ). The possible well-oriented composite diagrams µλ are shown in Figure 4.
54
...
...
n
n
...
...
n
n
Figure 4: Well-oriented composite diagrams.
where n ≥ 0 and all vertices not shown are at the ends of rays. The corresponding µ’s are shown in
Figure 5.
...
...
n+ 1
n+ 1
...
...
n+ 1
n+ 1
Figure 5: Markers of the composition factors of a standard module.
The first two of these correspond to the partitions ( , ), ( , ), respectively. The fourth
corresponds to the bipartition with both constituent partitions being the partition of (n + 1)2 with
n+ 1 equal parts. The third corresponds to the bipartition obtained from the fourth by removing the
two removable nodes.
VI.2. Characters of Simple Modules in Characterstic 0
We have a method for computing the first several terms of the characters of simples modules
in characteristic zero. Suppose we want to know the terms eai in ch L(λ) with `(a) ≤ n (call this
chnL(λ)).
Theorem VI.1.1 tells us how to write chn ∆(λ) in terms of chn L(µ). Fix a total order
55
refining the partial order on the set of bipartitions µ ≤ λ such that |µ| ≤ n. Then the matrix whose
entry in row µ and column ν is [∆(µ) : L(ν)] is upper unitriangular, hence invertible. Row λ (the
first row) of the the inverse matrix gives chn L(λ) in terms of chn ∆(µ), which we can compute using
the branching graph.
Example VI.2.1. Let m = m′ = 0. We compute ch5 L(∅,∅). Figure 6 shows the upper unitrian-
gular matrix described above.
(∅,∅) ( , )
(
,
) (
,
) (
,
) (
,
) (
,
)
. . .
(∅,∅) 1 1 0 0 0 0 0
( , ) 0 1 1 1 0 1 0(
,
)
0 0 1 0 1 1 0(
,
)
0 0 0 1 0 1 1(
,
)
0 0 0 0 1 0 0(
,
)
0 0 0 0 0 1 0(
,
)
0 0 0 0 0 0 1
...
Figure 6: Decomposition numbers of a standard module.
One can compute the entries in column ν by writing pν in terms of vµ. For example column(
,
)
is computed as follows. Vertices −1, 0, 1, 2 of the marker for ( , ) are shown below with
its left arc diagram (all vertices not shown are at the ends of rays).
So the markers for those µ such that ∆(µ) appears in the ∆-flag of P (ν) are obtained by replacing
the above four vertices with each of
These correspond to µ =
(
,
)
,
(
,
)
,
(
,
)
, ( , ), respectively.
56
We invert the upper left 4× 4 submatrix and find the first row is 1,−1, 1, 1. So
ch5 L(∅,∅) = ch5 ∆(∅,∅)− ch5 ∆ ( , ) + ch5 ∆
(
,
)
+ ch5 ∆
(
,
)
= (e∅ + e↓0↑0 + e↑0↓0 + e↓0↑0↓0↑0 + e↓0↑0↑0↓0 + e↑0↓0↑0↓0 + e↑0↓0↓0↑0 + e↓0↓1↑1↑0 + e↓0↓−1↑−1↑0
+ e↓0↑0↓0↑0 + e↓0↑0↑0↓0 + e↑0↑1↓1↓0 + e↑0↑−1↓−1↓0 + e↑0↓0↑0↓0 + e↑0↓0↓0↑0)
− (e↓0↑0 + e↑0↓0 + e↓0↑0↓0↑0 + e↓0↑0↑0↓0 + e↑0↓0↓0↑0 + e↑0↓0↑0↓0 + e↓0↓1↑1↑0 + e↓0↓−1↑−1↑0
+ e↓0↑0↓0↑0 + e↓0↑0↑0↓0 + e↑0↑1↓1↓0 + e↑0↑−1↓−1↓0 + e↑0↓0↑0↓0 + e↑0↓0↓0↑0
+ e↓0↓1↑0↑1 + e↓0↓−1↑0↑−1 + e↓0↑0↓1↑1 + e↓0↑0↓−1↑−1 + e↓0↑0↑−1↓−1 + e↓0↑0↑1↓1
+ e↑0↑1↓0↓1 + e↑0↑−1↓0↓−1 + e↑0↓0↑1↓1 + e↑0↓0↑−1↓−1 + e↑0↓0↓−1↑−1 + e↑0↓0↓1↑1)
+ (e↓0↓1↑0↑1 + e↓0↑0↓1↑1 + e↓0↑0↑1↓1 + e↑0↑1↓0↓1 + e↑0↓0↑1↓1 + e↑0↓0↓1↑1)
+ (e↓0↓−1↑0↑−1 + e↓0↑0↓−1↑−1 + e↓0↑0↑−1↓−1 + e↑0↑−1↓0↓−1 + e↑0↓0↑−1↓−1 + e↑0↓0↓−1↑−1)
= e∅
The relevant edges of the branching graph are shown in Figure 7.
57
(∅,∅)
( ,∅) (∅, )
(
,∅
)
( ,∅) ( , )
(
∅,
)
(∅, )
(
,
)
( , )
(
,
)
( , )
(
,
)
(
,
)
0 0
1 −1 0 0 −1 1
0 0
1 −1 −1 1
0 0
1
−1 −1
1
Figure 7: Branching graph (computation of character of standard module).
In this case we can actually see that chL(∅,∅) = e∅ because
⊕
a6=∅ 1a∆(∅,∅) is a submod-
ule of ∆(∅,∅) (δ = 0) and dim 1∅∆(∅,∅) = 1, so that L(∅,∅) = 1∅L(∅,∅) is one dimensional.
VI.3. Representations of the Walled Brauer Algebra
Recall the walled Brauer algebra Br,s(δ) = EndOB(δ)(↓r↑s) = 1↓r↑sOB1↓r↑s from the intro-
duction of this thesis. Since Br,s(δ) is an idempotent truncation of OB, Theorem II.4.4 describes
the simple Br,s(δ)-modules. First let Lr,s(λ) = 1↓r↑sL(λ) for λ ∈ Λ, and set Λr,s(δ) = {λ ∈ Λ :
Lr,s(λ) 6= 0}. Then the simple Br,s(δ)-modules are {Lr,s(λ) : λ ∈ Λr,s(δ)}. Using a result of Cox et
al. (see [CDDM]) we are able to describe the set Λr,s(δ).
Theorem VI.3.1. The set {Lr,s(λ) : λ ∈ Λr,s(δ)} is a complete set of inequivalent irreducible
58
Br,s(δ)-modules, where
Λr,s(δ) =

min(r,s)⋃
t=0
Λr−t,s−t δ 6= 0 or r = s = 0
min(r,s)⋃
t=0
Λr−t,s−t \ {(∅,∅)} δ = 0 and r + s > 0.
(VI.3.0.1)
Proof. It is clear that Lr,s(λ) = 0 unless λ ∈
⋃min(r,s)
t=0 Λr−t,s−t. We have also observed that
L(∅,∅) = 1∅L(∅,∅) is one dimensional if δ, so Lr,s(∅,∅) = 0 if r + s > 0. This shows that
Λr,s(δ) contained in the set on the right hand side of (VI.3.0.1). It remains to show these two sets
have the same size. This follows from [CDDM], where the authors show that the right hand set
labels the simple Br,s(δ)-modules by another method.
We can now show that most simple OB-modules are (globally) infinite dimensional.
Corollary VI.3.2. The simple module L(λ) is (globally) finite dimensional if and only if δ = 0 and
λ = (∅,∅).
Proof. We have already observed the “if” part of the statement. Suppose δ 6= 0 or λ 6= (∅,∅). Then
by Theorem VI.3.1 we have λ ∈ Λr+t,s+t(δ) for all t ≥ 0, where λ ` (r, s). Hence 1↓r+t↑s+tL(λ) 6= 0
for all t ≥ 0, showing that L(λ) has infinitely many nonzero weight spaces. So L(λ) is infinite
dimensional.
Remark VI.3.3. It would have been nice to find a proof of Theorem VI.3.1 without appealing to
[CDDM], but we have been unable to do this.
59
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