REPRESENTATIONS OF PARTITION CATEGORIES
by
MAX VARGAS
A DISSERTATION
Presented to the Department of Mathematics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
March 2023
DISSERTATION APPROVAL PAGE
Student: Max Vargas
Title: Representations of Partition Categories
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 Brundan Chair
Ben Elias Core Member
Alexander Kleshchev Core Member
Victor Ostrik Core Member
Michael Kellman Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded March 2023
ii
© 2023 Max Vargas
This work is licensed under a Creative Commons
Attribution-NonCommercial-NoDerivs (United States) License
iii
DISSERTATION ABSTRACT
Max Vargas
Doctor of Philosophy
Department of Mathematics
March 2023
Title: Representations of Partition Categories
We explain a new approach to the representation theory of the partition
category based on a reformulation of the definition of the Jucys-Murphy elements
introduced originally by Halverson and Ram and developed further by Enyang.
Our reformulation involves a new graphical monoidal category, the affine partition
category, which is defined here as a certain monoidal subcategory of Khovanov’s
Heisenberg category. We use the Jucys-Murphy elements to construct some special
projective functors, then apply these functors to give self-contained proofs of results
of Comes and Ostrik on blocks of Deligne’s category Rep(St). We then study a
restriction functor Rep(St)→ Rep(St−1) and prove a conjecture of Comes and Ostrik
involving this functor. Finally, we use the restriction functor to verify a criterion of
Benson, Etingof, and Ostrik, thereby identifying the abelian envelope of Rep(St) with
the Ringel dual of the category of locally finite-dimensional Par t-modules.
This dissertation includes published co-authored material.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Max Vargas
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR, USA
Massachusetts Institute of Technology, Cambridge, MA, USA
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2023, University of Oregon
Bachelor of Science, Mathematics, 2018, Massachusetts Institute of Technology
AREAS OF SPECIAL INTEREST:
Representation Theory
Machine Learning
PROFESSIONAL EXPERIENCE:
Graduate Employee, University of Oregon, 2018-2023
Research Intern, Pacific Northwest National Labs, 2022-2023
GRANTS, AWARDS AND HONORS:
Promising Scholar Award, University of Oregon, 2018
PUBLICATIONS:
Brundan, J., & Vargas, M., A New Approach to the Representation Theory of
the Partition Category. Journal of Algebra, 601 (2022), pp. 198–279.
v
ACKNOWLEDGEMENTS
Foremost I must thank my advisor, Jonathan Brundan. Not only for your
mathematical lessons and guidance throughout the years, but also the unending
patience, kindness, and positivity you have shown.
My family — mother, father, brother, and sister. You have all set examples
for me to follow and have led me to boundless opportunity through your love and
support. It is thanks to you that I have all that I do. And to Boomer, woof woof
rRRrr bark woof rRRrRr bark.
Lastly, I want to thank Jules, Corey, Carol, Mitkins, and LobsterCucumber for
the remaining things not mentioned here.
.
vi
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2. Organization and main results . . . . . . . . . . . . . . . . . 5
II. FOUNDATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1. Categories and their path algebras . . . . . . . . . . . . . . . . 9
2.2. Restriction and induction along functors . . . . . . . . . . . . . 12
2.3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4. Monoidal categories . . . . . . . . . . . . . . . . . . . . . . 16
2.5. Induction product . . . . . . . . . . . . . . . . . . . . . . . 18
2.6. Projective functors . . . . . . . . . . . . . . . . . . . . . . 21
2.7. The symmetric category . . . . . . . . . . . . . . . . . . . . 23
2.8. Triangular decomposition of the partition category . . . . . . . . . 28
2.9. Classification of irreducible modules and highest weight structure . . 33
III. BLOCKS OF THE PARTITION CATEGORY . . . . . . . . . . . . 38
3.1. Schur-Weyl duality . . . . . . . . . . . . . . . . . . . . . . 38
3.2. Heisenberg category . . . . . . . . . . . . . . . . . . . . . . 41
3.3. The affine partition category . . . . . . . . . . . . . . . . . . 44
3.4. Action of APar on kSt-Modfd . . . . . . . . . . . . . . . . . 49
3.5. Jucys-Murphy elements for partition algebras . . . . . . . . . . . 58
3.6. Central elements . . . . . . . . . . . . . . . . . . . . . . . 61
3.7. Harish-Chandra homomorphism . . . . . . . . . . . . . . . . . 67
3.8. “Blocks” . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
vii
Chapter Page
3.9. Special projective functors . . . . . . . . . . . . . . . . . . . 78
3.10. Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.11. Proof of Theorem 3.9.5 . . . . . . . . . . . . . . . . . . . . . 91
IV. RESTRICTION FUNCTOR . . . . . . . . . . . . . . . . . . . . 103
4.1. Phantom partitions . . . . . . . . . . . . . . . . . . . . . . 103
4.2. Restriction and induction functors . . . . . . . . . . . . . . . . 110
4.3. A filtration on restriction . . . . . . . . . . . . . . . . . . . . 113
4.4. The Comes-Ostrik conjecture . . . . . . . . . . . . . . . . . . 123
V. THE ABELIAN ENVELOPE . . . . . . . . . . . . . . . . . . . 127
5.1. Review of abelian envelopes . . . . . . . . . . . . . . . . . . . 127
5.2. Ringel duality and the abelian envelope . . . . . . . . . . . . . 129
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . 134
viii
CHAPTER I
INTRODUCTION
Throughout this dissertation, fix an algebraically closed field k of characteristic
0 as well as a parameter t ∈ k. Chapters II and III both contain material that has
been published and co-authored with Jonathan Brundan in [BV22].
1.1 Overview
In [Del07], Deligne introduced a class of diagrammatic monoidal categories which
generalize a number of classical groups. His motivation was to construct examples
of tensor categories which do not exhibit the property of moderate growth. That is,
he wanted his categories to contain objects X for which the number of composition
factors ofX⊗n should be a super-exponential function of n. The first of these examples
is the category Rep(St), which may be defined as the additive Karoubi envelope of
the partition category, Par t, presented below. When t is a natural number, the usual
category Rep(St) of finite-dimensional representations of the symmetric group is a
certain quotient category of Rep(St) (its semisimplification). For values of t 6∈ N,
Rep(St) can be thought to interpolate between the usual categories Rep(St). From a
representation theoretic perspective, however, Rep(St) remains most interesting when
t ∈ N; otherwise this category is a semisimple abelian category.
This dissertation focuses primarily on the underlying category Par t, which we
now proceed to define via a monoidal presentation. This approach is based on Comes
[Com20, Thm. 2.11]; see also [LSR21, Prop. 2.1]. We use the string calculus for strict
monoidal categories with the convention that vertical composition f ◦ g is given by
stacking f on top of g and horizontal composition f ? g is given by placing f to the
left of g.
1
Definition 1.1.1. The partition category Par t is the strict k-linear monoidal category
generated by one object | and the morphisms
: | ? | → | ? | , : | ? | → | , : | → | ? | , •◦ : | → 1 , •◦ : 1→ | (1.1.1)
subject to the following relations, as well as the ones obtained from these by horizontal
and vertical flips:
= , = , (1.1.2)
= , •◦ = •◦ , (1.1.3)
= , = , (1.1.4)
•◦ = , = , (1.1.5)
◦•
= , = t1. (1.1.6)
•◦
We will sometimes denote the object |?a simply by a. Note |?0 = 1.
Traditionally, e.g. in [CO11, Def. 2.11] or [Del07, § 8], the partition category is
thought of in terms of set partitions instead of generators and relations as above. In
particular, in loc. cit., the morphism spaces HomPar t(a, b) are shown to have bases
given by set partitions of {1, . . . , a, 1′, . . . , b′}. To establish the connection between
their approach and ours, we make use of partition diagrams. By a b × a partition
diagram, we mean the diagram of a morphism f ∈ HomPar t(a, b) built from vertical and
horizontal compositions of the generators in Definition 1.1.1 which has no “floating”
components (any such component can be removed at the cost of the scalar t by
the “dimension relation” (1.1.6)). Labeling the boundary points (from right to left)
along the bottom and top rows of f by 1, . . . , a and 1′, . . . , b′, such a partition diagram
encodes a set partition of {1, . . . , a, 1′, . . . , b′}. Here is an example of a 9×7 partition
2
diagram.
9′ 8′ 7′ 6′ 5′ 4′ 3′ 2′ 1′
•◦ (1.1.7)
7 6 5 4 3 2 1
This partition diagram determines the set partition
{1, 4, 1′, 2′, 3′, 4′, 6′, 8′} t {2, 6} t {3, 5, 9′} t {7, 5′} t {7′}.
There is an equivalence relation on the set of partition diagrams where two diagrams
are equivalent if they determine the same partition of their boundaries. For example,
the morphism inside Par t represented by (1.1.7) is equal to the one represented by
the tidier diagram below because they determine the same partition on the labels of
the endpoints.
9′ 8′ 7′ 6′ 5′ 4′ 3′ 2′ 1′
•◦
(1.1.8)
7 6 5 4 3 2 1
From the defining relations, any diagrams which are equivalent in this sense are
equal as morphisms in Par t. In fact, the converse holds and one obtains a basis for
HomPar t(a, b) by fixing a set of representatives for the equivalence classes of b × a
partition diagrams. The special case when a = b = 0 implies that
EndPar t(1) = k. (1.1.9)
In this dissertation we take a module-theoretic approach to the representation
theory of Par t, working in terms of the⊕associated path algebra:
Part := HomPar t(a, b)
a,b∈N
3
Our analysis takes advantage of the fact that Part has a split triangular decomposition,
following the definition in [BS, Rmk. 5.32]. This is discussed in §2.8 and the key
principle is that for any partition diagram, there is an equivalent partition diagram
which is a composition of various “merges” at the bottom, then crossings, then “splits”
at the top as in (1.1.8). It follows that the category Part-Modlfd of locally finite-
dimensional Part-modules is an upper finite highest weight category in the sense
of [BS]. In particular, there are standard and costandard modules, indecomposable
projective modules have standard flags satisfying BGG reciprocity, and so on. In fact,
Par t is a monoidal triangular category in the sense of Sam and Snowden [SS22] who
have also developed these ideas in a general setting.
With standard modules in hand, our story follows the ideas presented by
Okounkov and Vershik in their approach to the representation theory of the symmetric
groups [OV05]. In the main part of the thesis, we study an induction functor D
induced by the monoidal operation |?? on Par t given by “multiplication with the
generating object”. This plays analogy to the induction functors in the setting of
S
symmetric groups ind n+1S (?) = kSn+1⊗? : Rep(Sn) → Rep(Sn+1). We introduce an
new monoidal category, the affine partition category, in order to better understand
the Enyang-Jucys-Murphy elements of [Eny13]. These let us split the functor D into
indecomposable constituents. Together with some facts about a new family of central
elements of Par t, this affords a new and self-contained analysis of the block structure
of Par t. This was originally worked out by Comes and Ostrik, although their proof
ultimately appealed to results about the partition algebras due to Martin [CO11,
§ 6.3].
The final chapters study a well-known restriction functor F tt−1 : Rep(St) →
Rep(St−1). After reinterpreting this in terms of modules over path algebras to
4
obtain a functor Rtt−1 : Part−1-Mod → Part-Mod in the other direction, we prove a
conjecture of Comes and Ostrik. Namely, restriction gives an equivalence between the
principal blocks of the corresponding categories [CO11, Conj. 6.8]. Finally, we study
the behavior of Rtt−1 on tilting modules. This allows us to apply a recent theorem
from [BEO23] to identify the abelian envelope of Rep(St) with the Ringel dual of
Part-Modlfd in the sense of [BS].
1.2 Organization and main results
Now we go into more detail regarding the layout of the thesis and formulate some
of the main results more precisely. Chapter II is dedicated mostly to introducing
the general techniques that will be used in the sequel. We set up a dictionary to
pass from the framework of k-linear categories and functors to that of modules over
their associated path algebras. Using the theory of highest weights granted by the
split triangular decomposition of Part, we quickly re-prove a classic result of Deligne
which was also proven by Comes and Ostrik using different methods: the isomorphism
classes of irreducible, indecomposable projective, indecomposable injective, standard,
and costandard modules are all indexed by the set of integer partitions, P . Those
modules corresponding to a partition λ ∈ P are denoted by L(λ), P (λ), I(λ),∆(λ),
and ∇(λ), respectively.
The main content of this disseration starts with chapter III. We pass from the
functor |?? : Par t → Par t to the induction functor D : Part-Modlfd → Part-Modlfd,
and then show that it respects modules with a filtration by standard objects.
Specifically, we establish the following combinatorial rule for determining the sections
of D∆(λ). Comes and Ostrik produced a similar result for generic parameter values
in their proof to classify blocks [CO11, Prop. 5.15].
5
Theorem (See Th. 3.9.1). For λ ∈ P, there is a filtration 0 = V0 ⊆ V1 ⊆ V2 ⊆
V3 = D∆(λ) such that ⊕ ( )
V3/V ∼2 = ∆ λ+ a ,
a∈add(λ)
∼ ⊕ ⊕ ( )V2/V1 = ∆(λ)⊕ ∆ (λ− b ) + a ,
⊕ b∈(rem(λ) a∈a)dd(λ− b )
V ∼1/V0 = ∆ λ− b ,
b∈rem(λ)
where add(λ) is the set of addable boxes to λ and rem(λ) is the set removable boxes
from λ.
This chapter also discusses the affine partition category, which we construct as a
certain subcategory of Khovanov’s Heisenberg category. A key feature of APar is its
two new generating morphisms: the ‘left dot’ • and the ‘right dot’ • . It turns out
Par t is a “cyclotomic” quotient of APar and the images of the left and right dots under
this homomorphism APar → Par t produce elements in the partition category which
are closely related to Enyang-Jucys-Murphy elements for the partition algebras. Much
the same as how Jucys-Murphy elements for the symmetric group algebras provide
S
endomorphisms (i.e., natural transformations) of the induction functors ind n+1S , then
Enyang-Jucys-Murphy elements⊕provide endomorphisms of D. From this we find a
functorial decomposition D = a,b∈kDb|a into summands. Each Db|a also respects
standardly-filtered modules, with a more refined combinatorial rule than the one
provided above.
We also construct a family of central elements for the partition algebra in order
to study a Harish-Chandra homomorphism Z(Part) → Z(Sym) from the center of
Part to the center of its Cartan subalgebra. This Harish-Chandra homomorphism
allows us to recover Deligne’s result that Part is semisimple if and only if t 6∈ N. In
6
the non-semisimple case, we use the functors Db|a to rediscover the block structure
of Part-Modlfd. The classification of blocks is summarized below, but more detailed
statements lie throughout §3.10.
Theorem (See Th. 3.10.5). The locally unital algebra Part is semisimple if and
only if t 6∈ N. In the case t ∈ N, the non-simple blocks of Part are in bijection with
isomorphism classes of irreducibles in kSt-Modfd. All of the non-simple blocks are
Morita equivalent, having infinitely many isomorphism classes of irreducible modules
parametrized by N. If L(n),∆(n), and P (n) are the nth irreducible, standard, and
indecomposable projective of some non-simple block, then:
(i) For each n ≥ 0, ∆(n) is of length two with head L(n) and socle L(n+ 1).
(ii) P (0) is isomorphic to ∆(0), while for n ≥ 1 the module P (n) has a two step
∆-flag with top section ∆(n) and bottom section ∆(n− 1).
(iii) For each n ≥ 1, P (n) is self-dual with irreducible head and socle isomorphic to
L(n) and completely reducible heart radP (n)/ socP (n) ∼= L(n− 1)⊕ L(n+ 1).
The theorem immediately implies that each non-simple block is equivalent to the
category of finite-dimensional modules over the algebra defined by the following quiver
with relations:
This equivalence was already proven in [CO11, Th.6.4], though the approach here is
independent of the results of Martin.
In chapter IV, we turn our attention to the restriction functor Rtt−1 :
Part−1-Modlfd → Part-Modlfd. Since t is now changing, we use ∆t(λ) to mean the
7
standard Part-module corresponding to λ, and similarly for other families mentioned
above. The main result in this chapter describes the effect of Rtt−1 on standard
modules:
Theorem (See Th. 4.3.10). For λ ∈ P, ther⊕e is a sh(ort exact) sequence
0→ ∆t(λ)→ Rtt−1∆t−1(λ)→ ∆t λ+ a → 0
a∈add(λ)
From this point, the remainder of the dissertation focuses on the consequenes
of this filtration. The first is an affirmative answer to the conjecture of Comes
and Ostrik involving the principal blocks of Part-Modlfd and Part−1-Modlfd — the
indecomposable subcategories containing the irreducible module L(∅) corresponding
to the empty partition.
Theorem (See Th. 4.4.4). The restriction functor Rtt−1 induces an equivalence
between the principal blocks of Part−1-Modlfd and Part-Modlfd.
Finally, chapter V gives an alternate description of the abelian envelope of
Rep(St). This was originally constructed in [CO14] by considering the heart of
a certain t-structure inside the homotopy category of Rep(St), but more general
constructions have appeared recently (eg. as in [BEO23, Cou21, HS22]). In the case
of Part-Modlfd, we show that the tilting modules familiar in highest weight theory are
the same as the splitting objects of [BEO23, § 2.2]. Proceeding to check the critera
provided in [BEO23, Thm. 2.42] regarding a characterization of abelian envelopes,
this gives our last result:
Theorem (See Th. 5.2.3). The Ringel dual of Part-Modlfd is the abelian
envelope of Rep(St).
8
CHAPTER II
FOUNDATIONS
Fix an algebraically closed field k of characteristic 0. Many of the definitions in
this chapter make sense over any field, but several results require these assumptions
so we fix them now for simplicity. We also establish the standing assumption that
all categories will be k-linear (and small) and algebras will be over k, unless specified
otherwise. Functors between these categories are also assumed to be k-linear. This
chapter summarizes the general background and theoretical techniques to be used
throughout the rest of the dissertation. This includes a recollection of k-linear
categories and their path algebras, monoidal categories, and finally the triangular
decomposition of the partition category. In the final section, we use the machinery of
this triangular decomposition to provide a quick classification of irreducible modules
for the partition category, first proven by Deligne in his seminal paper [Del07].
Most sections of this chapter (except §§2.1 and 2.4) contain previously published
co-authored material (with J. Brundan) appearing in [BV22].
2.1 Categories and their path algebras
Here we discuss a dictionary between k-linear categories and locally unital k-
algebras. Let A be a locally unital k-algebra. That is, A is a non-unital k-algebra
which is equipped with distinguished system of mutually orthogonal idempotents
{ei ∈ A | i ∈ I} indexed by some set I s⊕o that
A = ejAei.
i,j∈I
From A, one can construct a k-linear category C (A) whose objects are given by the
indexing set I and whose morphisms are given by HomC(A)(i, j) := ejAei for i, j ∈ I.
The identity morphism of the object i ∈ I is given by ei, and composition is induced by
multiplication: for any two morphisms a ∈ HomC(A)(i, j) and b ∈ HomC(A)(j, k), the
9
composition b◦a is given by the product ba. Whenever A is locally finite-dimensional,
in the sense that ejAei is finite-dimensional for all i, j ∈ I, then C (A) is locally finite,
in the sense that HomC(A)(i, j) is finite-dimensional for all i, j ∈ I.
Conversely, if C is a k-linear category then we can construct a locally unital
k-algebra A(C ) which we⊕call the path algebra of C . Letting O(C ) denote the object
set of C , define A(C ) := X,Y ∈O(C) HomC (X, Y ). The distinguished idempotents of
A(C ) are the identity morphisms 1X for all X ∈ O(C ). Given a ∈ HomC (X, Y ) and
b ∈ HomC (W,Z), the product ba isdefined as below.b ◦ a if Y = Wba = 0 else.
The full algebra structure on A(C ) is obtained by linearly extending the above rule.
Notice that for any X, Y ∈ O(C ), 1YA(C )1X = HomC (X, Y ). If C is locally finite,
then A(C ) is locally finite-dimensional.
Under these constructions, notice that we have C (A(C )) = C and A(C (A)) = A
for any k-linear categories C and locally unital k-algebras A. So, the data of A is
fundamentally equivalent to the data contained in the original category C . Henceforth
we will drop the notation A(C ) for the path algebra of C , opting instead to just use
A (or some other symbol whenever appropriate).
N⊕ow fix a k-linear category C with path algebra A. Given a left A-module
M = X∈O(C) 1XM , there is an associated covariant k-linear functor F : C → Vec
from C to the category of k-vector spaces. On objects X ∈ O(C ), F (X) := 1XM .
Given a morphism a ∈ HomC (X, Y ), the linear map F (a) : 1XM → 1YM is obtained
by acting on the left with a. This construction can be performed in the opposite
direction too. Starting with a (k-linear) covariant functor F : C → Vec, there is an
associated A-module M built as follows. Define for each X ∈ O(C ) the vector space
10
⊕
1XM := F (X). Then M := X∈O(C) 1XM . Given a ∈ 1YA1X and m ∈ 1ZM , the
left A-module structure on M is given by
F (a)(m) if X = Za ·m = 0 else.
The data of the module M is equivalent to the data of the original functor F . That
is to say, these constructions provide isomorphisms between the category A-Mod of
left A-modules and the category of k-linear covariant functors C → Vec. Because of
this, we will refer to the latter category as the category of left C -modules, denoted
C -Mod.
Imitating the above constructions with right A-modules and contravariant
functors C → Vec yields an isomorphism between the category of right A-modules
and the category of contravariant functors from C to Vec. Following as above,
contravariant functors C → Vec will be called right C -modules and the category
of such modules will be denoted Mod-C .
Restricting our attention to the category Vec fd of finite-dimensional k-vector
spaces, let C -Modlfd (resp. Modlfd-C ) be the category of covariant (resp.
contravariant) functors C → Vec fd. Also let A-Modlfd (resp. Modlfd-A) be the
category of locally finite-dimensional left (resp. right) A-modules: those A-modules
M for which each subspace 1XM (resp. M1X) is finite-dimensional for all X ∈ O(C ).
Then there is an equivalence C -Modlfd ' A-Modlfd (resp. Modlfd-C ' Modlfd-A).
Finally, there is the category A-Proj (resp. Proj-A) of finitely generated
projective left (resp. right) A-modules. If A is locally finite-dimensional then this is a
subcategory of A-Modlfd (resp. Modlfd-A). Under the contravariant (resp. covariant)
Yoneda embedding, A-Proj (resp. Proj-A) is equivalent to the Karoubi envelope
Kar(C ) of C — this is the idempotent completion of the additive envelope Add(C ).
11
2.2 Restriction and induction along functors
Consider a⊕functor F : A → B betwee⊕n two locally finite categories with path
algebras A = X,Y ∈O(A) 1YA1X and B = X,Y ∈O(B) 1YB1B. Precomposition with
F allows us to naturally restrict B-modules G : B → Vec to recover A-modules
G ◦ F : A → Vec. This easy construction on the categorical level translates to the
algebra level to give a restriction functor
resF : B-Mod→ A-Mo⊕d. (2.2.1)
To describe this functor, consider a B-module⊕M = X∈B 1XM . Then define
resF M = 1FM := 1F (Y )M. (2.2.2)
Y ∈O(A)
The left action of A on 1FM is defined so that a ∈ A1Y acts on the summand
1F (Y )M by the linear map F (a) and zero on all other summands. For a B-module
homomorphism φ : M → N , we obtain the restricted A-module morphism resF (φ) :
1F (M)→ 1F (N). For the summand corresponding to Y ∈ O(A), resF (φ) just applies
the evident linear map 1F (Y )M → 1F (Y )N,m 7→ φ(m). It is easy to see that resF is
an exact functor.
An analogous construction can be applied to get an exact functor between
categories of right modules:
F res : Mod-B → Mod-A. ⊕ (2.2.3)
In particular, F res sends a right B-module M to M1F := Y ∈O(A) M1F (Y ). An
alternate notation for these functors (e.g. used by Sam and Snowden in [SS22]) is F ∗
and (F op)∗, respectively.
⊕ View B itself as a (B,B)-bimodule and restrict on the right to get B1F =
X∈O(A) B1F (X). This is a (B,A)-b⊕imodule, and moreover, the functor resF :
B-Mod → A-Mod is isomorphic to X∈O(A) HomB(B1FX ,−). The tensor-Hom
12
adjunction in the setting of module categories for locally unital algebras (eg. as
in [BS, Lem. 2.2]) gives a left adjoint to resF , namely
indF := B1F ⊗A − : A-Mod→ B-Mod. (2.2.4)
Having already remarked that resF is exact, it follows that indF is right exact and
sends projective A-modules to projective B-modules. Choosing X ∈ O(A) and
examining indF A1X , we have
indF A1X := B1F ⊗A A1 ∼X = B1F (X).
From this observation along combined with the fact that left adjoints are
cocontinuous, it follows that if M is a finitely generated A-module, then indF M
i⊕s a finitely generated B-module. Alternatively, for any such M there is a surjection
i∈I A1X →M for I a finite set. Then apply the right exactness of indF .i
Her⊕e is another construction, starting with a left restriction to get the module
1FB = X∈O(A) 1FXB. This is a (A,B)-bimodule. Since resF is also isomorphic
to 1FB ⊗B −, the tensor-Hom adjuction in this setting gives a right adjoint, the
coinduction along F : ⊕
coindF := HomA(1FB1Y ,−) : A-Mod→ B-Mod
Y ∈O(B)
The functor coindF is right exact and sends injectives to injectives, being a right
adjoint to the exact functor resF . In [SS22, §3.6], the same functors were studied
under the names F! and F∗, respectively. Similar constructions can be made with
right modules instead of left modules to recover functors F ind and F coind. These
are called (F op)! and (F
op)∗ in [SS22], respectively.
Lemma 2.2.1. Let F : A → B be a functor as above.
(1) If B1F is a projective right A-module then indF and F coind are exact functors.
13
(2) If 1FB is a projective left A-module then F ind and coindF are exact functors.
Proof. This follows from the definitions of these functors and basic properties of
projective modules in abelian categories.
Lemma 2.2.2. Let A be a category (with path algebra A) having additive envelope
 = Add(A) (with path algebra Â). The natural inclusion I : A →  extends to an
equivalence Kar(I) : Kar(A)→ Kar(Â) . Hence
indI : A-Mod→ Â-Mod
is an equivalence.
Proof. The first statement follows since Kar(I) is fully faithful and dense. By
Yoneda, indI : A-Proj → Â-Proj is an equivalence. It immediately follows that
indI : A-Mod → Â-Mod is an equivalence (see [BD17, Cor. 2.5]) But this is true by
combining the first statement with the Yoneda equivalence.
It follows in the case of Lemma 2.2.2 that the right adjoint, resI , is also an
equivalence. So the right adjoint of resI (being coindI) will also be an equivalence
and ind ∼I = coindI .
Suppose that F,G : A → B are functors. A natural transformation α : F ⇒ G
induces natural transformations resα : resF ⇒ resG, indα : indG ⇒ indF and coindα :
coindG ⇒ coindF . In particular, restricting to weight spaces, we can explicitly define
these natural transformations as follows. Starting with resα, for any B-module M
and any Y ∈ O(A), we have:
resα(M) : 1F (Y )M → 1G(Y )M, m 7→ αY ·m (2.2.5)
14
For indF and coindF , take any A-module N and objects X ∈ O(A) and Y ∈ O(B).
Then we have for each 1F (X)-subspace
indα(M) : B1G(X) ⊗A 1XN → B1F (X) ⊗A 1XN (2.2.6)
b⊗m 7→ b · αX ⊗m
coindα(M) : HomA(1G(X)B1Y , N)→ HomA(1F (X)B1Y , N) (2.2.7)
ϕ 7→ (b 7→ ϕ(αX · b))
Similarly, α induces natural transformations αres : Gres ⇒ F res, αind : F ind ⇒ Gind
and αcoind : F coind ⇒ Gcoind. Assuming for simplicity1 that A = B, so that F
and G are k-linear endofunctors of A , these constructions define k-linear monoidal
functors
res rev∗ : End k(A)→ End k(A-Mod) , ind∗, coind∗ : End opk(A) → End k(A-Mod),
(2.2.8)
∗res : End k(A)op → End k(Mod-A)rev, ∗ind, ∗coind : End k(A)→ End k(Mod-A).
(2.2.9)
Here, End k(A) denotes the strict k-linear monoidal category of (k-linear)
endofunctors and natural transformations, “op” means the opposite category with
the same monoidal product, and “rev” means the same category with the reversed
monoidal product.
2.3 Duality
Continue with A and B be the path algebras of locally-finite categories A and
B, respectively. There is a contravariant functor
?~ : A-Mod→ Mod-A (2.3.1)
1To formulate analogs of (2.2.8) and (2.2.9) without this assumption, one needs to work in the
strict 2-category of k-linear categories.
15
⊕ ⊕
taking V = X∈O 1XV to V
~ := ∗X∈O (1XV ) , the direct sum of the linearA A
duals of the “weight spaces” 1XV . The restriction of this to locally finite-
dimensional modules is an equivalence, with quasi-inverse given by the restriction
of the analogously-defined duality functor
~? : Mod-A→ A-Mod (2.3.2)
in the other direction. To obtain a duality (= contravariant auto-equivalence) on
A-Modlfd from (2.3.1) and (2.3.2), one also needs a k-linear equivalence σ : A →
Aop. Restriction along σ gives equivalences resσ : Modlfd-A → A-Modlfd and σres :
A-Modlfd → Modlfd-A, hence, we obtain the duality functor
?©σ := resσ ◦?~ = ~? ◦ σres : A-Modlfd → A-Modlfd. (2.3.3)
Given a functor F : A → B, we obviously have that
?~ ◦ res ∼F = F res ◦?~ (2.3.4)
as functors from B-Mod to Mod-A. We deduce that
~? ◦ F ind ∼= coindF ◦~?, ~? ◦ F coind ∼= ind ~F ◦ ? (2.3.5)
as functors from Mod-A to B-Mod.
2.4 Monoidal categories
By a monoidal category, we mean a (k-linear) category C equipped with a
bifunctor ? : C ×C → C , k-linear in both variables of course, along with associativity
and unit contraints satisfying the pentagon and triangle axioms (see [EGNO15]).
Typically in this dissertation we require that EndC (1) ∼= k. Usually our monoidal
categories will also be strict monoidal, defined diagrammatically by generators and
relations. Note that we use the symbol ? for the monoidal product, reserving the
more traditionally-used ⊗ for the tensor product ⊗k over the ground field k.
16
Given another category A , we say that A is a (strict) C -module category if there
is a (strict) monoidal functor Ψ : C → End k(A) (see [EGNO15] for an alternate
definition). A left (resp. right, two-sided) tensor ideal I of C is the data of subspaces
I(X, Y ) ≤ HomC (X, Y ) for all X, Y ∈ O(C ), such that these subspaces are closed
in the obvious sense under the monoidal product on the left (resp. right, two-sides)
as well as composition either before or after with any morphism. Then C/I is the
category with the same objects as C and morphisms that are the quotient spaces
HomC (X, Y )/I(X, Y ). If I is a left (resp. right) tensor ideal, then C/I is a left
(resp. right) C -module category. If I is a two-sided tensor ideal, then C/I is again a
monoidal category.
An object X ∈ O(C) has a left dual if there is some X∨ ∈ O(C) with evaluation
and coevaluation morphisms ev : X∨X ?X → 1 and coevX : 1→ X?X∨ satisfying the
zig-zag relations. Nameley, the zig-zag relations state that the following compositions
are the identity on X and X∨, respectively.
−c−oevX −X−−?−id→X idX ? X∨ ? X −−X−?−e−v→X X (2.4.1)
∨ −id−X−?−c−oe−v→X ∨ ∨ −e−vX−−? i−dX X ? X ? X →X X∨ (2.4.2)
There is another notion that X has a right dual if there is some ∨X with morphisms
X ev : X ?
∨X → 1 and X coev : ∨1→ X ?X satisfying similar zig-zag identities. We
say that C is rigid if every object has a left and right dual.
In adopting the diagrammatic notation for categorical calculations in this
dissertation, the identity morphism of X is often represented by |X . Evaluation
and coevaluation are depicted as caps and cups, where the “phase change” between
X and X∨ occurs at the critical point.
X X∨
evX = , coevX =
X∨ X
17
The zig-zag equations 2.4.1 turn into the following, explaining the terminology.
= , = (2.4.3)
X X X∨ X∨
A related notion is that of a pivotal category; this is when C is a rigid monoidal
∼
category equipped with the data of isomorphisms αX : X −→ (X∨)∨ natural in X ∈
O(C ). Moreover, the morphisms αX must respect the monoidal structure of C , in
that αX?Y = αX ? αY for all X, Y ∈ O(C ). For a pivotal category, the notions of left
and right duals coincide. In such a category, assuming also that EndC (1) ∼= k, the
dimension of an object X is defined by the value of the morphism evX∨ ◦(αX⊗1X∨)◦
coevX ∈ k.
By a symmetric monoidal category, we mean a monoidal category equipped with
isomorphisms σX,Y : X ? Y → Y ? X satisfying the hexagon relation, and moreover,
σY,X ◦σX,Y = idX?Y for all X, Y ∈ O(C). See [EGNO15] for the full definition. A rigid
monoidal category C which is also symmetric can readily be equipped with a pivotal
structure. Such a structure can be described by identifying, for any X ∈ O(C ), its
left and right duals. In order to perform this identification, it is enough to prescribe
the evaluation map X ? X∨ → 1 and coevaluation map 1 → X∨ ? X. They are
diagrammatically defined as follows:
X X∨
evX∨ = , coevX∨ =
X∨ X
In other words, we have evX∨ = evX ◦σX,X∨ and coevX∨ = σX∨,X ◦ coevX . In this
case, the dimension of X is the value of the morphism 1→ 1 : evX ◦σX,X∨ ◦ coevX .
2.5 Induction product
Let C be a strict monoidal category with path algebra C. The monoidal product
? on C extends canonically to the Karoubian envelope, Kar(C ). This section describes
a monoidal product ?fwhich makes C-Mod into a (no longer strict) monoidal category.
18
It turns out that C-Proj is closed under ?fand this product agrees with the extension
of ? to Kar(C ) through the contravariant Yoneda equivalence. The algebraist’s
formulation of the definition for ?f is given in the next paragraph; see also [SS15,
(2.1.14)], [SS22, §3.10], where ?fis called Day convolution. Using ?f, we can make the
split Grothendieck group K0(C) of the category C-Proj into a ring with multiplication
[P ][Q] := [P ?fQ]. (2.5.1)
Its identity element is the isomorphism class of the distinguished projective module
C11, where 1 ∈ O(C ) is the unit object.
Here is the detailed definition of ?f. Let C  C be the k-linearization of the
Cartesian product C × C . The objects in C  C are pairs (X1, X2) ∈ O(C ) × O(C ),
and the morphism space from (X1, X2) to (Y1, Y2) is HomC (X1, Y1)⊗ HomC (X2, Y2).
We denote its path algebra by ⊕
C  C = 1Y1C1X1 ⊗ 1Y2C1X2 .
X1,X2,Y1,Y2∈O(C)
Multiplication in C  C is the obvious “tensor-wise” product just like for a tensor
product of algebras. If C is locally finit⊕e, so too is C C. Given V1, V2 ∈ C-Mod, let
V1  V2 = 1X1V1 ⊗ 1X2V2
X1,X2∈O(C)
be their tensor product over k viewed as a left C  C-module in the obvious way.
In fact, this defines a functor  : C-Mod  C-Mod → C  C-Mod. The monoidal
product on C is a functor ? : C  C → C⊕. Let
C1? = C1X1?X2
X1,X2∈O(C)
be the (C,C C)-bimodule obtained by restricting the right C-module C along this
functor. Induction along ?, that is, the functor ind? = C1?⊗CC : C  C-Mod →
C-Mod from (2.2.4), is left adjoint to the restriction functor res? from (2.2.1). Then
19
the induction product is the composition
?f:= ind? ◦  : C-Mod C-Mod→ C-Mod. (2.5.2)
Thus, for V1, V2 ∈ C-Mod, we have that V ?f1 Vf 2
= C1? ⊗CC (V1  V2). Associativity
of ? (up to natural isomorphism) follows from “transitivity of induction”, i.e., the
associativity of tensor products of modules over locally unital algebras. We obviously
have that
C1 ?fX1 C1 ∼X2 = C1X1?X2 (2.5.3)
for X1, X2 ∈ O(C ). This justifies our earlier assertion that ?f extends the monoidal
product ? on Kar(C ). It also follows that V ?f1 V2 is finitely generated if both V1 and
V2 are finitely generated. The induction product ?f is right exact in both arguments,
but in general it is not left exact.
Lemma 2.5.1. If C1? is a projective right C C-module then the induction product
?f is biexact.
Proof. This follows from Lemma 2.2.1.
Finally suppose that C is a strict symmetric monoidal category, so that there is
given a symmetric braiding σ. From this, we obtain a braiding indσ making C-Mod
into a k-linear symmetric monoidal category too.
Remark 2.5.2. There is a second convolution product ?f which we call the
coinduction product. This is defined by replacing ind? with coind? in (2.5.2). It is easy
to understand on injective rather than projective modules. It will not often be used
subsequently, but note that the induction and coinduction products are interchanged
by duality.
20
2.6 Projective functors
Suppose that C is a strict monoidal category and A is a strict C -module category,
denoting their path algebras by C and A as usual. The data of the functor Ψ : C →
End k(A) is equivalent to the data of a strictly associative and unital k-linear monoidal
functor ? : C  A → A . For f ∈ HomC (X,X ′), we sometimes denote the evaluation
of the natural transformation Ψ(f) on Y ∈ O(A) simply by fY : X ? Y → X ′ ? Y .
The definition of the induction product ? from (2.5.2) extends naturally to this
setting, thereby defining a functor
?f:= ind? ◦  : C-Mod A-Mod→ A-Mod (2.6.1)
which makes A-Mod into a (no longer strict) C-Mod-module category. For objects
X ∈ O(C ) and Y ∈ O(A), we have that
C1 ?fA1 ∼X Y = A1X?Y , (2.6.2)
i.e., ?f extends ? : C  A → A . Using ?f to define the action as in (2.5.1), the split
Grothendieck group K0(A) becomes a left module over the split Grothendieck ring
K0(C).
Now fix X ∈ O(C ) and consider the functor X? : A → A . There is an adjoint
pair of endofunctors (indX?, resX?) of A-Mod defined by induction and restriction
along X?: ⊕
indX? := A1X? ⊗A where A1X? := A1X?Y , (2.6.3)
Y⊕∈OA
resX? := 1X?A⊗A where 1X?A := 1X?YA. (2.6.4)
Y ∈OA
The general properties discussed earlier give that resX? is exact, and indX? is right
exact and sends (finitely generated) projectives to (finitely generated) projectives.
Thus, indX? restricts to a well-defined functor indX? : A-Proj → A-Proj. Note also
21
that
indX?(A1Y ) ∼= A1X?Y (2.6.5)
for all Y ∈ O(A). One can also interpret indX? as a special induction product, thanks
to the following lemma.
Lemma 2.6.1. For any X ∈ O(C ), we have that ind ∼X? = C1X ?f.
Proof. This follows from the chain of isomorphisms
A1X? ⊗ V ∼A = (A1? ⊗CA (C1X  A))⊗A V
∼= A1? ⊗CA ((C1X  A)⊗A V )
∼= A1? ⊗CA (C1X  V ) = C1X ?fV
for V ∈ A-Mod.
∼
Lemma 2.6.2. If X has a left dual Y in C then there is an isomorphism φ : 1X?A→
A1Y ? of (A,A)-bimodules given explicitlyby X · · ·  · · ·
ϕ  = ff (2.6.6)
· · · · · ·Y
Hence, the functors resX? and indY ? are isomorphic.
Proof. It is easily checked that φ is a bimodule homomorphism. It is an isomorphism
because it has a two-sided inverse ψ defined by · · ·g 

X
 · · ·ψ = g .
· · ·
Y · · ·
Corollary 2.6.3. If X has a left dual Y in C then (indX?, indY ?) and (resX?, resY ?)
are adjoint pairs of functors.
22
From the corollary, we deduce that if X is rigid, then both of the functors indX?
and resX? have both a right and a left adjoint. Moreover, as discussed earlier, both
of these functors are exact and they preserve finitely generated projectives. We will
refer to finite direct sums of direct summands of endofunctors of A-Mod of this sort
as projective functors.
2.7 The symmetric category
It is worth recalling the following basic example. The symmetric category Sym
is the free strict symmetric monoidal category on one object. Our analysis of
the partition category in chapter III is largely motivated by the Okounkov-Vershik
approach to the representation theory of symmetric groups which we review here
[OV05]. In string diagrams, we denote this generating object simply by |; then an
arbitrary object is the monoidal product |?n for some n ≥ 0. Morphisms in Sym are
generated by a single morphism depicted by the crossing
: | ? | → | ? | (2.7.1)
subject to the relations
= , = . (2.7.2)
Sometimes it is convenient to identify objects in Sym with natural numbers, so that the
object set {|?n |n ∈ N} of Sym is identified with N. For m,n ≥ 0, the morphism space
HomS ym(n,m) is {0} if m =6 n, while if m = n it consists of k-linear combinations of
string diagrams representing permutations in the symmetric group Sn, i.e., we have
that EndSym(n) = kSn. Note our general convention here is to number strings by
1, . . . , n from right to left, so that the transposition (1 2) ∈ Sn is represented by the
string diagram
· · · .
n 3 2 1
23
Let Sym be the path algebra of Sym .⊕Thus, we have that
Sym = kSn. (2.7.3)
n≥0
Since k is of characteristic zero, we deduce from Maschke’s theorem that Sym is
a semisimple locally unital algebra. In this case, the induction product ?f making
Sym-Mod into a monoidal category is nothing more than the usual induction product
on representations of the symmetric groups: we have that
S
V ?fW = ind n+mS (V W )n×Sm
for V ∈ kSn-Mod and W ∈ kSm-Mod viewed as Sym-modules using (2.7.3). In fact,
the induction product ?fand the coinduction product ?fon Sym-Mod are isomorphic
S
as ind n+m ∼ SS ×S = coind
n+m
S ×S (as always for finite groups).n m n m
Recall that the irreducible kSn-modules are the Specht modules S(λ)
parametrized by the set Pn of partitions λ = (λ1, λ2, . . . ) of n. Hence, the
irreducible⊔Sym-modules are the Specht modules S(λ) parametrized by all partitions
λ ∈ P = n≥0Pn. We sometimes write |λ| for the size λ1 + λ2 + · · · of a partition
λ ∈ P , and `(λ) for its length, that is, the number of non-zero parts. We will
often identify λ ∈ P with its Young diagram. For example, the partition (5, 32, 2) is
identified with
0 1 2 3 4
−1 0 1
−2 −1 0
−3 −2
The content of the node in row i and column j of a Young diagram is the integer
c = j − i (as above). Let add(λ) be the set consisting of the contents of the addable
nodes of λ, that is, the places in the Young diagram where a node can be added
to the diagram to obtain a new Young diagram. Similarly, let rem(λ) be the set of
24
contents of the removable nodes of λ, that is, the places in the Young diagram where
a node can be removed from the diagram to obtain a new Young diagram. Note that
all of the addable and removable nodes of a Young diagram are of different contents
(another of the benefits of working in characteristic zero). For a ∈ add(λ), let λ+ a
be the partition obtained by adding the unique addable node of content a to the
diagram. For b ∈ rem(λ), let λ − b be the partition obtained by removing the
unique removable node of content b from the diagram.
The combinatorial notions just introduced arise naturally on considering
branching rules for the symmetric group. In our setup, the sums over all n ≥ 0
S S
of the usual restriction and induction functors res n+1 and ind n+1Sn S = kSn n+1⊗kSn are
isomorphic to the functors
F := res| ? : Sym-Modfd → Sym-Modfd, E := ind| ? : Sym-Modfd → Sym-Modfd,
(2.7.4)
notation as in (2.6.3) and (2.6.4). This follows because the functor
· · · · · ·
| ? :Sym → Sym , g 7→ g . (2.7.5)
· · · · · ·
coincides with the natural inclusion Sn ↪→ Sn+1 on permutations g ∈ Sn ⊂ EndSym(n).
The canonical adjunction makes (E,F ) into an adjoint pair of functors. In fact, these
functors are are biadjoint, i.e., there is also an adjunction making (F,E) into an
adjoint pair. The effect of the functors F and E on the Specht module S(λ) is well
known: we have th⊕at ⊕
FS(λ) ∼= S(λ− b ), ES(λ) ∼= S(λ+ a ). (2.7.6)
b∈rem(λ) a∈add(λ)
We finally recall a bit about the Jucys-Murphy elements in Sym . One natural
way to obtain these is to start from the affine symmetric category ASym , which is the
strict k-linear monoidal category obtained from Sym by adjoining an extra generator
25
◦• subject to the equivalent relations
•◦ = ◦• + , •◦⊕ = •◦ + . (2.7.7)
The path algebra ASym is isomorphic to n≥0AHn where AHn is the nth degenerate
affine Hecke algebra. There is an obvious faithful strict k-linear monoidal functor
i : Sym → ASym . There is also a unique (non-monoidal) full k-linear functor
p : ASym → Sym (2.7.8)
such that p ◦ i = IdSym and ( )
p ··· •◦ = 0 (2.7.9)
n 2 1
for all n ≥ 1. For 1 ≤ j ≤ n, the jth Jucys-Murphy element of the symmetric group
Sn is ( )
j−1
x = p ··· •◦ ···
∑
j = (i j) ∈ kSn, (2.7.10)
n j 1
i=1
i.e., it is the sum of the transpositions “ending” in j. Whenever we use this notation,
it should be clear from context exactly which symmetric group we have in mind. Note
x1 = 0 always. We may also occasionally write x0, which should be interpreted as
zero by convention.
The Jucys-Murphy elements x1, . . . , xn generate a commutative subalgebra of
kSn known as the Gelfand-Tsetlin subalgebra. As concisely explained by [OV05],
for λ ∈ Pn, each Jucys-Murphy element acts diagonalizably on the Specht module
S(λ), and the Gelfand-Tsetlin character of S(λ) recording the dimensions of the
simultaneous generalized eigenspaces of x1, . . . , xn may be obtained from the contents
of standard λ-tableaux. Indeed, Young’s orthonormal basis {vT} for S(λ) indexed by
standard λ-tableaux T is a basis of simultaneous eigenvectors for x1, . . . , xn, with xj
26
acting on vT as the content contj(T) of the node labelled by j in T . We will assume
the reader is familiar with these ideas without giving any further explanation.
The functor p induces an isomorphism ASym I →∼/ Sym where I is the left tensor
ideal of ASym generated by the morphism •◦ . It follows that Sym is a strict left
ASym-module category. The functors E and F from (2.7.4) are also the induction
and restriction functors ind| ? and res| ? defined using this categorical action of ASym
on Sym . The advantage of passing from Sym to ASym here is that the object | of ASym
has the endomorphism defined by the dot, giving us a natural transformation
α := •◦ ? : | ?⇒ | ? .
Applying the general construction from (2.2.8) to this, we obtain endomorphisms
x := resα : F ⇒ F, x∨ := indα : E ⇒ E. (2.7.11)
Explicitly, on a kSn-module V , xV is the endomorphism of FV = resSnS V definedn−1
by multiplying on the left by xn ∈ kSn, while x∨V is the endomorphism of EV =
kSn+1⊗kSnV defined by multiplying the bimodule kSn+1 on the right by xn+1 ∈ kSn+1.
For c ∈ k, let Fc and Ec be the c eigenspaces of x : F ⇒ F and x∨ : E ⇒ E,
respectively — x and x∨ are diagonalizable as char(k) = 0. Since x∨ is the mate of
x and E and F are biadjoint, it follows that Ec and Fc are biadjoint endofunctors of
Sym-Modfd for each c ∈ k. The description of Gelfand-Tsetlin characters of Specht
modules from the previous paragraph is equivalent to the assertion that the functors
Ea and Fb take the Specht module S(λ) to exactly the summands S(λ + a ) and
S(λ − b ) in (2.7.6), or to zero if a ∈/ add(λ) or b ∈/ rem(λ), respectively. It follows
that ⊕ ⊕
F = Fb, E = Ea. (2.7.12)
b∈Z a∈Z
27
2.8 Triangular decomposition of the partition category
In this section we return to the partition category defined in the introduction,
expanding on its triangular structure as illustrated by the discussion surrounding
(1.1.7) and (1.1.8). As a preliminary, observe that Par t is a rigid monoidal category.
The justification of this fact requires only to specify the evaluation ev : | ? | → 1 and
coevaluation coev : 1→ | ?| morphisms for the single generating object of Par t. They
are defined below in terms of the generating morphisms provided in Definition 1.1.1.
◦•
ev = := , coev = := •◦ . (2.8.1)
In particular, it can easily be checked using (1.1.5) that these definitions satisfy the
zig-zag identites of (2.4.3).
Let c be a connected component in some partition diagram representing a
morphism in Par t. We call c an upward branch if c has at least two endpoints on
its top boundary and no endpoints on its bottom boundary, and a downward branch
if it has at least two endpoints on its bottom boundary but no endpoints at the top:
c = ··· or c = ··· .(upward) (downward)
We call c an upward leaf if it has exactly one endpoint at the top and no endpoints
at the bottom, and a downward leaf if it has no endpoints at the top and exactly one
at the bottom:
c = •◦ or c = ◦• .
(upward) (downward)
We refer to c as an upward tree if it has more than one endpoint at the top and exactly
one endpoint at the bottom, and a downward tree if it has exactly one endpoint at
the top and more than one endpoint at the bottom:
···
c = or c =
(upward) ··· (downward)
28
We say that c is a double tree if c has more than one endpoint at the top and more than
one endpoint at the bottom. In that case, it is equivalent to the composition of an
upward tree and a downward tree; for example, the rightmost connected component
in (1.1.8) is a double tree. Finally we say that c is a trunk if c has exactly one endpoint
both at the top and at the bottom:
c = .
Any connected component of a partition diagram can be represented either as an
upward branch, an upward leaf, an upward tree, a downward branch, a downward
leaf, a downward tree, a double tree, or a trunk.
Let f be an m× n partition diagram. We say f is
– a permutation diagram if all of its connected components are trunks, in which
case we must have that m = n;
– an upward partition diagram if its connected components are trunks, upward
branches, upward leaves and upward trees, in which case we must have that
m ≥ n;
– a downward partition diagram if its connected components are trunks, downward
branches, downward leaves and downward trees, in which case we must have
that m ≤ n.
Let f be an upward m × n partition diagram. We say that it is strictly upward if
m > n. Let c1, . . . , ck be the connected components of f that are either trunks or
upward trees, indexing them so that their bottom endpoints are in order from right
to left in f . We say that f is normally ordered if the rightmost of the top endpoints
of each of c1, . . . , ck are also in order from right to left in f . In other words, f is
normally ordered if it can be drawn so that the right edges of all of the upward trees
29
and trunks in f are non-crossing. Similarly, we define strictly downward and normally
ordered downward partition diagrams.
Now we can define some monoidal subcategories of Par t. Let Sym be the
symmetric category as defined in §2.7. There is a strict k-linear symmetric monoidal
functor
i◦t : Sym → Par t (2.8.2)
sending the generating object and the generating morphism of Sym to the generating
object and the generating morphism of Par t that is represented by the crossing. By the
discussion in §1.1 stating that Par t has basis consisting of partition diagrams, it follows
that this functor is faithful. We use it to identify Sym with a monoidal subcategory
of Par t. In other words, Sym is identified with the subcategory of Par t consisting
of all objects and all the morphisms which can be written as linear combinations of
permutation diagrams.
Next, let Par [ be the strict k-linear monoidal category generated by one object
| and the morphisms
: | ? | → | ? | , : | → | ? | , ◦• : 1→ | (2.8.3)
subject to the relations (1.1.2) to (1.1.4) and their flips in a vertical axis. We call
this the upward partition category. The cup can also be defined in Par [ as in (2.8.1).
Any upward partition diagram can be interpreted as a string diagram representing a
morphism in Par [. Moreover, the defining relations in Par [ imply that two upward
m×n partition diagrams which are equivalent in the sense that they define the same
partition of the set {1, . . . , n, 1′, . . . ,m′} labelling the endpoints are also equal as
morphisms in HomPar [(n,m). There is a strict k-linear monoidal functor
i[ [t : Par → Par t (2.8.4)
30
sending the generating morphisms of Par [ to the corresponding ones in Par t. Note
that any diagram built from compositions and monoidal products of the generating
morphisms (2.8.3) can be interpreted as an upward m× n partition diagram. Hence,
equivalence classes of upward m × n partition diagrams span HomPar [(n,m). Since
their images in HomPar t(n,m) are linearly independent too, this functor is faithful.
We use it to identify Par [ with a monoidal subcategory of Par t. In other words, Par [
is identified with the monoidal subcategory of Par t consisting of all objects and all
of the morphisms which can be written as linear combinations of upward partition
diagrams. Also let Par− be the monoidal subcategory of Par [ consisting of all objects
and all of the morphisms which can be written as linear combinations of normally
ordered upward partition diagrams.
Similarly to the previous paragraph, we define Par ], the downward partition
category, to be the strict k-linear monoidal category generated by one object | and
the morphisms that are the flips of (2.8.3) in a horizontal axis, subject to the relations
that are the flips of the ones for Par [. The cap can also be defined in Par ] as in (2.8.1).
Evidently, Par ] ∼= (Par [)op with isomorphism being defined by the flip σ in a horizontal
axis. There is a strict k-linear monoidal functor
i] : Par ]t → Par t (2.8.5)
sending the generating morphisms of Par ] to the corresponding ones in Par t. We have
that i]t = σ ◦ i[t ◦ σ, so we deduce from the previous paragraph that i
]
t is faithful too.
We use it to identify Par ] with a monoidal subcategory of Par t. In other words, Par ]
is identified with the monoidal subcategory of Par t consisting of all objects and all
of the morphisms which can be written as linear combinations of downward partition
diagrams. Also let Par + be the monoidal subcategory of Par ] consisting of all objects
31
and all of the morphisms which can be written as linear combinations of normally
ordered downward partition diagrams.
Finally we let Part be the path algebra of Par t. It is a locally unital algebra
with distinguished idempotents {1n |n ∈ N} arising from the identity endomorphisms
of the objects of Par t. We also have the path algebras Par[, Par−, Sym, Par+, Par]
of Par [,Par−, Sym ,Par +,Par ], which we may view as locally unital subalgebras of
Part via the embeddings (2.8.2), (2.8.4) and (2.8.5). The following theorem is the
triangular decomposition of Part.
⊕
Theorem 2.8.1. Let K := n≥0 k1n viewed as a locally unital subalgebra of Part.
Multiplication defines a linear isomorphism
Par− ⊗ ∼K Sym⊗K Par+ → Part. (2.8.6)
Hence, we also have isomorphisms
Par− ⊗ ∼Sym→ Par[K , (2.8.7)
Sym⊗ Par+K →
∼
Par], (2.8.8)
∼
Par[ ⊗ Par]Sym → Part. (2.8.9)
Proof. Any partition diagram is equivalent to a diagram that is the composition
of a normally ordered upward partition diagram, a permutation diagram, and a
normally ordered downward partition diagram; see (1.1.8) for an example of such
a decomposition. Moreover, equivalence classes of these sorts of diagrams give bases
for Part, Par
−, Sym and Par+. This implies that (2.8.6) is an isomorphism. Then
(2.8.7) to (2.8.9) follow as in [BS, Rem. 5.32].
32
Theorem 2.8.1 is all that is needed to see that the locally finite-dimensional
locally unital algebra ⊕
Part = 1mPart1n
m,n∈N
has a split triangular decomposition in the sense of [BS, Rem. 5.32]. Spelling this out
in the case of Part, we have:
– distinguished idempotents that are indexed by an upper finite poset. In this
case, N equipped with the ordering reverse to the usual one.
– locally unital subalgebras Par−, Sym, and Par+.
– subspaces Par[ = Par− · Sym and Par] = Sym · Par+ which are subalgebras,
and not merely subspaces.
– the multiplication map of (2.8.6) is an isomorphism.
– for n,m ∈ N, 1mSym1n is zero unless m = n, 1mPar−1 +n and 1nPar 1m are zero
unless n < m (with the reverse ordering), and 1 − +nPar 1n = 1nPar 1n = k1n
for all n ∈ N.
The subalgebras Par[ and Par] are referred to as the negative and positive Borel
subalgebras, respectively. Additionally, Sym = Par[ ∩ Par] is the Cartan subalgebra.
We stress that our imposition that the ordering on N be reversed gives an upper finite
poset, conforming to the general conventions of [BS].
2.9 Classification of irreducible modules and highest weight structure
As Part has a triangular decomposition with Cartan subalgebra Sym being
semisimple, we can appeal to the general results of [BS, §5.5] to obtain the
classification of irreducible Part-modules. Alternatively, this follows from the results
in [SS22, §5.5], but note that Sam and Snowden use the language of lowest weight
33
rather than highest weight categories. Since isomorphism classes of irreducible Part-
modules are in bijection with isomorphism classes of indecomposable projective Part-
modules, and the latter are identified with isomorphism classes of indecomposable
objects in Kar(Par t), the results discussed in this section are equivalent to the
classification obtained originally in [CO11, Th. 3.7].
The algebra Part is Z-graded with 1mPart1n being in degree m−n. The induced
gradings on the subalgebras Par[ and Par] make these into positively and negatively
graded algebras, respectively, with degree zero components in both cases being the
semisimple algebra Sym. It follows that the Jacobson radicals of Par[ and Par] are
the direct sums of their non-zero graded components. Moreover, the quotients by
their Jacobson radicals are naturally identified with Sym, i.e., there are locally unital
algebra homomorphisms
π[ :Par[  Sym, π] :Par]  Sym. (2.9.1)
Let infl] : Sym-Modfd → Par]-Modfd and infl[ : Sym-Modfd → Par[-Modfd be the
functor{s defined by restrict∣ion alon}g these {homomorphisms. Th∣ e modu}les
S[(λ) := infl[ S(λ) ∣ λ ∈ P , S](λ) := infl] S(λ) ∣ λ ∈ P (2.9.2)
give full sets of pairwise inequivalent irreducible modules for Par[ and Par],
respectively.
As in [BS, (5.13)–(5.14)], we define the standardization and costandardization
functors
j! := ind
Part
] ◦ infl] : Sym-Modfd → Part-Modlfd, (2.9.3)Par
j∗ := coind
Part
[ ◦ infl[ : Sym-⊕ModPar fd → Part-Modlfd, (2.9.4)
where indPart] := Par ⊗ and coindPartt Par] [ := n∈N HomPar[(Part1n, ?). From (2.8.6)Par Par
to (2.8.8) it follows that Part is projective both as a right Par
]-module and as a left
34
Par[-module, hence, these functors are exact. Then we define the standard and
costandard modules for Part by
∆(λ) := j Part ]!S(λ) = ind ] S (λ), ∇(λ) = j S(λ) = indPart [∗ [ S (λ), (2.9.5)Par Par
respectively.
Theorem 2.9.1. The Part-modules {L(λ) | λ ∈ P} defined from
L(λ) := hd ∆(λ) ∼= soc∇(λ)
give a complete set of pairwise inequivalent irreducible left Part-modules. Moreover,
Part-Modlfd is an upper finite highest weight category in the sense of [BS, Def. 3.34]
with weight poset (P ,), where  is the partial order on P defined by λ  µ if and
only if either λ = µ or |λ| > |µ|. Its standard and costandard objects are the modules
∆(λ) and ∇(λ), respectively.
Proof. This follows immediately from [BS, Cor. 5.39] using the triangular
decomposition from Theorem 2.8.1 and the semisimplicity of Sym; see also [SS22,
§5.5].
The fact established in Theorem 2.9.1 that Part-Modlfd is an upper finite highest
weight category has several significant consequences. As is the case for any category
equivalent to A-Modlfd for a locally finite locally unital algebra A (‘Schurian’ in the
language of [BS]), L(λ) has a projective cover we denote by P (λ) ∈ Part-Modlfd. Let
Part-Mod∆ be the exact subcategory of Part-Modlfd consisting of all modules with a
∆-flag, that is, a finite filtration whose sections are of the form ∆(λ) for λ ∈ P . For
any V ∈ Part-Mod∆, the multiplicity (V : ∆(µ)) of ∆(µ) as a section of some ∆-flag
in V is well-defined independent of the flag, indeed, it can be calculated from
(V : ∆(µ)) = dim HomPart(V,∇(µ)). (2.9.6)
35
This follows from the fundamental Ext-vanishing property of highest weight
categories, namely, that
dim ExtiPar (∆(λ),∇(µ)) = δt i,0δλ,µ (2.9.7)
for any λ, µ ∈ P and i ≥ 0; see [BS, Lem. 3.48]. The definition of highest weight
category gives that P (λ) has a ∆-flag, so that Part-Proj is a full subcategory of
Part-Mod∆. Moreover, from (2.9.6), one obtains the usual BGG reciprocity formula
(P (λ) : ∆(µ)) = [∇(µ) : L(λ)]. (2.9.8)
∼
The functor σ : Par → (Par )opt t defined by flipping diagrams in a horizontal
axis can also be viewed as a locally unital anti-involution of the algebra Part. It
interchanges the subalgebras Par[ and Par], and restricts to an anti-involution also
denoted σ on the subalgebra Sym. Let ?©σ be the duality on Sym-Modfd taking a
finite-dimensional left Sym-module to its linear dual viewed again as a left module
using the anti-automorphism σ. Since σ(g) = g−1 for a permutation g ∈ Sn ⊂ Sym,
this is the usual duality on each of the subcategories kSn-Modfd. It is well known
that the irreducible kSn-modules are self-dual, hence,
S(λ)©σ ∼= S(λ) (2.9.9)
for all λ ∈ P . There is also a duality ?©σ on Part-Modlfd defined as in (2.3.3). Similarly,
as σ interchanges Par[ and Par], we get contravariant equivalences also denoted ?©σ
between Par]-Modlfd and Par
[-Modlfd. Similarly to (2.3.4) and (2.3.5), we have that
?©σ ◦ infl[ ∼= infl] ◦?©σ , indPart ◦?©σ ∼=?©σ ◦ coindPart] [ . (2.9.10)Par Par
Hence:
j ◦?©σ ∼=?©σ ◦ j , j ◦?©σ! ∗ ∗ ∼=?©σ ◦ j! (2.9.11)
36
as functors from Sym-Modfd to Part-Modlfd. Then from (2.9.9) and (2.9.11), we
deduce that
∆(λ)©σ ∼= ∇(λ), ∇(λ)©σ ∼= ∆(λ), L(λ)©σ ∼= L(λ) (2.9.12)
for λ ∈ P .
The duality ?©σ will be called the Chevalley duality on Part-Modlfd, following the
language of [BS, Def. 4.49]
37
CHAPTER III
BLOCKS OF THE PARTITION CATEGORY
This chapter contains previously published co-authored material from [BV22].
We introduce an auxiliary monoidal category APar , the affine partition category.
We define this as a certain monoidal subcategory of the Heisenberg category Heis ,
exploiting an observation of Likeng and Savage from [LSR21]. We then use APar to
give a new approach to the definition of the Jucys-Murphy elements of Part. These
were first defined in the context of the partition algebra by Halverson and Ram [HR05]
and computed recursively by Enyang [Eny13]. We also construct more general central
elements.
The second half of this capter studies these central elements and their
images under an analog of the Harish-Chandra homomorphism. This affords us
a decomposition of Par t-Mod as a product of subcategories, which turn out to be
precisely the blocks. In fact, Part is semisimple if and only if t ∈/ N, while if t ∈ N
the non-simple blocks are in bijection with partitions of t. We also determine the
structure of the non-simple blocks and explicitly show that they are all equivalent,
recovering the results of Comes and Ostrik [CO11]. Our approach is similar to the
Okounkov-Vershik approach to representations of Sym as reviewed in §2.7.
3.1 Schur-Weyl duality
Recall the generators and relations for the partition category from Definition 1.1.1.
The following theorem of Deligne will play a key role in this section; see e.g. [Com20,
Th. 2.3] for a proof.
Theorem 3.1.1. Suppose that t ∈ N. Let Ut be the natural permutation
representation of the symmetric group St with standard basis u1, . . . , ut. Viewing
kSt-Modfd as a symmetric monoidal category via the usual Kronecker tensor product
38
⊗, there is a full symmetric monoidal functor ψt : Par t → kSt-Modfd sending the
generating ob(ject)| to Ut and defined on generating morphisms by
ψt( ) : Ut ⊗ Ut → Ut ⊗ Ut , ui ⊗ uj →7 uj ⊗ ui ,
ψt( ) : Ut ⊗ Ut → Ut, ui ⊗ uj →7 δi,jui ,
ψt( ) : Ut → Ut ⊗ Ut, ui 7→ ui ⊗ ui ,
ψt( ◦• ) : Ut → k , ui 7→ 1 ,
ψt ◦• : k→ Ut 1 7→ u1 + · · ·+ ut .
Furthermore, the linear map HomPar t(n,m) → Hom (U⊗n, U⊗mkSt t t ), f 7→ ψt(f) is an
isomorphism whenever t ≥ m+ n.
For the next corollary, we assume some basic facts about semisimplification of
monoidal categories; e.g., see [BEEO20, Sec. 2] which gives a concise summary of
everything needed here.
Corollary 3.1.2. When t ∈ N, the functor ψt induces a monoidal equivalence ψt
between the semisimplification of Kar(Par t) and kSt-Modfd. In particular, Part is
not a semisimple locally unital algebra in these cases.
Proof. The functor ψt extends canonically to a functor Kar(Par t) → kSt-Modfd. It
is well known that every irreducible kSt-module appears as a constituent of some
tensor power of Ut, hence, this functor is dense. Now the first statement follows from
the fullness of the functor ψt using [BEEO20, Lem. 2.6]; see also [Del07, Th. 2.18]
and [CO11, Th. 3.24]. Since Kar(Par t) has infinitely many isomorphism classes of
irreducible objects, it is definitely not equivalent to its semisimplification kSt-Modfd.
This shows that Kar(Par t) is not a semisimple Abelian category as it contains non-
zero negligible morphisms. Equivalently, the path algebra Part is not semisimple in
these cases, which is the second statement.
39
Remark 3.1.3. Continue to assume that t ∈ N. By the general theory of
semisimplification, the irreducible objects in the semisimplification of Kar(Par t)
correspond to the indecomposable projective Part-modules P (λ) of non-zero
categorical dimension. In [Del07, Prop. 6.4], Deligne showed that P (λ) has non-
zero categorical dimension if and only if t−|λ| > λ1−1, in which case the irreducible
object of the semisimpliciation arising from P (λ) corresponds under the equivalence
ψt to the irreducible kSt-module S(κ) where κ := (t− |λ|, λ1, λ2, . . . ).
Through the next few sections, our goal is to introduce the affine partition
category. In one sense, this is an extension of the partition category with extra
generators and relations. Though it would be nice to view Par t as embedding into its
affine version, our realization of the affine partition category is as a subcategory of
the Heisenberg category of [Kho14] and does not satisfy the final relation in (1.1.6)
which is dependent on t. So we introduce the generic partition category Par as the
strict monoidal category with the same generating object and generating morphisms
as Par t subject to all of the same relations except for the final relation in (1.1.6),
which is omitted. The morphism
•◦
T := ∈ End (1) (3.1.1)
•◦ Par
is strictly central in Par , so that Par can be viewed as a k[T ]-linear monoidal category.
For t ∈ k, let
evt : Par → Par t (3.1.2)
be the canonical functor taking T to t11. Using the basis theorem for Par t for infinitely
many values of t, one obtains a basis theorem for the generic partition category: each
morphism space HomPar (n,m) is free as a k[T ]-module with basis given by a set of
representatives for the equivalence classes of m × n partition diagrams. From this,
40
we see that evt induces an isomorphism k ⊗k ∼[T ] Par = Par t, where on the left hand
side we are viewing k as a k[T ]-module so that T acts as t. This point of view is
often useful since it can be used to prove a statement involving relations in Par t for
all values of t just by checking it for all sufficiently large positive integers, in which
case Theorem 3.1.1 can often be applied to reduce to a question about symmetric
groups. To make a precise statement, let
φt := ψt ◦ evt : Par → kSt-Modfd, (3.1.3)
assuming t ∈ N.
Lemma 3.1.4. If f ∈ HomPar (n,m) satifies φt(f) = 0 for infinitely many values of
t ∈ N then f = 0.
∑
Proof. We can write f = i pi(T )fi for polynomials pi(T ) ∈ k[T ] and fi running over
a set of representatives f∑or the equivalence classes of m×n partition diagrams. Since
φt(f) = 0 we have that i pi(t)φt(fi) = 0 for i∑nfinitely many values of t. By the final
assertion in Theorem 3.1.1, this implies that i pi(t) evt(fi) = 0 for infinitely many
values of t ≥ m+n. By the basis theorem in Par t, this means for each i that pi(t) = 0
for infinitely many values of t. Hence, pi(T ) = 0 for each i.
We note that the proof of Lemma 3.1.4 depends on our standing assumption that
the ground field k is of characteristic zero.
3.2 Heisenberg category
Next we recall the definition of the Heisenberg category Heis which was
introduced by Khovanov in [Kho14]. We follow the approach of [Bru18]; Khovanov’s
category is denoted Heis−1(0) in the more general setup developed there.
41
Definition 3.2.1 ([Bru18, Rem. 1.5(2)]). The Heisenberg category Heis is the strict
monoidal category with two generating objects ↑ and ↓ and five generating morphisms
:↑ ? ↑→↑ ? ↑ , : 1→↓ ? ↑ , :↑ ? ↓→ 1 , : 1→↑ ? ↓ , :↓ ? ↑→ 1,
subject to the following relations:
= , = , (3.2.1)
= , = , (3.2.2)
= 0, = 11, (3.2.3)
= − , = . (3.2.4)
Here, we have used the the sideways crossings which are defined from
:= , := .
It is also convenient to introduce the shorthand
•◦ := , (3.2.5)
which automatically satisfies the degenerate affine Hecke algebra relation as in (2.7.7):
•◦ = •◦ + , ◦• = •◦ + . (3.2.6)
Note by (3.2.3) that
•◦ = = 0. (3.2.7)
In addition, the following relations hold, so that Heis is strictly pivotal with duality
functor defined by rotating diagrams through 180◦:
= , = , (3.2.8)
•◦ := ◦• = •◦ , := = . (3.2.9)
42
Then we obtain further variations on (3.2.6) by rotating through 90◦ or 180◦ using
this strictly pivotal structure. One more useful consequence of the defining relations
is that
= , (3.2.10)
Heis → HeisopThere is also a symmetry σ : , which is the strict monoidal functor
that is the identity on objects and sends a morphism to the morphism obtained by
reflecting in a horizontal axis and then reversing all orientations of strings.
⊕ Khovanov constructed a categorical action of Heis on Sym-Modfd =
n≥0 kSn-Modfd, i.e., a strict monoidal functor
Θ : Heis → End k(Sym-Modfd). (3.2.11)
Explicitly, this takes the generating objects ↑ and ↓ to the induction functor E and
the restriction functor F , respectively, notation as in (2.7.4), and Θ takes generating
morphisms for Heis to the natural transformations defined on a kSn-module V as
(follow)s (where g is an element of the appropriate symmetric group):
: kSn+2 ⊗kSn+1 kSn+1 ⊗kSn V → kSn+2 ⊗kSn+1 kSn+1 ⊗kSn V,
V
( ) g ⊗ 1⊗ v 7→ g(n+1 n+2)⊗ 1⊗ v,
: kSn ⊗kSn−1 V → kSn+1 ⊗kSn V, g ⊗ v →7 g(n n+1)⊗ v,( )V  g2 ⊗ g1v if g = g2(n n+1)g1 for gi ∈ Sn,
: kSn+1 ⊗kSn V → kSn ⊗kSn−1 V, g ⊗ v 7→
( )V  0 otherwise,
: V → V, v 7→ (n−1 n)v,
V
( )V : kSn ⊗kSn−1 V → V, g ⊗ v 7→ g∑v,n
( )V : V → kSn ⊗kSn−1 V v 7→ (i n)⊗ (i n)v,
i=1
43

 gv if g ∈ Sn,( )V : kSn+1 ⊗kSn V → V, g ⊗ v 7→ 0 otherwise,
(( ))V : V → kSn+1 ⊗kSn V v 7→ 1⊗ v,
( ◦• ) : kSn+1 ⊗kSn V → kSn+1 ⊗kSn V, g ⊗ v →7 gxn+1 ⊗ v,V
•◦ : V → V, v 7→ xnv.
V
In the last two formulae, we have used the Jucys-Murphy elements xn+1 ∈ kSn+1
and xn ∈ kSn from (2.7.10), respectively; the natural transformations here are the
endomorphisms of E and F denoted x and x∨ just before (2.7.12). All of the other
formulae displayed here can also be found in [LSR21, §3]. Note in particular that the
clockwise bubble acts as multiplication by n on any V ∈ kSn-Modfd.
It is known moreover that the functor Θ is faithful. Indeed, in [Kho14], Khovanov
uses the functor Θ to prove a basis theorem for morphism spaces in Heis , and the
argument implicitly establishes the faithfulness of Θ over fields of characteristic zero.
We will not use this here in any essential way.
3.3 The affine partition category
Now the background is in place and we can make a new definition.
Definition 3.3.1. The affine partition category APar is the monoidal subcategory of
Heis generated by the object | :=↑ ? ↓ and the following morphisms
:= + , (3.3.1)
:= , := , (3.3.2)
•◦ := , •◦ := , (3.3.3)
• := •◦ + , • := •◦ + , (3.3.4)
• := + , • := + . (3.3.5)
44
We refer to the morphisms in (3.3.4) as the left dot and the right dot, and the
morphisms in (3.3.5) as the left crossing and the right crossing, respectively. The
other shorthands for the generating morphisms of APar introduced in Definition 3.3.1
are the same as the symbols used for generators of the partition category. This
is deliberate, indeed, the morphisms (3.3.1) to (3.3.3) generate a copy of the generic
partition category Par as a monoidal subcategory of Heis . This important observation
is due to Likeng and Savage; see Corollary 3.4.4 below. For now, we just need
the following, which is proved in [LSR21] by a direct calculation using the defining
relations in Heis .
Lemma 3.3.2 ([LSR21, Th. 4.1]). There is a strict monoidal functor
i : Par → APar (3.3.6)
sending the generating object and generating morphisms of Par to the generating object
and generating morphisms in APar denoted by the same diagrams.
Because of the symmetry of the generators of APar under rotation through 180◦,
the strictly pivotal structure on Heis restricts to a strictly pivotal structure on APar .
The left and right dots are duals, as are the left and right crossings. Moreover, the
cap and the cup making | into a self-dual object are given by the same formula (2.8.1)
as we had before in Par , hence, i is a pivotal monoidal functor. Note also that
◦•
T = = . (3.3.7)
•◦
Also, the symmetry σ on Heis restricts to σ : APar → APar op. This just reflects affine
partition diagrams in a horizontal axis, just like the earlier anti-automorphism σ on
Par t. Here are some further relations, all of which are easily proved using the defining
relations in Heis :
• •
• =
• , = , • • • ••◦ = ◦• , • = • . (3.3.8)
45
Of course, the horizontal and vertical flips of all of these also hold. The next two
lemmas establish some less obvious relations.
Lemma 3.3.3. The following relations hold in APar :
◦• ◦•
• = • = • , • = = • , (3.3.9)•◦ ◦• •
• = • , • = • , (3.3.10)
• •
= • = , = • = , (3.3.11)
• •
• • • •
= = , = = . (3.3.12)
• • • •
Proof. For each of (3.3.9) to (3.3.11), it suffices just to prove the first equality, and
then all the others follow using σ and duality to reflect in horizontal and/or vertical
axes. For (3.3.9), use (3.3.8) and (1.1.5). To prove (3.3.10), we expand as morphisms
in Heis to see that
(3.2.3) (3.2.6)• = •◦ + •◦ + = ◦• + +
(3.2.7)
(3.2.3)
= + = • .
For (3.3.11), we again expand the left hand side as a morphism in Heis :
(3.2.3) (3.2.3) (3.2.1)
= + = + = + = • .
• (3.2.4)
Finally, to prove (3.3.12), the second set of relations follows from the first set of
relations by composing on the bottom with a crossing and using (3.3.11). For the
first set of relations, it suffices to prove the first equality, the second then follows by
duality. Expanding both of the left crossings as morphisms in Heis produces a sum
of four terms, two of which are zero, so we obtain:
• (3.2.3) (3.2.3) (3.2.1)
= + = + = +
• (3.2.4)
46
(3.2.4)
= − + = = .
Corollary 3.3.4. As a monoidal category, the subcategory APar of Heis is generated
by the object | , the five undotted generators (3.3.1) to (3.3.3), and the left dot.
Proof. The relations (3.3.9) and (3.3.10) together show that the right dot and the left
and right crossings may be written in terms of the left dot and the other undotted
generating morphisms.
Lemma 3.3.5. The following relations hold:
• = • + • + • − • − • , (3.3.13)
•
• = • + + − • − • , (3.3.14)
•
• = • + • + • − • − • , (3.3.15)
• = • + • + • − • − • . (3.3.16)
Proof. To prove (3.3.13), we observe by composing on the bottom with the crossing
and using relations from Par plus (3.3.11) that the relation we are trying to prove is
equivalent to
• + • + = + • + • .
• •
Now we expand the left hand side in terms of morphisms in Heis using (3.2.3)
and (3.2.7), then we use (3.2.6) to commute the dot past a crossing in the first
and fourth terms:
• + • ◦•+ = + + ◦• + ◦• + + 3
• ◦•
47
= •◦ + + + •◦ + + + 3 .•◦
Similarly, the expansion of the right hand side is
+ • + • = + + + + •◦ + 3
• •◦ •◦
= •◦ + + + + + •◦ + 3 ,•◦
where we commuted the dot past a crossing just in the first term. These are equal.
To deduce (3.3.14), first apply duality to (3.3.13), i.e., rotate through 180◦. Then
compose on the top and bottom with a crossing and simplify using relations in Par
together with (3.3.11).
To prove (3.3.15), we rewrite its left hand side, replacing the right crossing with
a left dot using (3.3.10), then we apply (3.3.13) to push this left dot past the right
hand string:
• = • = • + • + • − • − • .
Now we simplify the five terms on the right hand side of the equation just displayed
to obtain the five terms on the right hand side of (3.3.15) (there is no need here to
expand in terms of morphisms in Heis). The following treats the first term:
• • (3.3.10)= = • .
The second and third terms are easy to handle, we omit the details. For the fourth
and the fifth terms, it suffices by symmetry to consider the fifth term, which we
rewrite as follows:
• (3.3.10) • (3.3.11)• = = = • .
Finally (3.3.16) follows easily from (3.3.15) on composing on the bottom with the
crossing of the leftmost two strings and using (3.3.11).
48
Corollary 3.3.6. The category APar has object set {|?n | n ∈ N} (which we often
identify simply with N) and morphisms that are linear combinations of vertical
compositions of morphisms in the image of i : Par → Heis together with the morphisms
· · · • (3.3.17)
n 2 1
for all n ≥ 1.
Proof. In view of Corollary 3.3.4, we just need to show that one can obtain the
endomorphism of n defined by the left dot on the mth string (m = 1, . . . , n) by
taking a linear combinations of compositions of morphisms in the image of i and the
given morphism (3.3.17) (in which the left dot is on the first string). This follows by
induction on m using relations (3.3.13) and (3.3.10).
Remark 3.3.7. We have not attempted to formulate or prove a basis theorem for
the morphism spaces in APar . Creedon and De Visscher do this by combining their
work with results of Khovanov [CD23]. They also show that APar is actually the full
monoidal subcategory is Heis generated by the object |.
3.4 Action of APar on kSt-Modfd
Suppose that t ∈ N. The restriction of the functor Θ from (3.2.11) to the
subcategory APar is a strict monoidal functor APar → End k(Sym-Modfd) sending
the generating object | to the endofunctor E ◦ F (induction after restriction). Since
E ◦ F takes kSt-modules to kSt-modules, the restriction of Θ gives strict monoidal
functors
Θt : APar → End k(kSt-Modfd). (3.4.1)
The functor Θt takes | to the endofunctor indStS ◦ res
St = kS ⊗
t−1 St−1 t kSt−1 of kSt-Modfd;
this should be interpreted as the zero functor in the case t = 0. The natural
49
transformations arising by applying Θt to the other generating morphisms of APar
may be computed using the formulae after (3.2.11) (taking n := t). Explicitly, one
ob(tains )the following for V ∈ kSt-Modfd and g, h ∈ St:
: kSt ⊗kSt−1 kSt ⊗kSt−1 V → kSt ⊗kSt−1 kSt ⊗kSt−1 V,
V
( ) g ⊗ h⊗ v →7 gh⊗ h−1 ⊗ hv,
• : kSt ⊗kSt−1 kSt ⊗kSt−1 V → kSt ⊗kSt−1 kSt ⊗kSt−1 V,
V
( ) g ⊗ h⊗ v →7 g ⊗ h⊗ (h−1(t) t)v,
• : kSt ⊗kSt−1 kSt ⊗kSt−1 V → kSt ⊗kSt−1 kSt ⊗kSt−1 V,
V
( ) g ⊗ h⊗ v 7→ gh⊗ (h
−1(t) t)⊗ v,
( ) : kSt ⊗kSt−1 kSt ⊗kSt−1 V → kSt ⊗kSt−1 V, g ⊗ h⊗ v 7→ δh(t),tgh⊗ v,V
( ) : kSt ⊗kSt−1 V → kSt ⊗kSt−1 kSt ⊗kSt−1 V, g ⊗ v 7→ g ⊗ 1⊗ v,V
( ◦• ) : kSt ⊗kSt−1 V → V, g ⊗ v 7→ gv,V ∑t
◦• : V → kSt ⊗kSt−1 V, v →7 (i t)⊗ (i t)v,
( )V ∑i=1t
• : kSt ⊗kSt−1 V → kSt ⊗kSt−1 V, g ⊗ v 7→ g(j t)⊗ v,
( )V ∑j=1t
• : kSt ⊗kSt−1 V → kSt ⊗kSt−1 V, g ⊗ v 7→ g ⊗ (j t)v.
V
j=1
Recall the natural kSd-module Ut from Theorem 3.1.1; in particular, U0 is the zero
module. Using the Kronecker product, we can consider Ut⊗ as an endofunctor of
kSt-Modfd. Also let trivSt be the trivial module.
50
Lemma 3.4.1. The functor Θt is monoidally isomorphic to the strict monoidal
functor
Φt : APar → End k(kSt-Modfd) (3.4.2)
which sends the generating object | to the endofunctor Ut⊗ and taking the generating
morphisms for APar to the natural transformations defined as follows on V ∈
kSt-M(odfd a)nd 1 ≤ i, j ≤ t:
( ) : Ut ⊗ Ut ⊗ V → Ut ⊗ Ut ⊗ V, ui ⊗ uj ⊗ v 7→ uj ⊗ ui ⊗ v,V
( • ) : Ut ⊗ Ut ⊗ V → Ut ⊗ Ut ⊗ V, ui ⊗ uj ⊗ v →7 ui ⊗ uj ⊗ (i j)v,V
( • ) : Ut ⊗ Ut ⊗ V → Ut ⊗ Ut ⊗ V, ui ⊗ uj ⊗ v →7 uj ⊗ ui ⊗ (i j)v,V
( ) : Ut ⊗ Ut ⊗ V → Ut ⊗ V, ui ⊗ uj ⊗ v 7→ δi,jui ⊗ v,V
( ) : Ut ⊗ V → Ut ⊗ Ut ⊗ V, ui ⊗ v 7→ ui ⊗ ui ⊗ v,V
( •◦ ) : Ut ⊗ V → V, ui ⊗ v 7→ v,V ∑t
•◦ : V → Ut ⊗ V, v →7 ui ⊗ v,
( )V ∑i=1t
• : Ut ⊗ V → Ut ⊗ V, ui ⊗ v 7→ uj ⊗ (i j)v,
( )V ∑j=1t
• : Ut ⊗ V → Ut ⊗ V, ui ⊗ v 7→ ui ⊗ (i j)v.
V
j=1
∼
Proof. There is an isomorphism kSt ⊗kSt−1 trivSt−1 → Ut, g ⊗ 1 →7 gut. Combining
this with the tensor identity, we obtain a natural kSt-module isomorphism
(t) ∼
(α1 )V : kSt ⊗kSt−1 V → Ut ⊗ V, g ⊗ v 7→ gut ⊗ gv (3.4.3)
(t) ∼
for V ∈ kSt-Modfd. This defines an isomorphism α1 : kSt⊗kSt−1 ⇒ Ut⊗. Let
(t) (t) (t) (t)
αn := α1 · · ·α1 be the n-fold horizontal composition of α1 . This is a natural
(t) ∼
isomorphism α ◦n ◦nn : (kSt⊗kSt−1) ⇒ (U⊗) whose value on a kSt-module V is given
51
e(xpli)citly by the map
α(t)n : gn ⊗ · · · ⊗ g1 ⊗ v →7 gnut ⊗ gngn−1ut ⊗ · · · ⊗ gngn−1 · · · g1ut ⊗ gngn−1 · · · gV 1v.
Now define Φt : APar → End k(kSt-Modfd) to be the strict monoidal functor taking
the object n(to (U) ⊗)◦nt , and defined on a morphism f ∈ HomAPar (n,(m) b)y Φt(f) :=(t)
αm ◦ (t)
−1 (t)
Θt(f)◦ αn . It is immediate from this definition that α(t) = αn ≥ : Θt ⇒n 0
Φt is an isomorphism of strict monoidal functors.
It remains to check that Φt as defined in the previous paragraph is equal to the
functor Φt defined on generating morphisms in the statement of the lemma. So we
need to check for each generating morphism f ∈ HomAPar ((n,m)) that the form(ula for(t) ◦ ◦ (t))−1Φt(f)V written in the statement of the lemma is equal to αm Θt(f)V αV n V
for V ∈ kSt-Modfd and t ∈ N. This is a routine but lengthy calculation. We just go
through a couple of the cases. (( )(t)) (◦ ◦ (t))−1If f is the crossing, we need to show that α2 Θt(f)V α2 (u ⊗u ⊗V V i j
v) =(u(j ⊗)ui ⊗ v. Now w(e co)nsid)er four cases. If t 6= i 6= j =6 t we have that(t) ◦ (t) −1α2 ΘV t(f)V ◦ α2 (ui(⊗(uj ⊗ v)V )(t))
= ( α2) ◦Θt(f)V ((i t)⊗ (j t)⊗ (j t)(i t)v)V(t)
= α2 ((i t)(j t)⊗ (j t)⊗ (i t)v)V
= uj ⊗ ui ⊗ v.
If i = j (w(e ha)ve that ( ) )(t)
α2 ◦
(t) −1
Θ
V t
(f)V ◦ α2 (u ⊗V i((uj ⊗ v) )(t))
= ( α2) ◦Θt(f)V ((i t)⊗ 1⊗ (i t)v)V(t)
= α2 ((i t)⊗ 1⊗ (i t)v)V
= uj ⊗ ui ⊗ v.
52
If i = t(6=( j w(t))
e have that ( ) )
α2 ◦
(t) −1
Θt(f)V ◦ α2 (ui(⊗(u ⊗ v)V V j )(t))
= ( α2) ◦Θt(f)V (1⊗ (j t)⊗ (j t)v)V(t)
= α2 ((j t)⊗ (j t)⊗ v)V
= uj ⊗ ui ⊗ v.
Finally i(f (i =6 t)= j we have that )(t) ( )◦ (t) −1α2 Θ (f) ◦ α (u ⊗V t V 2 V i((uj ⊗ v) )(t))
= ( α2) ◦Θt(f)V ((i t)⊗ (i t)⊗ v)V(t)
= α2 (1⊗ (i t)⊗ (i t)v)V
= uj ⊗ ui ⊗ v.
This completes the check in this case.
( If f is the left dot, we hav)e that( ( )(t)) ( (t))−1 ( (t))
α1 ◦Θt(f)V ◦∑α1 (ui ⊗ v) = α1 ◦∑Θt(f)V ((i t)⊗ (i t)v)V ( Vt ) V t(t)
= α1 ((i t)(j t)⊗ (i t)v) = (i t)(j t)ut ⊗ (i t)(j t)(i t)v.∑ Vj=1 j=1
If i = t this is tj=1 uj ⊗ (j t)v which is right. If i =6 t we pull out the j = i and j = t
terms of the sum, simp∑lify the three types of terms separately, then recombine to get
the desired expression tj=1 uj ⊗ (i j)v.
We now have in our hands monoidal functors φt from (3.1.3), i from (3.3.6), and
Φt from (3.4.2). Let
Act : kSt-Modfd → End k(kSt-Modfd) (3.4.4)
be the monoidal functor induced by the Kronecker product, i.e., Act(V ) = V⊗ for a
kSt-module V and Act(f) = f⊗ for a homomorphism f : V → V ′.
53
Lemma 3.4.2. For every t ∈ N, the following diagram commutes up to the obvious
canonical isomorphism of monoidal functors:
(3.4.5)
Proof. The composition Φt ◦ i takes the nth object of APar to (Ut⊗)◦n, while Act ◦φt
takes it to U⊗nt ⊗. Let
β(t)
∼
n : (U
◦n
t⊗) ⇒ U⊗nt ⊗
be the canonical isomorp(hism) between these functors defined by associativity of tensor
(t) (t)product. Then β = βn ≥ : Φt ◦ i ⇒ Act ◦φt is an isomorphism of monoidaln 0
functors. To see this, we need to check naturality. This follows because the five
formulae defining φt from Theorem 3.1.1 tensored on the right with a vector v are
exactly the same as the formulae defining Φt on these five generating morphisms from
Lemma 3.4.1.
Now we can prove the main theorem justifying the significance of the affine
partition category. Let
Ev : End k(kSt-Modfd)→ kSt-Modfd (3.4.6)
be the (non-monoidal) functor defined by evaluating on trivSt . There is an obvious
∼
isomorphism of functors Ev ◦Act ⇒ IdkSt-Mod defined on V by the isomorphismfd
V ⊗ trivSt → V, v ⊗ 1 7→ v.
Theorem 3.4.3. There is a unique (non-monoidal) functor
p : APar → Par (3.4.7)
54
such that p ◦ i = IdPar and 
p ··· • 

= ··· ◦•◦• . (3.4.8)
n 2 1 n 2 1
Moreover, for any t ∈ N, the following diagram of functors commutes up to natural
isomorphism:
(3.4.9)
The functor p also maps
··· • →7 T ··· , ··· • →7 ··· , ··· • →7 ··· . (3.4.10)
n 2 1 n 1 n 3 2 1 n 3 2 1 n 3 2 1 n 3 2 1
∈ N (t)Proof. For t , let γ : U⊗nn t ⊗
∼
triv ⊗nSt → Ut be the obvious isomorphism sending
uin ⊗ · · · ⊗ ui1 ⊗ 1 7→ uin ⊗ · · · ⊗ ui1 . We say that f ∈ HomAPar (n,m) is good if there
exists a morphism f̄ ∈ HomPar (n,m) such that ( )−1
φt(f̄) = γ
(t)
m ◦ Ev(Φt(f)) ◦ γ(t)n (3.4.11)
for all t ∈ N. If f is good, there is a unique f̄ such that (3.4.11) holds for all
t. To see this, suppose th(at f̄) and f̄ ′ both satisfy (3.4.11) for all t ∈ N. Then(t) (t) −1
φt(f̄) = γ
′
m ◦ Ev(Φt(f)) ◦ γn = φt(f̄ ), so that φt(f̄ − f̄ ′) = 0 for all t ∈ N. In
view of Lemma 3.1.4 this implies that f̄ = f̄ ′ as claimed.
Suppose that f ∈ HomAPar (n,m) and g ∈ HomAPar (l,m) are both good. Then
f ◦ g is good with f ◦ g :=(f̄ ◦ ḡ). This follows because−1 ( (t))−1 ( (t))−1
φt(f̄◦ḡ) = γ(t)m ◦Ev(Φ (f))◦ γ(t) (t)t m ◦γm ◦Ev(Φ (f))◦ γ = γ(t)t l m ◦Ev(Φt(f◦g))◦ γl .
Similarly, sums of good morphisms are good with f + g := f̄ + ḡ.
In this paragraph, we show that every morphism in APar is good. In view of the
previous paragraph, it suffices to show that some family of generating morphisms for
55
APar are all good. Hence, in view of Corollary 3.3.6, it is enough to show that i(f)
is good for every morphism f in Par and that the morphisms (3.3.17) are good for
all n. For f ∈ HomPar (n,m), the morphism i(f) is good with i(f) := f . This follows
from the following calculati(on u)sing Lemma 3.4.2: ( )
γ(t) (t)
−1 −1
m ◦ Ev(Φt(i(f))) ◦ γm = γ(t) ◦ Ev(Act(φ (f))) ◦ γ(t)m t m = φt(f).
Also the morphism f from (3.3.17) is good for every n. To see this, let f̄ be the
morphism on the right hand side∑of (3.4.8). Using the definition in Theorem 3.1.1,
φt(f̄) is the map uin⊗· · ·⊗ui1 7→
t
j=1 uin⊗· · ·⊗ui2⊗uj.∑Also using the definition in
Lemma 3.4.1, Ev(Φt(f)) is the map uin⊗· · ·⊗ui1⊗1 7→
t
j=1 uin⊗· · ·⊗ui2⊗uj⊗1.
(t)
On contracting the final ⊗1 using γn , these are equal, as required to prove that f is
good.
Now we can define a functor p making (3.4.9) commute (up to natural
isomorphism) for all t ∈ N. On objects, define p by declaring that p(n) = n for
each n ≥ 0. On a morphism f ∈ HomAPar (n,m), we define p(f) := f̄ . The checks
made so far imply that this( is a)well-defined functor satisfying (3.4.8). The equation(t)
(3.4.11) shows that γ(t) = γn ≥ : Ev ◦Φt ⇒ φt ◦ p is a natural isomorphism. Wen 1
have also already shown that p ◦ i = IdPar and that (3.4.8) holds. Thus, we have
established the existence of a functor p : APar → Par satisfying all of the properties
in the statement of the theorem. The uniqueness of p follows from Corollary 3.3.6.
It remains to check the three properties (3.4.10). These can be checked using the
commutativity of (3.4.9) in the same way as we just established (3.4.8). Alternatively,
and possibly quicker, they can be deduced directly from (3.4.8) using the relations
(3.3.9) to (3.3.11), respectively. We leave the details to the reader.
56
The faithfulness of i in the following corollary was already proved in two different
ways in [LSR21]. Our approach is similar in spirit to the first proof given in loc. cit.,
i.e., the argument used to prove [LSR21, Th. 5.2].
Corollary 3.4.4. The functor i : Par → APar is faithful and the functor p : APar →
Par is full.
Proof. This follows because p ◦ i = IdPar .
∼
Corollary 3.4.5. The functor p induces an isomorphism APar/I → Par where I is
the left tensor ideal of APar generated by the morphism • − ◦◦•• .
Proof. The left tensor ideal I is the data of subspaces I(n,m) of HomAPar (n,m) for
each m,n ≥ 0 which are closed under vertical composition on the top or bottom
with any morphism and closed under horizontal composition on the left with any
morphism. It is clear from (3.4.8) that p sends morphisms in I to zero, hence, p
induces a functor p̄ : APar/I → Par . This is surjective on objects and full. To see that
it is faithful, suppose that f + I(n,m) ∈ HomAPar/I(n,m) = HomAPar (n,m)/I(n,m)
is a morphism sent to zero by p̄, hence, p(f) = 0. In view of Corollary 3.3.6 and
the definition of I, we may assume that f = i(f̄) for some f̄ ∈ HomPar (n,m). Then
f̄ = p(i(f̄)) = p(f) = 0, so that f = i(f̄) = 0.
Composing the functor p : APar → Par with evaluation at any t ∈ k gives a full
functor
pt := evt ◦p : APar → Par t (3.4.12)
such that
··· • →7 ··· ◦◦ , ··· • 7→ t ··· , (3.4.13)
n 2 1 n 2 1 n 2 1 n 2 1
57
··· • →7 ··· , ··· • 7→ ··· . (3.4.14)
n 3 2 1 n 3 2 1 n 3 2 1 n 3 2 1
∼
Like in Corollary 3.4.5, the functor pt induces an isomorphism APar/It → Par t where
It is the left tensor ideal of APar generated by T − t11 and • − •◦•◦ .
3.5 Jucys-Murphy elements for partition algebras
Now we can explain how affine partition category is related to the works of
Enyang [Eny13] and Halverson-Ram [HR05]. These are concerned with the partition
algebra, which is the endomorphism algebra
Pn(t) := EndPar t(n) = 1nPart1n. (3.5.1)
By analogy, we define the affine partition algebra to be
APn := EndAPar (n) = 1nAPar1n. (3.5.2)
Let us denote the elements of APn defined by the left and right dots on the jth string
by XLj and X
R
j , and the elements defined by the left and right crossings of the kth
and (k + 1)th strings by SLk and S
R
k :
XLj := ··· • ··· , XRj := ··· • ··· , (3.5.3)
n j 1 n j 1
SLk := ··· • ··· , SRk := ··· • ··· (3.5.4)
n k+1 k 1 { n k+1 k ∣ 1 }
for 1 ≤ j ≤ n and 1 ≤ k ≤ n − 1. We note that XL, XR ∣j j j = 1, . . . , n are
algebraically independent, so they generate a free polynomial algebra of rank 2n
inside APn(t); this follows easily from the basis theorem for morphism spaces Heis
proved in [Kho14]. Taking the images of the elements (3.5.3) and (3.5.4) under the
functor pt from (3.4.12) gives us elements of Pn(t) denoted
xL := p (XL), xR := p (XR), sL L R Rj t j j t k k := pt(Sk ), sk := pt(Sk ). (3.5.5)
58
The notation here depends implicitly on the values of n and t, which should be clear
from the context. By (3.4.13) and (3.4.14), we have that xL1 = ··· •◦•◦ , xR1 = t, sL1 = 1
and sR1 = (1 2) ∈ Sn ⊂ Pn(t).
Theorem 3.5.1. Suppose that t ∈ N and let ψt : Pn(t) → EndkSt(U⊗nt ) be
the homomorphism induced by the functor φt from Theorem 3.1.1. The elements
xL, xR, sL, sRj j k k ∈ Pn(t) satisfy ∑t [ ]
ψ Lt(xj )(uin ⊗ · · · ⊗ ui1) = uin ⊗ · · · ⊗ ui ⊗ (i ij) ui ⊗ · · · ⊗ uj+1 j i2 ⊗ ui1 ,
i=1
∑ (3.5.6)t [ ]
ψ (xRt j )(uin ⊗ · · · ⊗ ui1) = uin ⊗ · · · ⊗ ui ⊗ (i ij j) ui − ⊗ · · · ⊗ ui2 ⊗ uj 1 i1 ,
i=1
[ (]3.5.7)
ψ (sLt k )(uin ⊗ · · · ⊗ ui1) = uin ⊗ · · · ⊗ ui ⊗ (ik ik k+1) ui − ⊗ · · · ⊗ uk 1 i2 ⊗ ui1 ,
[ (3].5.8)
ψ (sRt k )(uin ⊗ · · · ⊗ ui1) = uin ⊗ · · · ⊗ ui ⊗ (i i ) u ⊗ · · · ⊗ u ⊗ uk+2 k k+1 ik+1 i2 i1
(3.5.9)
for 1 ≤ i1, . . . , in ≤ t, where we are using the diagonal action of St on tensor powers
of Ut.
Proof. This follows from the commutativity of (3.4.9), (3.5.5) and the formulae in
Lemma 3.4.1.
Corollary 3.5.2. Identifying Pn(t) with the partition algebra in [Eny13] by reflecting
diagrams through a vertical axis to account for the fact that we number vertices from
right to left rather than from left to right, the elements (3.5.5) are related to the
elements L 1 , L1, . . . and σ 3 , σ2, . . . of the partition algebra Pn(t) defined in [Eny13]
2 2
59
according to the dictionary
xLj ↔ Lj, t− xRj ↔ L L Rj− 1 , sk ↔ σk+ 1 , sk ↔ σk+1. (3.5.10)
2 2
Hence, by [Eny13, Th. 5.5], the elements xLj and t−xRj are identified with the Jucys-
Murphy elements introduced originally by Halverson and Ram in [HR05].
Proof. Enyang’s elements are defined by a recurrence relation which is independent
of the value of the parameter t. Hence, his elements can be viewed as specializations
at T = t of corresponding elements of the generic partition algebra EndPar (n). To
identify them with our elements, we can use Lemma 3.1.4 to see that it suffices
to check that they act in the same way on U⊗nt for infinitely many values of the
parameter t ∈ N. This follows on comparing (3.5.6) to (3.5.9) to the formulae in
[Eny13, Prop. 5.2, Prop. 5.3].
Remark 3.5.3. Alternatively, one can prove Corollary 3.5.2 inductively, using the
recurrence relations in Lemma 3.3.5 which are equivalent to Enyang’s recurrence
relations [Eny13, (3.1)–(3.4)]. In fact, all of the relations derived in loc. cit. can now
be deduced easily using the relations in APar derived in the previous section.
Remark 3.5.4. Recently, Creedon [Cre21] has introduced a renormalization of the
Jucys-Murphy elements, which he denotes by N1, N2, . . . , N2n ∈ Pn(t). They are
defined in terms of the Enyang-Halverson-Ram elements simply by N t2j−1 := Lj− 1 −
2 2
and N := L − t2j j . The dictionary between Creedon’s elements and ours is2
xLj − t ↔ N t R2j, − xj ↔ N2j−1. (3.5.11)2 2
The motivation for such a renormalization will be discussed further in Remark 3.6.5
below.
60
3.6 Central elements
By the center of a (k-linear) category A , we mean the (unital) commutative
algebra Z(A) := Endk(IdA) of endomorphisms of the identity endofunctor of A . Thus,
an element z ∈ Z(A) is a tuple (zX)X∈O(A) such that zY ◦f = f ◦zX for all morphisms
f : X → Y inA . Equivalently, in te∏rms of the p∣∣ath algebra A, it is the algebra
Z(A) :=  ∣ z = (zX)X∈O(A) ∈ 1XA1X ∣∣ za = az for all a ∈ A , (3.6.1)
X∈O(A)
interpreting the products in the obvious way. We note that there is an algebra
isomorphism ∏
EndAAop(A)→
∼
Z(A), ζ →7 (ζ(1X))x∈O(A) ∈ 1XA1X , (3.6.2)
X∈O(A)
where the algebra on the left is the endomorphism algebra of the A  Aop-module
associated to the (A,A)-bimodule A. If A is locally finite-dimensional, then it
is a locally finite-dimensional A  Aop-module, hence, by [BS, Lem. 2.10], the
endomorphism algebra EndAAop(A) ∼= Z(A) is a pseudo-compact topological algebra
with respect to the pro-finite topology. That is, the topology of Z(A) is such that the
ideals of finite codimension form a base of neighborhoods of 0. Pseudo-compactness
means that Z(A) is isomorphic to l←im−Z(A)/J where the inverse limit is over all ideals
of finite codimension.
In the locally finite-dimensional case, Z(A) is isomorphic to the algebra C(A)∗
that is the linear dual of the cocenter C(A). The cocenter is a cocommutative
coalgebra isomorphic to CoendAAop(A) in⊕the notation of [BS, (2.15)]. To define
C(A) explicitly, note that the space D := ∗X,Y ∈O (1XA1Y ) is naturally an (A,A)-A
bimodule with 1YD1X = (1XA1Y )
∗. Also each 1XD⊕1X is a coalgebra as it is the
dual of the finite-dimensional algebra 1XA1X . Hence, X∈O 1XD1X is a coalgebra.A
61
Then the cocenter is ( ⊕ )/
C(A) := 1XD1X J (3.6.3)
X∈OA
where J is the coi{deal spann∣ed by the elements }
af − fa ∣X, Y ∈ OA, a ∈ 1XA1Y , f ∈ 1YD1X .
T⊕o identify C(A)∗ with Z(A∏), note that the linear dual of the coalgebra
X∈O 1XD1X is the a∏lgebra X∈O 1XA1X ; the annihilator J◦ of the coideal JA A
defines a subalgebra of X∈O 1XA1X which is exactly the center Z(A) according toA
the original definition (3.6.1).
In this subsection, we are going to construct a family of elements (z(r))r≥1 in
the center Z(APart) of the affine partition category APar t. We start by introducing
some convenient shorthand. Given a m(onom) ial x(rys)∈ k[x, y], we use the notation◦r ◦s
•xrys := • ◦ • (3.6.4)
to denote the element of EndAPar (|) on the right hand side, that is, it is the rth
power of the right dot (represented by x) composed with the sth power of the left dot
(represented by y). It then makes sense to label dots by polynomials f(x) ∈ k[x, y],
meaning the linear combination of the morphisms •xrys just as f(x) is the linear
combination of its monomials. We are also going to use generating functions in the
same way as explained in the context of Heis in [BSW20, §3.1]. For these, u will be a
formal variable which should always be interpreted by expanding as formal Laurent
series in k((u−1)), e.g., (u− x)−1 = u−1 + u−2x+ u−3x2 + · · · .
Let
◦• ◦•
©(u) := u1 − • (u−x)−1 −11 = u11 − • (u−y) ∈ u1 + u−11 End ( −1APar 1)[[u ]]. (3.6.5)
•◦ •◦
62
For r ≥ 0, the coefficient of u−r−1 in this formal Laurent series is − •◦•
•◦
xr ; the xr here
can be replaced by yr due to the third relation in (3.3.8). Also introduce the rational
function
(u− (x+ 1))(u− (x− 1))
αx(u) := ∈ k(x, u). (3.6.6)
(u− x)2
The expansion of this as a power series in k[x][[u−1]] is
α (u) = 1− (u− x)−2 = 1− u−2 − 2xu−3 − 3x2u−4 − 4x3u−5x − · · · , (3.6.7)
αx(u)
−1 = 1 + u−2 + 2xu−3 + (3x2 + 1)u−4 + (4x3 + 4x)u−5 + · · · . (3.6.8)
The following elementary lemma will play a fundamental role in the rest of the article.
It would be hard to formulate this without the aid of generating functions.
Lemma 3.6.1. The following bubble slide relations hold in APar :
© α (u) α (u)(u) = y • ©(u) , ©(u) = ©(u) • x . (3.6.9)
αx(u) αy(u)
Proof. The two equations are equivalent, so we just prove the first one. When working
with Heis , we adopt the notation of [BSW20, §3.1]: an open dot labelled by xr means
the rth power of the open dot in Heis , and (u) is the formal Laurent series from
[BSW20, (3.13)]. Under the embedding of APar into Heis , we have that
◦•
©(u+ 1) = (u+ 1)11 − •(u−(y−1))−1 = (u+ 1)1 (u−x)−11 − ◦• = 11 + (u).
◦•
The bubble slide relation for Heis from [BSW20, (3.18)] gives that
(u) = αx(u)◦• (u) = αx(u)•◦ ◦• α −1x(u) (u) .
According to (3.3.4), the label x on the open dot on the ↓ string translates into the
label x− 1 on a closed dot in APar , and the label x on the open dot on the ↑ string
translates into the label y − 1 on a closed dot in APar . So the relation just recorded
63
can be written equivalently as
© = αy−1(u) • © = αy(u+1)(u+ 1) (u+ 1) • ©(u+ 1) .
αx−1(u) αx(u+1)
Replacing u by u− 1 everywhere gives the desired relation.
The rational function αy(u)/αx(u) ∈ k(x, y, u) will also be important later on.
The low degree terms of its expansion as a power series in u−1 can be computed using
(3.6.7) and (3.6.8):
αy(u) [ ]
= 1+2(x−y)u−3 +3(x2−y2)u−4 + 4(x3 − y3) + 2(x− y) u−5 + · · · . (3.6.10)
αx(u)
For n ≥ 0∑, let αy(u) αy(u) αy(u)
C (u) = C(r)n n u
−r :=©(u)?1n?©(u)−1 = • · · · • • −1αx(u) αx(u) αx(u) ∈ 1nAPar1n[[u ]],
r≥0 n 2 1
(3.6.11)
where the final equality follows by applying the bubble slide relation repeatedly. Then
we define ∑ ∏
C(u) = C(r)u−r := (Cn(u))n≥0 ∈ 1nAPar1n[[u
−1]]. (3.6.12)
r≥0 n≥0
Note by (3.6.10) that C(0) = 1 and C(1) = C(2) = 0.
Theorem 3.6.2. C(u) ∈ Z(APar)[[u−1]].
Proof. The interchange law immediately gives that
· · · · · · · · ·
C (u) ©(u) · · · ©(u)−1m f f
· · · = = = · · ·
f f ©(u) · · · ©(u)−1 Cn(u)
· · · · · · · · ·
for any f ∈ HomAPar (n,m).
The proof of the following corollary is similar to an argument used to simplify
some analogous central elements in the quantum Heisenberg category in [MS22,
Prop. 4.3].
64
≥ (r) (r)
∏
Corollary 3.6.3. For each r 1, the element Z = (Zn )n≥0 ∈ n≥0 1nAPar1n
defined from∑n ( ) ( ) ( ) ( ) ( )
Z(r) := (XL)r − (XR r r)r = XL + · · ·+ XL − XR r − · · · − R rn i i 1 n 1 Xn
i=1
belongs to Z(APar) (notation as in (3.5.3) and (3.5.4)). Moreover, the elements
Z(1), Z(2), . . . generate the same subalgebra Z0(APar) of Z(APar) as the elements
C(3), C(4), . . . .
Proof. Let f(u) := αy(u)/αx(u) for short. Then define g(u) := f
′(u)/f(u) =
d (ln f(u)() to be its logarithmic derivative. Wedu )have that
g(u) = − 1 2 1
u− (x− 1)(+ −u− x u− (x+ 1) )
− − 1 2 − 1+
u− (y − 1) u− y u− (y + 1)
= 2 · 3(y − x)u−4 + 2 · 6(y2 − x2)u−5 + 2 · [10(y3 − x3) + 5(y(− x)]u
−6
) + · · · .
We deduce for r ≥ 1 that the u−r−3-coefficient of g(u) is equal to 2 r+2 (yr−xr) plus
2
a linear combination of terms (ys − xs) for 1 ≤ s < r with s ≡ r (mod 2).
The coefficients of the power series C ′(u)/C(u) are polynomials in the coefficients
of the series C(u). Hence, by the theorem, these coefficients are all central. To
compute them, we take logarithmic deriva∑tives of (3.6.11) to obtain the identityn
C ′n(u)/Cn(u) = ··· g(u)• ··· .
i=1 n i 1
Using the previous paragraph and the definition of Z(r), we deduce for r ≥ 1(tha)t the
central element defined by the u−r−3-coefficient of C ′(u)/C(u) is equal to 2 r+2 Z(r)
2
plus a linear combination of Z(s) for 1 ≤ s < r with s ≡ r (mod 2). Finally, induction
on r shows that each Z(r) is central.
The argument just given shows that each Z(r) lies in the subalgebra generated
by C(3), C(4), . . . . Conversely, by exponentiating an anti-derivative of the series
65
C ′(u)/C(u), one shows that each C(r) can be expressed as a polynomial in
Z(1), Z(2), . . . . Hence, the two families of elements generate the same subalgebra
of Z(APar).
Taking the im∑ages of C(u) and each Z
(r) under the functor pt from (3.4.12) give
c(u) = c(r)u−r := (cn(u))n≥0 ∈ Z(Part)[[u
−1]], (3.6.13)
r≥0 ∑
where c (u) = c(r)u−rn n := pt(Cn(u)),
( ) r≥0
z(r) := z(r)n ≥ ∈ Z(Part), where z
(r)
n := p
(r)
n 0 t
(Zn ). (3.6.14)
(r) (r)
The elements cn and zn belong to the center Z(Pn(t)) of the partition algebra Pn(t).
In terms o∑f th[e( Ju)cys-M( urp)h]y elements (3.5.5), we have thatn
z(r) = xL
r − xR r Ln i i = (x1 )r + · · ·+ (xLn)r − (xR1 )r − · · · − (xR)rn . (3.6.15)
i=1
(1)
From Corollary 3.5.2, it follows that zn equals zn−nt where zn is the central element
(1)
from [Eny13, Th. 3.10(2)]. In fact, zn is closely related to the central elements of
the group algebras kSt defined by sums of transpositions:
Lemma 3.6.4 ([Eny13, Prop. 5.4]). If t ∈∑N (1)then ψ (z ) : U⊗nt n t → U⊗nt is equal to
the endomorphism defined by the action of 1≤i<j≤t ((i j)− 1) ∈ Z(kSt).
(r)
Remark 3.6.5. After constructing the elements zn ∈ Z(Pn(t)) in the manner
explained above, we came across a recent paper of Creedon which constructs
similar central elements; see [Cre21, Th. 3.2.6]. To explain the connection,
recall that the rth supersymmetric power sum in variables x1, . . . , xn, y1, . . . , ym is
pr(x1, . . . , xn|y1, . . . , ym) = xr r r r1 + · · · + xn − y1 − · · · − ym. The expression on the
right hand side of (3.6.15) is p (xLr 1 , . . . , x
L
n |xR1 , . . . , xRn ). It is easy to see that these
elements belong to Z(Pn(t)) for all r ≥ 1 if and only if p (xLr 1 − t/2, . . . , xLn − t/2|xR1 −
66
t/2, . . . , xR − t/2) ∈ Z(P (t)) for all r ≥ 1. Moreover, p (xLn n r 1 − t/2, . . . , xLn − t/2|xR1 −
t/2, . . . , xRn − t/2) ∈ Z(Pn(t)) coincides with the rth supersymmetric power sum
pr(N2, N4, . . . , N2n|−N1,−N3, . . . ,−N2n−1) in Creedon’s renormalized Jucys-Murphy
elements from (3.5.11). Creedon showed that his elements are central in Pn(t) by a
direct check of relations. This gives an independent way to verify that the eleemnts
z(r)
(r)
= (zn )n≥0 belong to Z(Par t): Creedon’s calculations show that they commute
with all crossings (a surprisingly hard calculation), and after that it is easy to see
that they commute with all other generators of Par t.
Remark 3.6.6. In [CO11, Def. 4.5], Comes and Ostrik define another family of
central elements ωr(t) = (ωrn(t))n≥0 which lift the central elements of the group
(r)
algebras kSt defined by the sums of all r-cycles. We expect that our elements zn
and their elements ωrn(t) are closely related, but we do not know any explicit formula.
In particular, the Comes-Ostrik elements should generate the same subalgebra of
Z(Part) as our elements.
3.7 Harish-Chandra homomorphism
Although we just explain in the case of Par t, the general development in
this section is valid for any monoidal triangular category, replacing Sym with
the (semisimple) Cartan subcategory and replacing the set P of partitions by a
set parametrizing isomorphism classes of irreducible representations of the Cartan
subcategory.
According to the general definition∏(3.6.1), the center of the partition category
is a subalgebra of the unital algebra + −n≥0 1nPart1n. Let K (resp., K ) be the
left ideal (resp., right ideal) of Part generated by the strictly downward partition
diagrams (resp., the strictly upward partition diagrams). From the triangular basis,
it is easy to see that 1 K+n 1
−
n = 1nK 1n. We denote this by Kn. It is a two-sided
67
ideal of the finite-dimensional algebra 1nPart1n, and we have that
1nPart1n = kSn ⊕Kn. (3.7.1)
Equivalently, Kn is the two-sided ideal of 1nPart1n spanned by morphisms that factor
through objects m < n. By analogy with Lie theory, we define the Harish-Chandra
homomorphism∏ ∏
ĤC : 1nPart1n → kSn, (zn)n≥0 7→ (HCn zn)n≥0, (3.7.2)
n≥0 n≥0
where HCn : 1nPart1n  kSn is the projection along the direct sum decomposition
(3.7.1). It is obvious from (3.7.1) that the restriction of ĤC to Z(Part) defines an
algebra homomorphism ∏
HC : Z(Part)→ Z(Sym) = Z(kSn). (3.7.3)
n≥0
As each kSn is semisimple with its isomorphism classes of irreducible
representations parametrized by Pn, we can identify the algebra appearing on the right
hand side of (3.7.3) with the algebra k[P ] of all functions from the set P∏to the field k
with pointwise operations. Under this identification, the tuple (zn)n≥0 ∈ n≥0 Z(kSn)
corresponds to the function f : P → k such that f(λ) is the scalar that zn acts by
on the Specht module S(λ) for each λ ∈ Pn. Then the Harish-Chandra homorphism
becomes a homomorphism
HC : Z(Part)→ k[P ]. (3.7.4)
To describe HC more explicitly in these terms, let λ ∈ Pn be a partition. As we have
that EndPart(∆(λ))
∼= EndSym(S(λ)) ∼= k, an element z = (zn)n≥0 ∈ Z(Part) acts on
the standard module ∆(λ) as multiplication by a scalar denoted χλ(z). This defines
an algebra homomorphism
χλ : Z(Part)→ k. (3.7.5)
68
To compute χλ(z), note that it is the scalar by which zn acts on the highest weight
space 1n∆(λ), which is the scalar arising from the action of HCn(zn) ∈ Z(kSn) on
S(λ). It follows that
χλ(z) = HCn(zn)(λ) = HC(z)(λ). (3.7.6)
Recall that Z(Part) is a commutative pseudo-compact topological algebra with
respect to the profinite topology. Let Spec(Z(Part)) be its set of open (= finite-
codimensional) maximal ideals.
Lemma 3.7.1. Spec(Z(Part)) = {kerχλ | λ ∈ P}.
Proof. Points in Spec(Z(Part)) parametrize isomorphism classes of finite-dimensional
irreducible modules for Z(Part) Let Lλ be the irreducible Z(Part)-module associated
to χλ : Z(Part) → k. Then we need to show that any finite-dimensional irreducible
Z(Part)-module L is isomorphic to Lλ for some λ ∈ P . To see this, we find it easiest
to work equivalently in terms of irreducible comodules over the cocenter C := C(Part)
defined in (3.6.3). So let L be an irreducible C-comodule and L∗ be the dual comodule,
there being no need to distinguish between left or right since C is cocommutative. By
definition, C is a quotient of the coalgebra D that is the direct sum of the coalgebras
(1nPart1
∗
n) for all n ≥ 0. Since L∗ is isomorphic to a subcomodule of the regular
C-comodule, it follows that L∗ is isomorphic to a subquotient of the restriction of the
regular D-comodule to C. Hence, L∗ is isomorphic to a subquotient of (1nPar
∗
t1n)
for some n. So L is isomorphic to a subquotient of 1nPart1n. Now recall that the left
Part-module Part1n has a ∆-flag, and z ∈ Z(Part) acts on ∆(λ) as multiplication
by the scalar χλ(z). Hence, all composition factors of the finite-dimensional Z(Part)-
module 1nPart1n are of the form Lλ for λ ∈ P .
69
Let ≈t be the equivalence relation on P defined by
λ ≈t µ⇔ χλ = χµ. (3.7.7)
From Lemma 3.7.1, we see that the equivalence classes P/ ≈t parametrize the points
in Spec(Z(Part)).
Lemma 3.7.2. The image of HC : Z(Part) → k[P ] consists of of all functions
f ∈ k[P ] which are constant on ≈t-equivalence classes. Moreover, for each subset S
of P that is a union of ≈t-equivalence classes, there is a unique central idempotent
1S ∈ Z(Part) such that  1 if λ ∈ S,
HC(1S)(λ) =  (3.7.8)0 otherwise.
∏If S is a single equivalence class then 1S is a primitive idempotent, and Z(Part) =
S∈P/≈ 1SZ(Part).t
Proof. It is clear from (3.7.6) that any function in the image of HC is constant on
≈t-equivalence classes. Conversely, take a function f ∈ k[P ] which is constant on
equivalence classes. For an equivalence class S ∈ P/ ≈t, let LS be the irreducible
Z(Part)-module associated to the central character χλ (λ ∈ S). The previous lemma
shows that these give a full set of pairwise inequivalent irreducible finite-dimensional
Z(Part)-modules. It follows that the cocommutative coalgebra C(Part) decomposes
as a direct sum of indecomposable coideals ⊕
C(Part) = CS,
S∈P/≈t
where CS is the injective hull of LS. Then we consider the linear map θ : C(Part)→
C(Part) defined by multiplication by the scalar f(λ) (λ ∈ S) on the summand CS.
This is a comodule homomorphism. Now we use that
Z(Par ∗ ∼ opt) = C(Par t) = EndC(Part)(C(Part))
70
as holds for any coalgebra, e.g., see [BS, Lem. 2.2]. It implies that θ defines an element
of Z(Part). The image of this element under HC is the function f ∈ k[P ].
To prove the existence of the idempotent 1S for any S that is a union of ≈t-
equivalence classes, we apply the construction in the previous paragraph to obtain
1S ∈ Z(Part) such that 1S acts as the identity on the indecomposable summands CS′
of C(Part) for all ≈t-equivalence classes S ′ ⊆ S and as zero on all other summands.
This is an idempotent satisfying (3.7.8), and it is a primitive idempotent if and only
if S is a single equivalence class. We then∏have that
Z(Part) = 1SZ(Part)
S∈P/≈t
as this is the algebra decomposition that is dual to the decomposition of C(Part) as
the direct sum of its indecomposable coideals.
For S ∈ P/ ≈t, the primitive central idempotent 1S ∈ Z(Part) from Lemma 3.7.2
is not an element of Part, but we have that 1S = (1S,n)n≥0 for idempotents 1S,n =
1S1n = 1n1S ∈ 1nPart1n. Moreover, for∑a fixed n the idempotent 1S,n is zero for
all but fin⊕itely many S, so that 1n = S∈P/≈ 1S,n. The locally unital algebrast
1SPart = m,n≥0 1S,mPart1S,n are the blocks of the partition algebra Part, and we
have the block de⊕compositions ∏
Part = 1SPart, Part-Mod = 1SPart-Mod. (3.7.9)
S∈P/≈t S∈P/≈t
Representatives for the isomorphism classes of irreducible 1SPart-modules are given
by the modules L(λ) for all λ ∈ S.
Lemma 3.7.3. The following properties are equivalent:
(i) Part is semisimple.
(ii) All of the ≈t-equivalence classes are singletons.
71
(iii) HC : Z(Part)→ k[P ] is surjective.
(iv) HC : Z(Part)→ k[P ] is an isomorphism.
Proof. If (i) holds, then Part is a direct sum of locally unital matrix algebras indexed
by the se∏t P that labels its irreducible representations. Hence, its center is the direct
product λ∈P kλ. It follows easily that HC is an isomorphism, i.e., (iv) holds.
Obviously, (iv) implies (iii).
The equivalence of (ii) and (iii) follows from Lemma 3.7.2.
It remains to show that (ii) implies (i). Assuming (ii), Lemma 3.7.2 shows for
any λ ∈ P that there is a primitive central idempotent in Z(Part) which acts as the
identity on ∆(λ) and as zero on L(µ) for all µ 6= λ. We deduce that all composition
factors of ∆(λ) are isomorphic to L(λ). Since this is a highest weight module we
have that [∆(λ) : L(λ)] = 1, so actually ∆(λ) is irreducible. This is the case for all
λ ∈ P , so by BGG reciprocity we deduce that P (λ) = ∆(λ) = L(λ) for all λ, and (i)
holds.
Remark 3.7.4. When Part is semisimple, the standardization functor j! :
Sym-Modfd → Par t-Modlfd sends the irreducible Sym-modules S(λ) to the irreducible
Part-modules ∆(λ) = L(λ) for all λ ∈ P . It follows easily that j! is an equivalence of
categories in the semisimple case (although it is not a monoidal equivalence). Since
the center is a Morita invariant, it follows that Z(Sym) ∼= Z(Part) in the semisimple
case. Recalling that Z(Sym) ∼= k[P ], this gives another way to understand the
equivalence (i)⇒(iv) of Lemma 3.7.3.
Remark 3.7.5. As Part-Modlfd is an upper finite highest weight category, there is
also a canonical partial order on P , called the minimal order in [BS, Rem. 3.68],
which we denote here by t. By definition, this is the partial order generated by
72
the relation λ t µ if [∆(λ) : L(µ)] =6 0. As always for highest weight categories,
the equivalence relation ≈t defining the blocks of Part is the transitive closure of the
minimal order t. We will describe t explicitly in Corollary 3.10.7 below.
3.8 “Blocks”
In the previous section, we introduced an equivalence relation ≈t on P whose
equivalence classes parametrize the blocks of Part. The relation ≈t was defined in
terms of the central characters χλ : Z(Part) → k arising from the irreducible Part-
modules L(λ); see (3.7.7). On the other hand, in (3.6.13) and (3.6.14), we constructed
some explicit central elements of Part. Let∼t be the equivalence relation on P defined
from
λ ∼t µ⇔ χλ|Z0(Part) = χµ|Z0(Part) { ∣ (3.8.1})
where Z0(Part) is the subalgeb{ra of ∣∣Z(Par}t) generated by the elements c
(r) ∣ r ≥ 3
(equivalently, by the elements z(r) r ≥ 1 ). We refer to the ∼t-equivalence classes
as “blocks”. We obviously have that
λ ≈t µ⇒ λ ∼t µ, (3.8.2)
i.e., “blocks” are unions of blocks. Defining 1S as in Lemma 3.7.2, there are induced
“block” decompo⊕sitions ∏
Part = 1SPart, Part-Mod = 1SPart-Mod. (3.8.3)
S∈P/∼t S∈P/∼t
In this section, we are going to describe the relation ∼t in explicit combinatorial
terms.
Lemma 3.8.1. The images of the elements xL, xR, sL, sRj j k k ∈ 1nPart1n from (3.5.5)
under the Harish-Chandra homomorphism ĤC from (3.7.2) are
HC (xLn j ) = xj, HC
R
n(xj ) = t− j + 1, (3.8.4)
73
HCn(s
L
k ) = 1, HCn(s
R
k ) = (k k+1), (3.8.5)
where xj ∈ kSn is the Jucys-Murphy element from (2.7.10).
Proof. Applying HCn to the relations (3.3.15) and (3.3.16) (on the kth, (k+1)th and
(k + 2)th strings) we deduce that HC Ln(sk+1) = (k k+1 k+2) HC
L
n(sk )(k+2 k+1 k)
and HC Rn(sk+1) = (k k+ 1 k+ 2) HCn(s
R
k )(k+ 2 k+ 1 k). Now (3.8.5) follows by
induction on k, the base case k = 1 being immediate from (3.4.14). Note for this that
(k k+1 k+2)(k k+1)(k+2 k+1 k) = (k+1 k+2).
Applying HCn to the relations (3.3.13) and (3.3.14) (on the jth and (j + 1)th
strings), using also (3.3.10), we deduce that HCn(x
L
j+1) = (j j+1) HC
L
n(xj )(j j+1) +
HC (sR) and HC (xRn j n j+1) = (j j+1) HC
R
n(xj )(j j+1)−HC Ln(sj ). Now (3.8.4) follows
using (3.8.5) and induction on j, the base case j = 1 being immediate from (3.4.13).
Note for this that (j j+1)xj(j j+1) + (j j+1) = xj+1.
Lemma 3.8.2. For λ ∈ Pn, we have tha∏tn αconti(T)(u)χλ(c(u)) = (3.8.6)
αt−i+1(u)i=1
where T is some fixed standard λ-tableau and αx(u) is as in (3.6.6).
Proof. Note by (3.7.6) that χλ(c(u)) = HCn(cn(u))(λ) ∈ k[[u−1]]. To compute this, we
use Lemma 3.8.1 and the explicit formula for cn(u) = pt(Cn(u)) arising from (3.6.11)
to deduce that ∏n αx (u)
HCn(cn(u)) =
i ∈ Z(kSn)[[u−1]].
αt−i+1(u)i=1
To evaluate this at λ, we act on the basis vector vT from Young’s orthonormal basis
for S(λ), remembering that xivT = conti(T)vT.
Lemma 3.8.1 suggests some combinatorics of weights. Let P be the free Abelian
group on basis {Λc | c ∈ k}. Let εc := Λc − Λc+1 and αc := εc − εc−1. We define the
74
weight of a rational function f(u) ∈ k(u) to be∑

 Multiplicity of c   Multiplicity of c− wt f(u) := Λc ∈ P. (3.8.7)
c∈k as a pole of f(u) as a zero of f(u)
For example, wtαc(u) = −Λc−1 + 2Λc − Λc+1 = αc. For λ ∈ Pn, let wtt(λ) be the
weight of the rational function appearing on the right hand side of (3.8.6). As the
coefficients of the power series c(u) generate the subalgebra Z0(Part), the equivalence
relation ∼t defined by (3.8.1) satisfies
λ ∼t µ⇔ wtt(λ) = wtt(µ). (3.8.8)
This suggests using elements of P rather than ∼t-equivalence classes to index the
“blocks” from (3.8.3): for any γ ∈ P{, let ∣ }
S(γ) := λ ∈ P ∣ wtt(λ) = γ . (3.8.9)
Then define
prγ : Part-Mod→ Part-Mod (3.8.10)
to be the projection functor defined by multiplication by the central idempotent 1S(γ)
from Lemma 3.7.2. In other words, prγ projects a Part-module V to its largest
submodule all of whose irreducible subquotients are of the form L(λ) for λ ∈ P with
wtt(λ) = γ. The admissible γ ∈ P which parametrize “blocks” are the ones with
S(γ) 6= ∅; if S(γ) = ∅ then prγ is the zero functor.
Lemma 3∑.8.3. For λ ∈ Pn and any standard λ-tableau T , we have thatn
wtt(λ) = (αcont − α ) = (ε − ε ) + (ε − ε ) + · · ·+ (ε − ε )i(T) t−i+1 t−|λ| t λ1−1 −1 λk−k −k
i=1
(3.8.11)
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for any k ≥ `(λ). Moreover, given another partition µ ∈ P, we have that wtt(λ) =
wtt(µ) if and only if the infinite sequences (t−|λ|, λ1−1, λ2−2, . . . ) and (t−|µ|, µ1−
1, µ2 − 2, . . . ) are rearrangements of each other.
Proof. The first equality in (3.8.11) follows immediately from Lemma 3.8.2. To deduce
the second equality, take k ≥ `(λ). For 1 ≤ r ≤ k the contents of the nodes in the
rth row of the Young diagram of λ are 1 − r, 2 − r, . . . , λr − r, and we have that
α1−r + · · ·+ αλr−r = ελr−r − ε−r. Also αt + αt−1 + · · ·+ αt−n+1 = εt − εt−n. Now the
desired formula follows easily.
Rearranging the right hand side of (3.8.11) gives that εt−|λ| + ελ1−1 + ελ2−2 +
· · · + ελ −k = wtt(λ) + εk −1 + ε−2 + · · · + ε−k + εt for any k ≥ `(λ). Hence, we have
that wtt(λ) = wtt(µ) if and only if
εt−|λ| + ελ1−1 + ελ2−2 + · · ·+ ελk−k = εt−|µ| + εµ1−1 + εµ2−2 + · · ·+ εµk−k
for all k  0. This is clearly equivalent to saying that the infinite sequences (t −
|λ|, λ1 − 1, λ2 − 2, . . . ) and (t − |µ|, µ1 − 1, µ2 − 2, . . . ) may be obtained from each
other by permuting the entries.
The final assertion from Lemma 3.8.3 shows that ∼t is exactly the same as the
equivalence relation on partitions defined in [CO11, Def. 5.1]. The equivalence classes
of this relation were investigated in detail in [CO11, §5.3]. The following summarizes
the results obtained there. For the statement, we say that λ ∈ P is typical if it is
the only partition in its ∼t-equivalence class; otherwise we say that λ is atypical. Of
course, these notions depend on the fixed value of the parameter t.
Theorem 3.8.4 (Comes-Ostrik). If t ∈/ N then all partitions are typical. If t ∈ N
∼
then there is a bijection Pt → {atypical ∼t-equivalence classes} taking κ ∈ Pt to the
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∼t-equivalence class {κ(0), κ(1), κ(2), . . . } where
κ(n) := (κ1 + 1, . . . , κn + 1, κn+2, κn+3, . . . ) ∈ Pt+n−κn+1 , (3.8.12)
i.e., it is the partition obtained from κ by adding a node to the first n rows of its
Young diagram then removing its (n + 1)th row. Moreover, still assuming t ∈ N, a
partition λ ∈ P is typical if and only if t− |λ| = λi − i for some i ≥ 1.
Example 3.8.5. For any t ∈ N, the ∼t-equivalence class associated to κ = (t) ∈ Pt
is { }
S = ∅, (t+1), (t+1, 1), (t+1, 12), · · · .
For t ∈ N− {{0, 1}, the ∼t-equivalence class associated to κ = (1
t) ∈ Pt is
− }S = (1t 1), (2, 1t−2), (22, 1t−3), · · · , (2t−1), (2t), (2t, 1), (2t, 12), · · · .
As noted in [CO11, Cor. 5.23] (using a different argument for the forward
implication), the first assertion of Theorem 3.8.4 allows us to recover the following
well known result of Deligne [Del07, Th. 2.18]: Rep(St) is semisimple if and only if
t ∈/ N. In terms of the path algebra Part, Deligne’s result can be stated as follows.
Corollary 3.8.6 (Deligne). Part is semisimple if and only if t ∈/ N.
Proof. We already know that Part is not semisimple if t ∈ N by Corollary 3.1.2.
Conversely, if t ∈/ N, we apply the criterion from Lemma 3.7.3, noting that all ≈t-
equivalence classes are singletons thanks to (3.8.2) and the first part of Theorem 3.8.4.
Remark 3.8.7. When t ∈/ N, the above arguments show for λ, µ ∈ P with λ =6 µ
that there is a central element in the subalgebra Z0(Part) of Z(Part) which acts
by different scalars on the irreducible modules L(λ) and L(µ). It follows in these
cases that Z0(Part) is a dense subalgebra of the pseudo-compact topological algebra
77
Z(Par 1t) . We do not expect that this is the case when t ∈ N, but nevertheless
Z0(Part) is still sufficiently large to separate blocks. This will be established in
Corollary 3.10.6 below, which shows for any value of t that the relations ∼t and ≈t
coincide, so that “blocks” are blocks, and (3.8.3) is always the same decomposition
as (3.7.9); see also [CO11, Th. 5.3].
3.9 Special projective functors
From now on, we will primarily be interested in parameter values t ∈ N, so that
Part is not semisimple. Consider the atypical block {κ(0), κ(1), κ(2), . . . } associated
to κ ∈ Pt. From (3.8.12), it follows that κ(n) is obtained from κ(n−1) by adding
κn − κn+1 + 1 nodes to the nth row of its Young diagram, leaving all other rows
unchanged. The partition κ(0) is the smallest of all of the κ(n), hence, it is maximal
in the highest weight ordering from Theorem 2.9.1. It follows that
P (κ(0)) ∼= ∆(κ(0)). (3.9.1)
The indecomposable projectives ∆(κ(0)) are exactly the ones of non-zero categorical
dimension mentioned already in Remark 3.1.3, with the irreducible kSt-module
associated to the image of ∆(κ(0)) under the equivalence ψt between the
semisimplification of Kar(Par t) and kSt-Modfd being the Specht module S(κ). It
is also useful to note for t ∈ N and κ ∈ P (0) (1)t that the associated block {κ , κ , . . . } is
the set S(γ) from (3.8.9) for
γ := (εκ1 − εt) + (εκ2−1 − ε−1) + · · ·+ (εκt−t+1 − ε−t) ∈ P. (3.9.2)
This is follows easily using (3.8.11) and (3.8.12).
∏
1These algebras are certainly not equal since Z(Par ) ∼t = λ∈P kλ is of uncountable dimension.
78
In order to understand the structure of the atypical blocks more fully, we are
going to use the endofunctor | ? : Par t → Par t. Let
D := res| ? = 1| ?Part⊗Part : Part-Mod→ Part-Mod (3.9.3)
be the corresponding restriction functor from (2.6.4). This obviously preserves
locally finite-dimensional modules. The object | is self-dual so, by Lemma 2.6.2, the
restriction functor D is isomorphic to the induction functor ind| ?. By Corollary 2.6.3,
D is a self-adjoint projective functor, so it preserves finitely generated projectives
(and finitely cogenerated injectives). To make the canonical adjunction as explicit
as possible, we note that its unit and counit are induced by the bimodule
homomorphisms
m 1 m+1m 1 n+2 n+1 n 1
· · · · · ·
η : Part → 1| ?Part ⊗Part 1| ?Part, f 7→ f ⊗ · · · ,
· · · · · ·
n 1 n+1 n 1 n 1
(3.9.4)
m
m+1m 1 n+1n· · · 1 · · ·
1
··· f
ε : 1| ?Part ⊗Part 1| ?Part → Part, f ⊗ g →7 · · · .
· · · · · · g
n 1 n+1 1 · · ·
n+1 1
(3.9.5)
Using (2.3.4), it follows that D commutes with the duality ?©σ on Part-Modlfd.
Theorem 3.9.1. For λ ∈ P, there is a filtration 0 = V0 ⊆ V1 ⊆ V2 ⊆ V3 = D∆(λ)
such that
V /V ∼
⊕ ( )
3 2 = ∆ λ+ a ,
a∈add(λ)
∼ ⊕ ⊕ ( )V2/V1 = ∆(λ)⊕ ∆ (λ− b ) + a ,
b∈rem(λ) a∈add(λ− b )
79
∼ ⊕ ( )V1/V0 = ∆ λ− b .
b∈rem(λ)
Proof. This follows from Lemma 3.11.2 below, which constructs the filtration
explicitly. The proof of Lemma 3.11.2 is valid over fields of positive characteristic.
Now we are going to use the affine partition category APar to decompose the
endofunctor D as a direct sum of special projective functors Db|a. The approach here
is analogous to the way the affine symmetric category ASym was used to decompose
E and F as direct sums of Ea and Fb in (2.7.12). As noted at the end of §3.3, Par t
is isomorphic to the quotient of APar by a left tensor ideal. Hence, Par t is a strict
APar -module category. The self-adjoint functor D is also the restriction functor res| ?
arising from this categorical action of APar on Par t. Now the left and right dots give
us natural transformations
α := • ? : | ?⇒ | ?, β := • ? : | ?⇒ | ? .
Applying the general construction from (2.2.8) to these, we obtain commuting
endomorphisms
x := resα : D ⇒ D, y := resβ : D ⇒ D. (3.9.6)
LetDb|a be the summand ofD that is the simultaneous generalized eigenspace of x and
y of eigenvalues a and b, respectively. Explicitly, D = res| ? is defined by tensoring
with the bimodule 1| ?Part, and the endomorphisms x and y of D are induced by
the bimodule endomorphisms ρ and λ of 1| ?Part given by left multiplication by
xRm+1 and x
L
m+1, respectively, on the summand 1m+1Part of 1| ?Part for each m ≥ 0.
Then, Db|a is the functor defined by tensoring with the summand of 1| ?Part that
is the simultaneous generalize⊕d eigenspaces of ρ and λ for the eigenvalues a and b,
respectively. As 1m+1Part = n≥0 1m+1Part1n with each 1m+1Part1n being finite-
80
dimensional, these endomorphisms are lo⊕cally finite, so we have that
D = Db|a. (3.9.7)
a,b∈k
Lemma 3.9.2. For a, b ∈ k, the endofunctor Db|a commutes with the duality ?©σ, i.e.,
D ©σ ©σb| ∼a◦? =? ◦Db|a.
Proof. This follows from the fact that D commutes with the duality ?©σ , and σ fixes
both the left dot and the right dot.
Lemma 3.9.3. For a, b ∈ k, the endofunctors Db|a and Da|b are biadjoint.
Proof. The adjunction (Da|b, Db|a) is induced by the self-adjunction of D. The unit
η̄ of adjunction comes from the bimodule homomorphism that is the composition of
the unit η from (3.9.4) with the projection onto the generalized a and b eigenspaces
of ρ and λ on the left tensor factor and the generalized b and a eigenspaces of ρ and
λ on the right tensor factor. The counit ε̄ of adjunction comes from the composition
of the counit ε from (3.9.5) with the inclusion of the generalized b and a eigenspaces
of ρ and λ on the left tensor factor and the generalized a and b eigenspaces of ρ and
λ on the right tensor factor. To check the zig-zag identities, one just needs to use the
relations
• = • , • = • , • = • , • = • ,
i.e., the fact that the left and right dots are duals.
When a 6= b, Lemma 3.9.3 can also be proved a bit more easily using the
description of Db|a given in the following lemma, since the projection functor prγ
commutes with ?©σ thanks to (2.9.12).
Lemma 3.9.4. Let prγ be the pro⊕jection functor defined by (3.8.10). If a 6= b then
D | ∼b a = prγ+αa−α ◦D ◦ pr .b γ
γ∈P
81
⊕
Also pr ◦D ◦ pr ∼
⊕
γ∈P γ γ = a∈kDa|a.
Proof. Take a module V in the “block” parametrized by γ ∈ P , so that wtt(λ) = γ
for all irreducible subquotients of V . We need to show that Db|aV is in the “block”
parametrized by γ+αa−αb. Since Db|a is exact, we may assume that V is irreducible,
so V = L(λ) for λ ∈ P with wtt(λ) = γ. The module DV = 1| ?Part ⊗ ∼Part V = 1| ?V
is generated by the finite-dimensional vector spaces 1m+1V for all m ≥ 0. Hence,
D R Lb|aV is generated by the simultaneous generalized eigenspaces of xm+1 and xm+1 on
1m+1V of eigenvalues a and b, respectively. Consequently, if L(µ) is an irreducible
subquotient of Db|aV , then c(u) must act on L(µ) in the same way as | ? cm(u) acts
on a simultaneous eigenvector v ∈ 1m+1V for xRm+1 and xLm+1 of eigenvalues a and
b. Also cm+1(u) acts on v ∈ V as multiplication by χλ(c(u)), the rational function
displayed on the right hand side of (3.8.6). Using (3.6.9), we deduce that
αa(u)
χµ(c(u)) = × χλ(c(u)).
αb(u)
Hence, wtt(µ) = wtt(λ) + αa − αb.
Our main combinatorial result about the functors Db|a is as follows.
Theorem 3.9.5. For λ ∈ P and a, b ∈ k, there is a filtration 0 = V0 ⊆ V1 ⊆ V2 ⊆
V3 = Db|a∆(λ) such that
∼  ∆(λ+ a ) if a ∈ add(λ) and b = t− |λ|V3/V2 =  0 otherwise,
 ∆(λ)⊕∆(λ) if t− |λ| = a = b ∈ rem(λ)
∼ ∆(λ) if t− |λ| =6 a = b ∈ rem(λ) or t− |λ| = a = b ∈/ rem(λ)V2/V1 =  ( ) ∆ (λ− b ) + a if a 6= b ∈ rem(λ) and a ∈ add(λ− b )0 otherwise,
82

∼  ∆(λ− b ) if a = t− |λ|+ 1 and b ∈ rem(λ)V1/V0 = 0 otherwise.
In particular, when t ∈ Z, the functor Db|a is zero unless both a and b are integers.
Proof. See §3.11 below.
The following corollary is an immediate consequence of the theorem, but actually
it has a much easier proof which we include below.
Corollary 3.9.6. For λ ∈ P and a, b ∈ k with a 6= b, there is a filtration 0 = V0 ⊆
V1 ⊆ V2 ⊆ V3 =Db|a∆(λ) such that
∼  ∆(λ+ a ) if a ∈ add(λ) and b = t− |λ|V3/V2 =  0 ( ) otherwise,
∼  ∆ (λ− b ) + a if b ∈ rem(λ) and a ∈ add(λ− b )V2/V1 =  0 otherwise,
∼  ∆(λ− b ) if a = t− |λ|+ 1 and b ∈ rem(λ)V1/V0 = 0 otherwise.
Direct proof avoiding Theorem 3.9.5. Let γ := wtt(λ). By Lemma 3.9.4, we can
computeDb|a∆(λ) by applying prγ+α −α to the ∆-flag forD∆(λ) from Theorem 3.9.1.a b
This produces a module with a ∆-flag consisting of all ∆(µ) in the original ∆-flag
such that wtt(µ)−wtt(λ) = αa − αb. It just remains to compute wtt(µ)−wtt(λ) for
the various possible µ. If µ = λ + c for c ∈ add(λ) then, by a computation using
the first equality from (3.8.11), we have that wtt(µ)−wtt(λ) = αc−αt−|λ|; for this to
equal αa−αb we must have b = t−|λ| and c = a. If µ = λ− d for d ∈ rem(λ) then,
by a similar computation, wtt(µ) − wtt(λ) = αt−|λ|+1 − αd; for this to equal αa − αb
we must have d = b and a = t− |λ|+ 1. Finally if µ = (λ− d ) + c for d ∈ rem(λ)
83
and c ∈ add(λ − d ) then wtt(µ) − wtt(λ) = αc − αd; for this to equal αa − αb we
must have c = a and d = b.
3.10 Blocks
We assume throughout the section that t ∈ N. We are going to describe
the structure of the atypical “blocks,” revealing in particular that they are
indecomposable, hence, they are actually blocks. Recall from Theorem 3.8.4 that
the atypical “blocks” are parametrized by partitions κ ∈ Pt, with the irreducible
modules in the “block” being the ones labelled by the partitions {κ(0), κ(1), . . . }. This
is the set S(γ) from (3.8.9) where γ ∈ P is obtained from κ according to (3.9.2).
The first step is to show that all of the atypical “blocks” are equivalent to each
other. The proof of this uses the special projective functors Db|a with a =6 b. These are
the ones which can be defined just using information about central characters rather
than requiring the Jucys-Murphy elements; cf. Lemma 3.9.4 and Corollary 3.9.6. In
view of Remark 3.6.6, this sort of information was already available to Comes and
Ostrik in an equivalent form, and indeed they were also able to prove a similar result
by an analogous argument; see [CO11, Lem. 5.18(2)] and [CO11, Prop. 6.6].
Lemma 3.10.1. Let κ and κ̃ be partitions of t such that κ̃ is obtained from κ by
moving a node from the first row of its Young diagram to its (r + 1)th row for some
r ≥ 1. Let a := κr+1 − r + 1 and b := κ1. Then for all n ≥ 0 we have that
D ∆(κ(n)) ∼= ∆(κ̃(n)b|a ) and D (n) ∼ (n)a|b∆(κ̃ ) = ∆(κ ).
Proof. Let γ, γ̃ ∈ P be defined from κ and κ̃ according to (3.9.2). From this formula
it follows that γ̃ = γ + αa − αb where a = κr+1 − r + 1 and b = κ1 as in the
statement of the lemma. Note that a 6= b. So we can apply Lemma 3.9.4 to see
that D (n)b|a∆(κ ) = prγ+α −α (D∆(κ
(n))) and D (n)a|b∆(κ̃ ) = prγ−α +α (D∆(κ̃
(n))).
a b a b
84
Now we use this description to show that D (n)b|a∆(κ ) ∼= ∆(κ̃(n)). The proof that
Da|b∆(κ̃
(n)) ∼= ∆(κ(n)) is similar and we leave this to the reader.
Fix n ≥ 0 and let Bn be the set of µ ∈ P which are obtained from κ(n) by
removing a node, removing a node then adding a different node, or adding a node.
Bearing in mind that a 6= b, the standard modules ∆(µ) for µ ∈ Bn include all of the
ones which are sections of the ∆-flag from Theorem 3.9.1 which could possibly be in
the same block as ∆(κ̃(n)). Now it suffices to show for m ≥ 0 that κ̃(m) ∈ Bn if and
only if m = n. There are four cases to consider.
Case one: n = 0. We have that κ(0) = (κ2, κ3, . . . , κr+1, . . . ) and κ̃
(0) =
(κ2, κ3, . . . , κr+1+1, . . . ), which is κ
(0) with one node added to the rth row of its Young
diagram. We definitely have that κ̃(0) ∈ B0. All other µ ∈ B0 satisfy |µ| ≤ |κ̃(0)|.
Since all κ̃(m) with m > 0 have |κ̃(m)| > |κ̃(0)|, none of these belong to B0.
Case two: 1 ≤ n < r. We have that κ(n) = (κ1+1, κ2+1, . . . , κn+1, . . . , κr+1, . . . ) and
κ̃(n) = (κ1, κ2+1, . . . , κn+1, . . . , κr+1+1, . . . ), which is κ
(n) with a node removed from
the first row and a node added to the rth row of its Young diagram. We definitely
have that κ̃(n) ∈ B . For m < n, κ̃(m) is of smaller size than κ(n)n and its rth row is of
length κr+1 + 1. This cannot be obtained from κ
(n) by removing a node since κ(n) has
rth row of length κr+1. So it does not belong to Bn. For m > n, κ̃
(m) is of greater
size than κ(n) and its first row is of length κ . This cannot be obtained from κ(n)1 by
adding a node since κ(n) has first row of length κ1 + 1. So again it does not belong
to Bn.
Case three: n = r. We have that κ(n) = (κ1 + 1, κ2 + 1, . . . , κr + 1, κr+2, . . . ) and
κ̃(n) = (κ1, κ2 + 1 . . . , κr + 1, κr+2, . . . ), which is κ
(n) with a node removed from the
first row of its Young diagram. We definitely have that κ̃(n) ∈ B . The κ̃(m)n with
m < n have |κ̃(m)| ≤ |κ̃(n)| − 1 = |κ(n)| − 2 so are not elements of Bn. The κ̃(m) with
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m > n have (r + 1)th row of length κr+1 + 2, so these are not elements of Bn either
since this is at least two more than the length of the (r + 1)th row of κ(n).
Case four: n > r. We have that κ(n) = (κ1 + 1, κ2 + 1, . . . , κr+1 + 1, . . . ) and
κ̃(n) = (κ1, κ2 + 1, . . . , κ
(n)
r+1 + 2, . . . ), which is κ with a node removed from its first
row and a node added to its (r + 1)th row. We definitely have that κ̃(n) ∈ Bn. The
κ̃(m) with m > n are of greater size than κ(n) and have first row of length κ1; these
cannot be obtained by adding a node to κ(n). The κ̃(m) with r + 1 ≤ m < n are
of smaller size than κ(n) and have (r + 1)th row of length κr+1 + 2; these cannot be
obtained by removing a node from κ(n). The κ̃(m) with m ≤ r have first row of length
≤ κ1 and (r + 1)th row of length κr+2, whereas these two rows of κ(n) are of lengths
κ1 + 1 and κr+1 + 1 > κr+2, so these are not elements of Bn.
Theorem 3.10.2 (Comes-Ostrik). Let κ and κ̃ be partitions of t, denoting the
associated ∼t-equivalence classes by S := {κ(0), κ(1), . . . } and S̃ := {κ̃(0), κ̃(1), . . . }.
There is an equivalence of categories
Σ : 1SPart-Mod→ 1S̃Part-Mod
between the corresponding “blocks” such that ΣL(κ(n)) ∼= L(κ̃(n)) for all n ≥ 0. The
functor Σ is a composition of the special projective functors Db|a (a =6 b), hence, it is
a projective functor.
Proof. We may assume that κ̃ is obtained from κ by moving a node from the first
row of its Young diagram to its (r + 1)th row for some r ≥ 1. Thus, we are in
the situation of Lemma 3.10.1. The lemma gives us functors Db|a : 1SPart-Mod →
1S̃Part-Mod and Da|b : 1S̃Part-Mod→ 1SPart-Mod such that D ∆(κ(n)) ∼= ∆(κ̃(n)b|a )
and D (n)| ∼a b∆(κ̃ ) = ∆(κ(n)). These functors are also biadjoint thanks to Lemma 3.9.3.
It follows easily that they are quasi-inverse equivalences of categories as claimed in the
86
theorem. In more detail, the unit and counit of one of the adjunctions gives natural
transformations Da|b ◦Db|a ⇒ Id and Id ⇒ Db|a ◦Da|b. We claim that these natural
transformations are isomorphisms. They are non-zero, hence, they are isomorphisms
on all standard modules. The functors are exact and indecomposable projectives
have finite ∆-flags, so it follows that the natural transformations are isomorphisms
on all indecomposable projectives. Then we get that they are isomorphisms on an
arbitrary module by considering a two step projective resolution and applying the
Five Lemma.
The next lemma does use the functors Db|a in the case a = b, i.e., it definitely
requires the full strength of Theorem 3.9.5 rather than merely Corollary 3.9.6.
Lemma 3.10.3. Let κ ∈ P (0) (1)t and S := {κ , κ , . . . } be the corresponding ∼t-
equivalence class. For each n ≥ 0, there is an endofunctor Πn : Part-Mod →
Part-Mod such that Π ∆(κ
(m)
n ) = 0 for m 6= n, n+ 1, and moreover there exist short
exact sequences 0 → ∆(κ(n)) → Π ∆(κ(n)) → ∆(κ(n+1)n ) → 0 and 0 → ∆(κ(n)) →
Πn∆(κ
(n+1)) → ∆(κ(n+1)) → 0. The functor Πn is a composition of the special
projective functors Db|a (a, b ∈ Z), hence, it is a projective functor.
Proof. In view of Theorem 3.10.2, it suffices to prove the lemma in the special case
that κ = (t), when S = {∅, (t + 1), (t + 1, 1), (t + 1, 12), . . . } as in Example 3.8.5.
Then we take Π0 := D0|t ◦ · · · ◦Dt−1|1 ◦Dt|0 and Πn := D−n|−n for n > 0. Now it is
just a matter of applying Theorem 3.9.5 to see that these functors have the stated
properties.
The situation for Π0 is the most interesting. To understand this, let u := d t e and2
v := b t c. Then one checks that Dv+1|u−1 ◦ · · · ◦Dt−1|1 ◦Dt|0(∆(∅)) ∼= ∆((u)); each of2
these functors adds a single node to the first row of the Young diagram. After that we
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apply Dv|u to get a module with a two step ∆-flag, with a copy of ∆((u+ 1)) at the
top and a copy of ∆((v)) at the bottom. Note this is obtained from Theorem 3.9.5
in a slightly different way according to whether u = v (i.e., t is even) or u = v + 1
(i.e., t is odd). Also, this is now a module in an atypical block. Finally we apply
D0|t ◦D1|t−1 ◦ · · ·Dv−1|u+1 to end up with the desired two step ∆-flag with a copy of
∆(κ(1)) = ∆((t + 1)) at the top and ∆(κ(0)) = ∆(∅) at the bottom; each of these
functors adds a single node to the first row of the Young diagram labelling the module
at the top and removes a node from the Young diagram labelling the module at the
bottom. This is what Π0 is meant to do to ∆(∅). A similar argument shows that
Π0∆((t + 1)) has a ∆-flag with the same two sections. It is also easy to check that
Π ∆(κ(m)0 ) = 0 for m > 1, indeed, Dt|0 already annihilates these standard modules.
The functors Πn = D
(n)
−n|−n for n > 0 are easier to analyze. Noting that κ =
∆((t+1, 1n−1)), the module Π ∆(κ(n)n ) has a two step ∆-flag with ∆(κ
(n+1)) = ∆((t+
1, 1n)) at the top and ∆(κ(n)) at the bottom; this uses the t − |λ| = a = b ∈/ rem(λ)
case from Theorem 3.9.5. Similarly, Πn∆(κ
(n+1)) has a ∆-flag with the same two
sections. Finally, one checks that Πn∆(κ
(m)) = 0 for m =6 n, n+ 1.
Remark 3.10.4. In the proof of the next theorem, we will show that the functor Πn
from Lemma 3.10.3 satisfies Π ∆(κ(n)) ∼= Π ∆(κ(n+1)) ∼= Π L(κ(n+1)) ∼= P (κ(n+1)n n n )
for all n ≥ 0.
Now we can prove the main result about blocks. This can also be deduced from
[CO11, Th. 6.10], but the proof of that appealed to results of Martin [Mar96] in
order to obtain the precise submodule structure of the indecomposable projectives,
whereas we are able to establish this by exploiting the highest weight structure and
the Chevalley duality ?©σ .
88
Theorem 3.10.5. Let κ ∈ Pt and S := {κ(0), κ(1), . . . } be the corresponding ∼t-
equivalence class.
(i) For each n ≥ 0, the standard module ∆(κ(n)) is of length two with head L(κ(n))
and socle L(κ(n+1)).
(ii) The indecomposable projective module P (κ(0)) is isomorphic to ∆(κ(0)), while
for n ≥ 1 the module P (κ(n)) has a two step ∆-flag with top section ∆(κ(n)) and
bottom section ∆(κ(n−1)).
(iii) For each n ≥ 1, P (κ(n)) is self-dual with irreducible head and socle isomorphic
to L(κ(n)) and completely reducible heart radP (κ(n))/ socP (κ(n)) ∼= L(κ(n−1))⊕
L(κ(n+1)).
Proof. To improve the readability, we write simply P (n),∆(n) and L(n) in place
of P (κ(n)),∆(κ(n)) and L(κ(n)). For n ≥ 0, Lemma 3.10.3 shows that the module
Pn := Πn−1 ◦ · · ·Π1 ◦ Π0(∆(0)) has a two step ∆-flag with top section ∆(n) and
bottom section ∆(n − 1). Since ∆(0) is projective by the minimality observed in
(3.9.1) and each Πi is a projective functor, Pn is projective. Since Pn has L(n) in
its head, it must contain the indecomposable projective P (n) as a summand, so we
either have that P (n) ∼= Pn if Pn is indecomposable, or P (n) ∼= ∆(n) otherwise. In
the former case, (P (n) : ∆(m)) = δm,n + δm,n−1, while (P (n) : ∆(m)) = δm,n in
the latter situation. Now we apply BGG reciprocity to deduce for any m ≥ 0 that
[∆(m) : L(n)] = δn,m + δn,m+1 if Pn is indecomposable and [∆(m) : L(n)] = δn,m
otherwise. Hence, for each m ≥ 0, we either have that ∆(m) ∼= L(m), or ∆(m) is of
composition length two with composition factors L(m) and L(m+ 1).
We claim for any n ≥ 0 that ∆(n) ∼= L(n) if and only if ∆(n + 1) ∼= L(n + 1).
Suppose first that ∆(n) ∼= L(n). Since Πn commutes with duality by Lemma 3.9.2,
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this implies that Πn∆(n) is self-dual. But this module has a two step ∆-flag with top
section ∆(n+ 1) and bottom section ∆(n) ∼= L(n). The only way such a module can
be self-dual is if ∆(n+1) ∼= L(n+1) (and the module must be completely reducible).
Conversely, suppose for a contradiction that ∆(n+ 1) ∼= L(n+ 1) but ∆(n) 6∼= L(n).
Then ∆(n) is of length two with composition factors L(n) and L(n + 1), so that
P (n + 1) has a two step ∆-flag with top section ∆(n + 1) ∼= L(n + 1) and bottom
section ∆(n). Since Πn+1∆(n) = 0 according to Lemma 3.10.3 and Πn+1 is exact,
we must have that Πn+1L(n + 1) = 0. Since ∆(n + 1) ∼= L(n + 1), this implies that
Πn+1∆(n+ 1) = 0, which contradicts Lemma 3.10.3.
From the claim, we see that if ∆(n) is irreducible for any one n ≥ 0, then it is
irreducible for all n ≥ 0. Since all atypical “blocks” are equivalent by Theorem 3.10.2,
it follows in that case that the standard modules ∆(λ) for all λ ∈ P are irreducible.
This implies that the minimal ordering t from Remark 3.7.5 is trivial, hence, the
blocks are trivial and Part is semisimple, which contradicts Corollary 3.8.6. Thus,
we have proved that ∆(n) must be of length two for every n ≥ 0, and (i) is proved.
Property (ii) follows immediately from (i) and BGG reciprocity as noted earlier.
It remains to prove (iii). Take n ≥ 1. By Lemma 3.10.3, we have that Πn−1∆(n+
1) = 0. Since Πn−1 is exact and L(n + 1) is a composition factor of ∆(n + 1),
it follows that Πn−1L(n + 1) = 0 too. From this, we deduce that Πn−1∆(n) ∼=
Πn−1L(n). By Lemma 3.10.3 again, Πn−1∆(n − 1) has the same composition length
as Πn−1∆(n) ∼= Πn−1L(n). Also ∆(n − 1) has L(n) as a constituent. Using the
exactness of Πn−1 again, we must therefore have that Π ∼n−1∆(n− 1) = Πn−1L(n). As
observed earlier in the proof, this module is isomorphic to P (n), so using that L(n)
is self-dual and Πn−1 commutes with duality, we now see that P (n) is self-dual. We
also know that it has length four with irreducible head L(n), [P (n) : L(n)] = 2 and
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[P (n) : L(n − 1)] = [P (n) : L(n + 1)] = 1. The only possible structure is the one
claimed.
Corollary 3.10.6 (Comes-Ostrik). All “blocks” of Part-Mod are indecomposable,
hence, they coincide with the blocks.
Corollary 3.10.7. The minimal ordering t from Remark 3.7.5 is the partial order
such that κ(m)  κ(n)t for each κ ∈ Pt and m ≤ n, with all other pairs of partitions
being incomparable.
In general, in an upper finite highest weight category, the standard objects can
have infinite length. Our final corollary, which is also noted in [SS22, Rem. 6.4],
shows that this is not the case in Part-Modlfd. Consequently, the full subcategory
consisting of all modules of finite length has enough projectives and injectives, indeed,
this subcategory is an essentially finite highest weight category in the sense of [BS,
Def. 3.7].
Corollary 3.10.8. The locally unital algebra Part is locally Artinian, i.e., the left
ideals Part1n and the right ideals 1nPart are of finite length for all n ≥ 0.
Proof. Theorem 3.10.5 shows that all indecomposable projective left Part-modules
are of finite length, hence, all finitely generated projectives are of finite length too.
This includes all of the left ideals Part1n. Since there is a duality ?
©σ , it also follows
that all fintely cogenerated injective left Part-modules are of finite length. This
includes all of the duals (1nPar )
~
t , hence, each 1nPart is of finite length as a right
module.
3.11 Proof of Theorem 3.9.5
It just remains to prove Theorem 3.9.5. In fact, we will prove the following
slightly stronger result, from which Theorem 3.9.5 follows easily on applying the
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functors involved to the Specht module S(λ). To state this stronger result, let j! :
Sym-Modfd → Part-Modlfd be the standardization functor from (2.9.3), Ea and Fb
be the refined induction and restriction functors from (2.7.12), Db|a be the special
projective functor from (3.9.7), and prc : Sym-Modfd → Sym-Modfd be the functor
defined by multiplication by the identity element of the symmetric group Sc if c ∈ N,
i.e., it is the projection onto kSc-Modfd followed followed by the inclusion of kSc-Modfd
into Sym-Modfd, or the zero functor if c ∈ k− N.
Theorem 3.11.1. For a, b ∈ k, there is a filtration of the functor Db|a ◦ j! :
Sym-Modfd → Part-Modlfd by subfunctors 0 = S0 ⊆ S1 ⊆ S2 ⊆ S3 ⊆ S4 = Db|a ◦ j!
such that
S ∼4/S3 = j! ◦ Ea ◦ prt−b,
S3/S ∼2 = j! ◦ prt−a ◦ prt−b,
S2/S ∼1 = j! ◦ Ea ◦ Fb,
S1/S ∼0 = j! ◦ prt−a ◦Fb.
(Recall that a subfunctor S of a functor T : Sym-Modfd → Part-Modlfd is a functor
S : Sym-Modfd → Part-Modlfd such that SV is a submodule of TV for all V ∈
Sym-Modfd and Sf = Tf |SV for all f ∈ Hom ′Sym(V, V ); then the quotient T/S is the
obvious functor with (T/S)(V ) := TV/SV .)
The proof will take up the rest of the subsection. We begin by constructing a
filtration of the functor D◦j! : Sym-Modfd → Part-Modlfd. Note that D◦j ∼! = M⊗Sym
where M is the (Part, Sym)-bimodule
M := 1| ?Part ⊗Par] infl] Sym. (3.11.1)
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We also have the (Part, Sym)-bimodules
N4 = Part ⊗Par] infl](Sym1| ?), (3.11.2)
N3 := Part ⊗ ]Par] infl Sym, (3.11.3)
N2 := Part ⊗Par] infl](Sym1| ? ⊗Sym 1| ?Sym) (3.11.4)
N1 := Part ⊗ ]Par] infl (1| ?Sym). (3.11.5)
The functors Sym-Modfd → Part-Modlfd defined by tensoring with N4, N3, N2 and
N1 are isomorphic to j! ◦ E, j!, j! ◦ E ◦ F and j! ◦ F , respectively.
For m ≥ n ≥ 0, let Bm,n be the basis for 1mPar−1n defined by representatives
for the equivalence classes of normally ordered upward partition diagrams. By
Theorem 2.8.1, the vector space M is isomorphic to 1| ?Par
− ⊗K Sym, hence, it has
basis { ∣
f ⊗ g ∣ }m ≥ 0, n ≥ 0,m+ 1 ≥ n, f ∈ Bm+1,n, g ∈ Sn . (3.11.6)
For any f ∈ Bm+1,n, let c(f) be the connected component of the diagram containing
the top left vertex. In the language from §2.8, this component could be a trunk, an
upward tree, an upward leaf, or an upward branch. Then we introduce the following
subspaces of M :
– Let M1 be the subspace of M spanned by all f ⊗ g in this basis such that c(f)
is a trunk.
– Let M2 be the subspace spanned by all f ⊗ g such that c(f) is either a trunk
or an upward tree.
– Let M3 be the subspace spanned by all f ⊗ g such that c(f) is either a trunk,
an upward tree, or an upward leaf.
– Let M0 := 0 and M4 := M .
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The following is a generalization of Theorem 3.9.1.
Lemma 3.11.2. The subspaces 0 = M0 ⊂ M1 ⊂ M2 ⊂ M3 ⊂ M4 = M
are sub-bimodules of the (Part, Sym)-bimodule M . Moreover, there are bimodule
∼
isomorphisms θi : Ni →Mi/Mi−1 for each i = 1, . . . , 4.
Proof. The fact that each Mi is a sub-bimodule of M is easily checked by vertically
composing a basis vector f ⊗ g with an arbitrary partition diagram on the top and
with any permutation diagram on the bottom. One just needs to note that the action
on top involves res| ?, so that the top left vertex is untouched. This implies that the
type c(f) does not change if it is a trunk or an upward leaf, while if it is an upward
tree it can only be changed to another upward tree or to a trunk.
We show in this paragraph that there is a bimodule isomorphism
· · · · · · · · · · · ·
θ1 : N1 →M1, f ⊗ g 7→ f ⊗ g (3.11.7)
· · · · · · · · · · · ·
for any m ≥ 0, n > 0, f ∈ 1mPart1n−1 and g ∈ Sn. This is a well-defined bimodule
homomorphism. By Theorem 2.8.1, N1 is isomorphic as a vector space to Par
− ⊗K
1| ?Sym, hence, it ha{s basis∣ }
f ⊗ g ∣m ≥ n− 1 ≥ 0, f ∈ Bm,n−1, g ∈ Sn . (3.11.8)
The vector space M1 has basis given by all f1 ⊗ g for m + 1 ≥ n > 0, f1 ∈ Bm+1,n
and g ∈ Sn such that c(f1) is a trunk. As it is normally ordered, any such f1 is of the
form
· · ·
f1 = f
· · ·
for a unique f ∈ Bm,n−1. Moreover, f1 ⊗ g = θ1(f ⊗ g) for every g ∈ Sn. It follows
that θ1 takes a basis for N1 to a basis for M1, so it is an isomorphism.
94
Next we show that there is a bimodule isomorphism
· · · · · · · · · · · · · · ·
f
θ2 : N2 →M2/M1, f ⊗ g ⊗ h 7→ g ⊗ h +M1 (3.11.9)
· · · · · · · · · · · · · · ·
for m ≥ 0, n > 0, f ∈ 1mPart1n and g, h ∈ Sn. Again, this is a well-
defined bimodule homomorphism. By Theorem 2.8.1, N2 is isomorphic to Par
− ⊗K
Sym1| ? ⊗Sym 1| ?Sym. Also kSn is free as a right kSn−1-module with basis given by
{(i i+1 · · · n) | 1 ≤ i ≤ n}, which is a set of Sn/Sn−1-cosets. It follows that N2 has
basis{ ∣ }
f ⊗ (i i+1 · · · n)⊗ g ∣m ≥ n > 0, f ∈ Bm,n, 1 ≤ i ≤ n, g ∈ Sn . (3.11.10)
The vector space M2/M1 has a basis given by all f2⊗ g+M1 for m+ 1 ≥ n > 0, f2 ∈
Bm+1,n and g ∈ Sn such that c(f2) is an upward tree. Any such f2 is equal to
· · ·
f
f2 = n ·· i ·· 1
· · ·
for a unique f ∈ Bm,n and a unique 1 ≤ i ≤ n (the index of the string at which the
component c(f2) meets the bottom of f). Moreover, f2⊗g = θ2(f⊗(i i+1 · · · n)⊗g)
for each g ∈ Sn. It follows that θ2 takes a basis for N2 to a basis for M2/M1, so it is
an isomorphism.
The isomorphism θ3 is defined by
· · · · · · · · · · · ·
θ3 : N →M /M , f ⊗ g 7→ •◦3 3 2 f ⊗ g +M2 (3.11.11)
· · · · · · · · · · · ·
for m ≥ 0, n ≥ 0, f ∈ 1mPart1n and g ∈ Sn. This is obviously a well-defined
bimodule homomorphism. It is an isomorphism because it takes the basis
{f ⊗ g |m ≥ n ≥ 0, f ∈ Bm,n, g ∈ Sn} (3.11.12)
for N3 to the basis for M3/M2 consisting of all f3 ⊗ g + M2 for m + 1 > n ≥ 0, f3 ∈
Bm+1,n and g ∈ Sn such that c(f3) is an upward leaf.
95
Finally, we construct the isomorphism θ4. The vector space N4 has basis
{f ⊗ g |m− 1 ≥ n ≥ 0, f ∈ Bm,n+1, g ∈ Sn+1}. (3.11.13)
We define the linear map
· · · · · · · · · · · ·
θ4 : N4 →M4/M3, f ⊗ g 7→ f·· ·· ⊗ g′ +M3, (3.11.14)· · · · · · n+1 i 1· · · · · ·
where f ⊗ g is a vector from the basis for N4 just displayed, and g′ ∈ Sn and 1 ≤ i ≤
n + 1 are defined from the equation g = (i i+1 · · · n+1)g′. To see that this linear
map is actually a bimodule isomorphism, we construct a bimodule homorphism in
the other direction and show that it is a two-sided inverse of θ4. Consider the map
· · · · · · · · · · · ·
φ : M → N4, f ⊗ g 7→ f ⊗ g (3.11.15)
· · · · · · · · · · · ·
for m ≥ 0, n ≥ 0, f ∈ 1m+1Part1n and g ∈ Sn. It is easy to show that this is a
well-defined bimodule homomorphism. Moreover, M3 ⊆ kerφ since applying φ to
any basis vector f ⊗ g ∈ M3 produces a downward leaf, a cap or a downward tree
which can be pushed across the tensor to act as zero on infl] Sym. Hence, φ induces
a homomorphism φ̄ : M4/M3 → N4. It remains to check that φ̄ ◦ θ4 and θ4 ◦ φ̄ are
both identity morphisms, which is straightforward.
In the next two lemmas, we finally need to make some explicit calculations with
the relations involving the left and right dots in the affine partition category. However,
we are working now with Par t, not with APar , so all string diagrams from now on
should be interpreted as the canonical images of these morphisms in APar under
the functor pt : APar → Part from (3.4.12). We will also use the notation from
(2.7.10) for an open dot on the interior of a string, meaning the canonical image of
this morphism in ASym under the functor p : ASym → Sym from (2.7.8). This is quite
different from an open dot at the end of a string!
96
Lemma 3.11.3. Suppose that m ≥ 0, n ≥ 0, f ∈ 1mPart1n and g ∈ Sn.
(i) The following holds in the bimodule M = 1 ]| ?Part⊗Par] infl Sym for i = 1, . . . , n:
· · ·
f · · · · · · · · ·
⊗ g ≡ •◦ f ⊗ g (mod M2).
··· • · · · · · · · · · · · ·
n i 1
(ii) The following holds in the bimodule M for i = 0, 1, . . . , n (the case i = 0 is
when there are no strings to the right of the dangling dots):
· · ·
f · · · · · · · · ·
⊗ g ≡ (t− i) •◦ f ⊗ g (mod M2).
···◦•• · · · · · · · · · · · ·
n i 1
Proof. (i) We proceed by induction on i = 1, . . . , n. The base case i = 1 follows from
(3.4.13). For the induction step, we take i > 1 and assume the result has been proved
for i− 1. Then we apply (3.3.13) to commute the left dot past the string to its right.
This produces a sum of five terms. Ordering these terms in the same way as they
appear on the right hand side of (3.3.13), the induction hypothesis can be applied to
the first term, to produce the right hand side that we are after. It remains to show
that the other four terms lies in M2. These terms are as follows:
· · · · · ·
f · · · f · · ·
g g
··· •
⊗ + ⊗
··· · · · · · · • ··· · · ·
n i 1 n i 1
· · · · · ·
f · · · f · · ·
− • ⊗
g − ⊗ g .
· · · ··· · · · · · · • ··· · · ·
n i 1 n i 1
The second and third terms here are zero already in M because, in both of them, the
diagram to the left of the tensor is equivalent to a diagram with a downward tree at
the bottom. It remains to show that the first and the fourth terms lie in M2. For the
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fourth term, we note that
· · · · · ·
f f
= .
· · · • ··· •· · · ···
n i 1 n i 1
The left dot can now be absorbed into the morphism f , changing it to some other
morphism f ′. The result is a linear combination of morphisms in all of which the
top left vertex is connected to the bottom edge, so that the connected component
containing this vertex is either a tree or a trunk, and it belongs to the sub-bimodule
M2. The reason the first term lies in M2 is very similar, one just needs to rewrite
the right crossing using (3.3.10), and then it is easy to see that the top left vertex is
again connected to the bottom edge.
(ii) Again we proceed by induction. The base case i = 0 follows from (3.4.13) using
that T = t11. For the induction step, we consider some i > 0. Then we apply
(3.3.14) to commute the right dot past the string to its right. This produces a sum
of five terms. This time, the induction hypothesis can be applied to the first term, to
produce the vector that we are after but scaled by (t− i+ 1) rather than the desired
(t− i). The remaining four terms are as follows:
· · · · · ·
f
• · · ·
f · · ·
⊗ g + ⊗ g
··· · · · · · · ···• · · · · · ·
n i 1 n i 1
· · · · · ·
f · · · f · · ·
− ⊗ g − ⊗ g .
· · · •··· · · · · ·•·•◦ ··· · · ·
n i 1 n i 1
In the first term here, the left dot is some morphism in Part, which has the effect
of changing f to some other morphism f ′. After doing that, it is clear that the top
left vertex is still connected to the bottom edge, so the first term lies in M2. For the
second and third terms, the left and right dots can be commuted across the tensor
98
using (3.8.4), then again we see that these morphisms lie in M2 since the top left
vertex is connected to the bottom edge again. For the final term, we note that
(3.3.11) (3.3.10) (1.1.3)••◦ = = = • .•◦ • ◦• • (1.1.5)
Making this substitution in the middle of the picture reveals that the final term is
exactly the expression studied in (i). On applying the conclusion of (i), we deduce
that it contributes exactly the needed correction to complete the proof.
Lemma 3.11.4. Consider the bimodule endomorphisms ρ and λ of M defined on
1 Rm+1Part1n⊗kSn by the left action of xm+1⊗1n and xLm+1⊗1n, respectively, for each
m,n ≥ 0. These endomorphisms preserve each of the sub-bimodules Mi (i = 1, 2, 3, 4),
hence, ρ and λ induce endomorphisms also denoted ρ and λ of each of the subquotients
Mi/Mi−1. Moreover, for each i, the isomorphism θi from Lemma 3.11.2 satisfies
θi ◦ ρi = ρ ◦ θi, θi ◦ λi = λ ◦ θi, (3.11.16)
where ρi, λi : Ni → Ni are defined as follows:
(i) ρ1 and λ1 are the bimodule endomorphisms of N1 defined on the subspace
1mPart1n−1⊗kSn by the left actions of (t−n+1)1m⊗1n and 1m⊗xn, respectively,
for each m ≥ 0, n > 0.
(ii) ρ2 and λ2 are the bimodule endomorphisms of N2 defined on 1mPart1n⊗kSn⊗
kSn by the right action of 1n ⊗ xn ⊗ 1n and the left action of 1m ⊗ 1n ⊗ xn,
respectively.
(iii) ρ3 and λ3 are both equal to the bimodule endomorphism of N3 defined on
1mPart1n ⊗ kSn by multiplication by (t− n).
(iv) ρ4 and λ4 are the bimodule endomorphisms of N4 defined on 1mPart1n+1⊗kSn+1
by the right actions of 1n+1 ⊗ xn+1 and (t− n)1n+1 ⊗ 1n+1, respectively.
99
Proof. (i) Recall the definition of θ1 from (3.11.7). Take a vector f⊗g in the basis for
N1 from (3.11.8). By (3.8.4), we have that x
L
n ≡ xn (mod Kn) and xRn ≡ (t−n+1)1n
(mod Kn) where Kn is the two sided ideal of 1nPart1n from (3.7.1). Since the strictly
downward partition diagrams which generate K+ are zero on infl] Sym, it follows that
• · · · · · · · · · · · ·
ρ(θ1(f ⊗ g)) = f ⊗ g = f ⊗ g
· · · · · · • · · · · · ·
· · · · · ·
= (t− n+ 1) f ⊗ g = θ1(ρ1(f ⊗ g)),
· · · · · ·
• · · · · · · · · · · · ·
λ(θ1(f ⊗ g)) = f ⊗ g = f ⊗ g
· · · · · · • · · · · · ·
· · · •◦· · ·
= f ⊗ g = θ1(λ1(f ⊗ g)).
· · · · · ·
This shows as the same time that ρ and λ both leave M1 invariant.
(ii) Recall the definition of θ2 from (3.11.9). The argument for λ is similar to in (i).
It follows from the calculation
• · · · · · · · · · · · ·
f f
λ(θ2(f ⊗ g ⊗ h)) = g ⊗ h +M1 = g ⊗ h +M1
· · · · · · • · · · · · ·
· · · ( )•◦ · · · · · · · · · ◦• · · ·
f
= g ⊗ h +M1 = θ2 f ⊗ g ⊗ h
· · · · · · · · · · · · · · ·
= θ2(λ2(f ⊗ g ⊗ h)),
where f ⊗ g ⊗ h is one of the basis vectors for N2 from (3.11.10). For ρ, we instead
have that
• · · ·
f · · · · · ·f · · ·
ρ(θ2(f ⊗ g ⊗ h)) = g ⊗ h +M 1
= g ⊗ h +M1
· · · · · · • · ·· · · · · · · · · ·= θ f2 g ⊗ · · · ⊗ h 
• · · · · · ·
100
 · · · · · ·= θ f ⊗ · · · ⊗ h 

2( g◦• · · · · · · )
· · · · · · · · ·
= θ2 f ⊗ g ⊗ h = θ2(ρ2(f ⊗ g ⊗ h)).
· · · •◦ · · · · · ·
(iii) Recall the definition of θ3 from (3.11.11). Note that ρ = λ by the third relation
from (3.3.8). For ρ, we need to show that ρ(θ3(f ⊗ g)) = (t − n)θ3(f ⊗ g) for
any basis vector f ⊗ g ∈ 1mPart1n ⊗ kSn ⊂ N3 from (3.11.12). This follows from
Lemma 3.11.3(ii) taking i = n.
(iv) Instead of working with θ4 from (3.11.14), it is easier to use the inverse map φ̄
induced by the homomorphism φ : M → N4 from (3.11.15). We need to show that
φ◦ρ = ρ4◦φ. This follows from the following calculations for f⊗g ∈ 1m+1Part1n⊗kSn
and m,n ≥ 0: ( )
• · · · · · • · · · · · · · · · · ·
φ(ρ(f ⊗ g)) = φ f ⊗ g = f ⊗ g = f ⊗ g
· · · · · · · · · · · · • · · · · · ·
· · · ◦• · · · · · · · · ·
= ( f ⊗ g = f ⊗ g = ρ4(φ(f ⊗ g)),· · · · · · ) · · · •◦ · · ·
• · · · · · · • · · · · · · · · · · · ·
φ(λ(f ⊗ g)) = φ f ⊗ g = f ⊗ g = f ⊗ g
· · · · · · · · · · · · • · · · · · ·
· · · · · ·
= (t− n) f ⊗ g = λ4(φ(f ⊗ g)).
· · · · · ·
Proof of Theorem 3.11.1. The functor Db|a ◦ j! is defined by tensoring with the
bimodule M that is the simultaneous generalized a eigenspace of the endomorphism
ρ and generalized b eigenspace of the endomorphism λ defined in Lemma 3.11.4.
Lemma 3.11.2 defines a filtration of M with sections M ∼i/Mi−1 = Ni for i = 1, . . . , 4.
Then Lemma 3.11.4 shows that the endomorphisms ρ and λ preserve this filtration,
hence, the filtration of M induces a filtration of the summand M . Moreover, for each
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i, M i/M i−1 is isomorphic to the summand N i of N defined by the simultaneous
generalized a-eigenspace of the endomorphism ρi and generalized b eigenspace of
the endomorphism λi. By the descriptions of ρi and λi, it follows that N i⊗Sym is
isomorphic to the functor j! ◦Ea ◦ prt−b, j! ◦ prt−a ◦ prt−b, j! ◦Ea ◦ Fb or j! ◦ prt−a ◦Fb
for i = 4, 3, 2, 1, respectively. It remains to observe that Sym is semisimple, so
every Sym-module is flat. This means that the filtration of M induces a filtration
0 = S0 ⊆ S1 ⊆ S2 ⊆ S3 ⊆ S4 = Db|a◦j! such that S ∼i = M i/M i−1⊗ ∼Sym = N i⊗Sym.
102
CHAPTER IV
RESTRICTION FUNCTOR
There is a well-known ‘restriction’ functor F tt−1 : Rep(St) → Rep(St−1) which
interpolates between Deligne’s categories for different parameters of t ∈ k. Comes
and Ostrik conjectured that this restriction functor provides an equivalence of
categories between the nontrivial principle blocks of Rep(St) and Rep(St−1) whenever
t ∈ Z>0. Recall that Deligne’s category Rep(St) is monoidally equivalent to the
full subcategory Part-Proj of Par t-Modlfd consisting of finitely generated projectives.
In this light, the first portion of this chapter provides an interpretation of this
restriction functor in terms of path algebras and bimodules over them. To this
end, we introduce a new intermediate category Par×t−1 along with functors Par t →
Add(Par×t−1) and Par t−1 → Add(Par×t−1). The composition of these functors, at
the level of module categories, will allow us to induce, then restrict, locally finite-
dimensional Par t−1-modules to get locally finite-dimensional Par t-modules and vice-
versa. This construction gives a module-theoretic version of the restriction functor
Rtt−1 : Part−1-Modlfd → Part-Modlfd. After studying Rtt−1 and its interaction
with standardly-filtered modules, we show that it gives an equivalence between the
principal blocks of Part−1-Modlfd and Part-Modlfd, thereby proving the Comes-Ostrik
conjecture.
4.1 Phantom partitions
Fix some t ∈ k. Throughout this chapter, it is sometimes useful to distinguish
between objects of Par t (or Part-Mod) and of Par t−1 (or Part−1-Mod, or any other
category Par u for another value of u ∈ k). In particular, we use subscripts Xt ∈
O(Par t) when there may be potential confusion for which category an object X is
contained in. We also fix the notations |t and 1t ∈ O(Par t) for the monoidal generator
103
and unit object of Par t, actually reserving the unscripted | and 1 to mean the monoidal
generator and unit of Par t−1.
Definition 4.1.1. The phantom partition category Par×t−1 is the strict symmetric
monoidal category obtained from Par t−1 by adjoining a new generating object × and
generating morphisms:
· ×
: × → 1, : 1→ ×
× ·
subject to(the r)elat(ion t)hat these morphisms are tw(o-sid)ed (inver)ses of each other:
× × · × · · × ·
= ◦ = = id×, × = ◦ = = id1,
× · × × · × · ·
(4.1.1)
Remark 4.1.2. The dots appearing in these new morphisms are purely decorative
and serve only to help distinguish the top or bottom of a diagram.
Introducing new nomenclature, a phantom partition diagram is a diagram
obtained by taking a partition diagram, f , and inserting some (perhaps zero) symbols
× along the top and bottom rows of f . Given a phantom partition diagram f , say
that f is a partition diagram with phantoms if it contains at least one symbol ×. In
the other case, we say f is phantomless.
Gi⋃ven n ∈ N, let W n be the set of words in the letters {|,×} of length n and let
W := W nn∈N be the set of all words. Then O(Par
×
t−1) = W . Suppose w and w
′
are any two words with k(w) and k(w′) of the letters being |, respectively. Then the
isomorphism between and × ′induces an isomorphism Hom (|⊗k(w), |⊗k(w ) −→∼1 Par t−1 )
HomPar× (w,w
′). This map is described on a basis of partition diagrams by inserting
t−1
phantoms × in the unique way along the bottom and top rows to spell the words w
and w′, respectively. Hence, the path algebra of Par×t−1 inherits a basis consisting of
phantom partition diagrams.
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The restriction functor of the Comes and Ostrik conjecture is defined using
Deligne’s categories Rep(St) and Rep(St−1). In order to understand the induced
functor at the level of modules over Part and Part−1, we pass through additive
envelopes. Introducing the object | := | ⊕ × ∈ Add(Par×t−1), the following special
matrices are crucial for building the restriction functor in this setting. These matrices
are morphisms in Add(Par×t−1), i.e., matrices of morphisms in Par×t−1.
 
 0 0 0 
 0  
 
× 0 0 × 0 
= : | → |, =   : | ? | → | ? |
×
0 ×  0 × 0 0 × 
0 0 0 ××××
  (4.1.2)
  
 0 
 0 0 0 =   0 0 : | ? | → |, =   : | → | ? |×
0 0 0 × ×  0 0 × ×
0 ×
  (4.1.3)( )
◦  ◦ = ◦ · : | → 1, ◦ =   : 1→ |× ×
·
(4.1.4)
Remark 4.1.3. In the above definitions, we implicitly make the following
identification
| ? | = (| ⊕ ×)?2 = (| ? |)⊕ (| ?×)⊕ (× ? |)⊕ (× ?×).
More generally, the summands of |?n are words w ∈ W n for any n ∈ N. The ordering
of the summands comes from the strict monoidal structure on Add(Par×t−1) which is
105
inherited by that on Par×t−1, described as follows. Consider objects X1⊕ . . .⊕Xn, Y1⊕
. . .⊕ Y ,X ′ ⊕ . . .⊕X ′m 1 n′ and Y ′ ′1 ⊕ . . .⊕ Ym′ ∈ O(Add(Par×t−1)) along with morphisms
f : X1 ⊕ · · ·Xn → Y1 ⊕ Ym and g : X ′1 ⊕ · · · ⊕X ′n′ → Y ′1 ⊕ · · · ⊕ Y ′n′ . The monoidal
product is defined on objects by ⊕
(X1 ⊕ · · · ⊕Xn) ? (X ′1 ⊕ · · · ⊕X ′ ′n′) = Xi ? Xj,
1≤i≤n
1≤j≤n′
where the summands on the right hand side are ordered lexicographically with respect
to the indices (i, j). To compute the product f ? g, recall that f is a m × n matrix
whose (p, q)-entry is given by a m⊕orphism (f)p,q :⊕Xq → Yp, and similarly for g. Then
f ? g : X ? X ′i j → Yi ? Y ′j
1≤i≤n 1≤i≤m
1≤j≤n′ 1≤j≤m′
is a (mm′)× (nn′) matrix whose entries are given by the monoidal products fp,q ?gr,s.
We refer to this monoidal structure on Add(Par×t−1) as the Kronecker structure as
the product f ? g is given by the Kronecker product of matrices. Following the
usual convention with diagrams, the Kronecker product of morphisms introduced in
(4.1.5)-(4.1.8) will be denoted by horizontal juxtaposition.
Remark 4.1.4. Obviously Add(Par×t−1) contains Par
×
t−1 as the full subcategory of
objects being all words w ∈ W (rather than finite direct sums of words). It also
contains Par t−1 as the full subcategory whose objects are |?n = |︸·︷·︷·︸| for all n ∈ N.
n
So there are faithful inclusion functors
I : Par t−1 → Add(Par×t−1), J : Par× ×t−1 → Add(Par t−1)
106
In performing calculations, it is useful to express the blue morphisms of (4.1.2)-
(4.1.3) in terms of the following elementary matrices:   0 
 0 0 0  0 0 0 0= : | → |, =   : | ? | → | ? | (4.1.5)0 0  0 0 0 0
0 0 0 0  

0
 0 0 0=   0 0: | ? | → | , = 
0 0 0 0  
 : | → | ? | (4.1.6)0 0
0 0
( )  ◦ ◦◦ = 0 : | → 1, ◦ =   : 1→ | (4.1.7)
0
· ( )  · × 0
= 0 : | →  1, =   : | → 1 (4.1.8)
× × · ×
·
These matrices are just another set of special morphisms inside of Add(Par×t−1).
They allow us to write the blue morphisms of (4.1.2)-(4.1.3) as sums of Kronecker
products of these. Compositions and Kronecker products of any matrices in (4.1.5)-
(4.1.8) are again elementary matrices.
| := | + × , := + × + × ××× × × + ×× (4.1.9)
◦ := ◦ + ×× , ◦ := ◦ + ×× (4.1.10)
:= + ×× × = +
× ×
× (4.1.11)
107
Example 4.1.5. As an example, consider the following matrix built from Kronecker
products and compositions of those in (4.1.5)-(4.1.8).
×
× ×
This is a matrix with rows indexed by W 4 and columns indexed by W 3 with a single
non-zero entry: ( )
× ×
= .
× × × ×
|×||,×|×
Lemma 4.1.6. The object | ⊕ × ∈ O(Add(Par×t−1)) is a special symmetric Frobenius
object of dimension t.
Proof. This is straightforward and is mostly relation-checking. The fact that | ⊕ ×
has the appropriate dimension is simply due to the additivity of dimension. We claim
that | ⊕ × is a special Frobenius object whose structure maps are given by (4.1.9)-
(4.1.11). To prove this, it amounts to checking that these morphisms satisfy the
relations (1.1.2)-(1.1.6). This is straightforward, so we do not scribe the calculations
for each relation — only for (1.1.3).
= (( ( ) ◦ ( ))◦(( ) ) )
= +( ( ×× × |)+( ×× ) )
◦ ( ( | + × + × × ××× × + )×( + ××) )
( ◦ + × + ×× × )+ ×××× | + ××
= +( × + ××× × + ××××× )
◦ ( + × + × + ×× × ×× × × ×××× × ×× + × + × × + ×× + ×××)
◦ + ×× +
×
× +
××
×× +
× × × ×× ×××
× + ×× + × × + ×××
× × ××
= + + +
×× × ×××
108
= (( ) ◦ ( ) ) ( ( ) ( ) )
= ( + × + × + ×× ) ◦ ( | + × + ×× × ×× × × × )
= + × + × + ×× ◦ + × + × + ××× × ×× ×× × ×××
× × ××
= + + +
×× × ×××
By Lemma 4.1.6 and the universal property of Par t, there is a symmetric tensor
functor F tt−1 : Par t → Add(Par×t−1) given from the assignment |t 7→ |t−1 ⊕× and
F(t ( |)) = | , F tt−1 t−1(( )) = (4.1.12)
F t ◦t−1 = ◦ , F tt−1 ◦ = ◦ (4.1.13)
F t tt−1 ( ) = , Ft−1 ( ) = (4.1.14)
Remark 4.1.7. It will be useful to know how F tt−1 is defined in terms of the Kronecker
products and compositions of elementary matrices in (4.1.5)-(4.1.8), generalizing
(4.1.9)-(4.1.11). Suppose for convenience that f ∈ Par t is a partition diagram. Then
F tt−1(f) is the sum of all matrices obtained by erasing connected components of f ,
replacing the erased boundary points on the bottom and top edges by the symbol ×
and coloring th(e result)ing picture red. As an example,
× × ××
F tt−1 ◦ = ◦ + + ◦ +× ◦× ×
× × ×× × ××× × ×××
+ + + ◦ +
× × ×× ×× ×××
In order to handle some of the calculations in §4.3 involving general partition
diagrams with arbitrary amounts of connected components, it will be convenient to
introduce a bit more notation to deal with the sums described above. Let C(f)
denote the set of connected components of f and let S ⊆ C(f) be any subset. Define
109
f [S ×] as the diagram obtained from erasing those connected components c ∈ S and
replacing the top and bottom boundary points by ×. Then F tt−1(f) is given by the
following formula. ∑
f× := F t [S ×]t−1(f) = f (4.1.15)
S⊆C(f)
4.2 Restriction and induction functors
Having obtained functors F tt−1 : Par t → Add(Par×t−1) and I : Par t−1 →
Add(Par×t−1), restriction and induction allows us to construct a new functor from
the category of locally finite-dimensional left Part-modules to the category of locally
finite-dimensional left Part−1-modules. In the formalism from §2.2, inducing along I
then restricting along F tt−1 defines a functor R
t
t−1 : Part−1-Modlfd → Part-Modlfd on
module categories.
Rtt−1 := resF t ◦ indI : Part−1-Modlfd → Par− t-Modlfd (4.2.1)t 1
Under this definition, it is obvious that Rtt−1 has a right adjoint given by the functor
resI ◦ coindF t and a left adjoint resI ◦ indF t .t−1 t−1
Since the functor I : Par t−1 → Add(Par×t−1) is given by composing an equivalence
(Par ×t−1 → Par t−1) with the inclusion J : Par×t−1 ↪→ Add(Par×t−1), Lemma 2.2.2
provides that indI itself is an equivalence. So it has quasi-inverse given by its right
adjoint resI . Consequently, coindI is an equivalence by the same reason and then
ind ∼I = coindI .
Lemma 4.2.1. The functor Rtt−1 is exact.
Proof. This follows since Rtt−1 is a composition of exact functors.
110
Recall the Chevalley duality functor ?©σ from §2.9. Following (2.3.4) and (2.3.5),
Rtt−1 commutes with duality:
res ◦ ind ◦?©σF t ∼= res ©σt ◦? ◦ coindt− I F − I1 t 1
∼=?©σ ◦ resF t ◦ coindt− I1
∼=?©σ ◦ resF t ◦ indt− I1
So there is an isomorphism
Rt ©σ ∼ ©σ tt−1◦? =? ◦Rt−1 (4.2.2)
Let At−1 be the path algebra of Add(Par×t−1). The functors F tt−1 and I provide
At−1 with a (Part, Part−1)-bimodule structure. Then define
M := 1F t A− t−11I . (4.2.3)t 1
Lemma 4.2.2. There is an isomorphism of functors
Rt (−) ∼t−1 = M ⊗Part−1 −.
Proof. Recall that there are natural isomorphisms below, for any N ∈ Part−1-Mod
and any N ′ ∈ At−1-Mod.
indI N = A
′ ∼
t−11I ⊗Part−1 N, resF t N = 1F t A ′− − t−1 ⊗At 1 t 1 t−1 N
Hence for any N ∈ Part−1-Mod,
Rtt−1N
∼= 1F t A− t−1 ⊗At−1 At−11I ⊗Part−1 Nt 1 ⊕  ⊕ ∼=  1F t A   t−11X ⊗At−1 1YAt−11I ⊗− Part 1 t−1 N
X∈O⊕(Add(Pa(r×t−1)) ) Y ∈O(Add(Par×t−1))∼= 1F t At−11X ⊗1 At−11 (1XAt−11I)⊗− X X Par Nt 1 t−1
X∈O(Add(Par×t−1))
∼= 1F t A 1 ⊗ Nt− t−1 I Par1 t−1
∼= M ⊗Part−1 N
111
Going back to the definition restriction as in (2.2.2) and (2.2.3), observe that
there is a vector space decomposition as below where 1n := 1|?n . Therefore, M has a
vector space decomposition ⊕
M ∼= 1nAt−11m. (4.2.4)
n,m∈N
Viewing M as a left Part-module, we can cut with any of the local units 1n ∈ Part,
n ∈ N. Obviously, 1nM = 1nM .
Lemma 4.2.3. Let m,n ∈ N and let v = |︸·︷·︷·︸| ∈ Wm. With the lexicographical
m
ordering on W n, the subspace 1nA1m of M has a basis consisting of elementary
column vectors (0, . . . , 0, fw, 0, . . . , 0)
T of length |W n| = 2n with a single nonzero
entry corresponding to some word w. Fixing w ∈ W n and collecting the basis elements
whose nonzero entries lie in the w-th entry, those fw form a basis of HomPar× (m,w).t−1
Such a basis for 1nM1m is identified with a set of phantom partition diagrams whose
top row is a word of length n and bottom row is v.
Proof. Starting with (4.2.4), fix some n,m ∈ N. By definition,
?m
1 ?nnAt−11m = HomAdd(Par× (| , | )t−1)
= HomAdd(Par× )((|?m, (| ⊕ ×)?nt−1 ⊕ ))
= HomAdd(Par× ) |
?m, w
t−1
w∈Wn
∼ ⊕= Hom ?mAdd(Par× ) (| , w)t−1
w∈Wn
∼ ⊕= Hom ?mPar× (| , w)t−1
w∈Wn
112
All that remains to note is that each summand HomPar× (|?m, w) has basis consistingt−1
of phantom partition diagrams f with bottom row spelling v = | · · · | and top row
spelling the word w.
Remark 4.2.4. For any t ∈ k and s ∈ N, one can just as well define more
general restriction functors F tt−s with the assistance of a new category Par
×,s
t−s. This
category Par×,st−s can be constructed from Par t−s by adjoining s new generating objects
×t−1,×t−2 . . . ,×t−s which are all isomorphic to 1t−s, and then taking the additive
envelope. Then the functor F tt−s : Par t → Add(Par
×,s
t−s) is defined by sending |t to
|t−s ⊕ ×t−s ⊕ · · · ⊕ ×t−1. Similar to (4.1.15), applying this functor to any partition
diagram f results in a sum over all ways of erasing connected components of f while
replacing the boundary points of an erased component by phantoms with a common
index. Then one obtains functors Rtt−s : Part−s-Modlfd → Part-Modlfd.
For a pair s, r ∈ N, F t−st−s−r has an canonical extension to the additive envelope,
named with the same symbol by an abuse of notation:
F t−st−s−r : Add(Par
×,s
t−s)→ Add(Par
×,s+r
t−s−r)
This functor sends |t−s 7→ |t−s−r ⊕ ×t−s−r ⊕ · · ·×t−s−1 and ×t−i 7→ ×t−i for i =
1, . . . , s. It is easy to see that there is a natural isomorphism F t ∼= F t−st−s−r t−s−r ◦ F tt−s.
Consequently, Rtt−s◦Rt−s ∼ t tt−s−r = Rt−s−r. Later on, the functors R−1 will be of particular
interest for t ∈ Z≥0.
4.3 A filtration on restriction
This section studies Rtt−1 and its behavior on standard modules. Let ∆t :
Sym-Modlfd → Part-Modlfd denote the standardization functor (2.9.3). Fixing m ∈
N, the main goal of this section is understanding Rtt−1∆t−1(kSm) as a (Part, Sym)-
bimodule.
113
By (2.8.7), along with the fact that Par− has a k-basis comprised of normally
ordered upwards partition diagrams, it follows that ∆t−1(kSm) has basis indexed
indexed by pairs (f, σ) where σ ∈ Sm and f is a normally ordered upwards partition
diagram.
Lemma 4.3.1. Fix m ∈ N. Then Rtt−1∆t−1(kSm) has a basis over k indexed by pairs
(f, σ) where σ ∈ Sm and f ∈M1m is a phantom partition diagram whose underlying
partition diagram is normally ordered and upwards. Hence, as a free right Sm-module,
Rtt−1∆t−1(kSm) has basis consisting of phantom parition diagrams whose underlying
partition diagram is normally ordered and upwards.
Proof. Given w ∈ W n for some n ∈ N, let 1w be the idempoten∑t which is the
identity on the summand w of |. There is a decomposition 1n = w∈Wn 1w into
mutually orthogonal idempotents. From this, we recover the following decomposition
of Rtt−1∆t−1(kSm) making use of Lemma 4.2.3.
Rt ∼t−1∆t−1(kSm) = (M⊕⊗Part−1)∆t−1(kSm)
= ( 1nM ⊗Pa)rt−1 ∆t−1(kSm)⊕n ⊕
= 1wM ⊗Part−1 ∆t−1(kSm)
n∈N w∈Wn
∼⊕ ⊕ ( )= 1wM ⊗Part−1 ∆t−1(kSm)
⊕n∈N w⊕∈Wn ( )
= 1 M ⊗ ]w Part−1 Part−1 ⊗Par] infl kSm
⊕n∈N w⊕∈Wn∼ ( )= 1wM ⊗ ]Par] infl kSm
n∈N w∈Wn
114
Fixing some n ∈ N and w ∈ W n, let k(w) denote the number of letters | appearing
∼
in w. Then there is an isomorphism of right Part−1-modules γ : 1wM −→ 1k(w)Part−1
defined by erasing all phantom ×’s. Making use of the triangular decomposition
Part− ∼1 = Par− ⊗K Sym⊗ Par+K , we recover an isomorphism
1wM ⊗Par] infl] kS ∼m = 1 −k(w)Par ⊗K Par] ⊗Par] infl] kSm
∼= 1 −k(w)Par 1m ⊗ kSm.
Observing that 1k(w)Par
−1m ⊗ kSm⊕is no⊕nzero( if and only if k(w) ≥) m,
Rtt−1∆t−1(kS ∼m) = 1k(w)Par−1m ⊗ kSm
n∈N w∈Wn
k(w)≥m
In conclusion, for every word w with k(w) ≥ m, there is a nonzero summand
1 Par−k(w) 1m ⊗ kSm which has basis indexed by pairs (f, σ) where σ ∈ Sm and f
is a normally ordered upwards partition diagram. Since the isomorphism γ was just
given by erasing the phantom ×’s, the result follows.
The notation Bm will denote a chosen basis for ∆t(kSm) corresponding to pairs
(f, σ) with σ ∈ S −m and f ∈ Par 1m a normally ordered upwards partition diagram.
The basis element corresponding to some (f, σ) is just the composition fσ ∈ Bm.
Similarly, B×m will denote a chosen basis for R
t
t−1(kSm) corresponding to pairs (f, σ),
this time with f ∈ M1m a phantom partition diagram which is normally ordered
and upwards. Once again, fσ ∈ B×m is the basis element corresponding to some pair
(f, σ). Diagrammatically,
f
fσ = ···
σ
Notice that the bases Bm and B
×
m do not depend on t.
Remark 4.3.2. There is always an evident inclusion Bm → B×m given by sending
some fσ to the same diagram in B×m. There is also another map ζ : B
×
m → Bm, where
115
ζ(fσ) is the (phantomless) partition diagram obtained by erasing all phantoms; ζ just
picks out the underlying partition diagram of fσ.
Example 4.3.3. The weight space 15R
t
t−1∆t−1(kS ∼2) = 15M12⊗kS2 has the following
vector-space decomposition, ⊕
1 t5Rt−1∆t−1(kS ) ∼2 = 1k(w)∆t−1(kS2)
w∈W 5
k(w)≥2
A k-basis for the summand corresponding to the word |×||× is given by the following
set of diagrams, where σ runs over S2, for a total of 12 basis elements.
× ◦× ×◦ × ◦× × × × × × × ×
, , , , ,
σ σ σ σ σ σ
The proofs of the next few lemmas make use of several linear maps, the first
of which is defined here. These will only be homomorphisms over k, not as Part-
modules. There is a surjection ϕ : Rtt−1∆t−1(kSm)→ ∆t(kSm) defined as follows. Let
fσ ∈ B×m be a basis element corresponding to a pair (f, σ) as in Lemma 4.3.1. Thenfσ if fσ is phantomlessϕ(fσ) =  (4.3.1)0 otherwise
Proposition 4.3.4. For m ∈ N, there is an injection of Part-modules
ι
0→ ∆t(kSm)→− Rtt−1∆t−1(kSm)
Proof. Consider the Sym-modules 1m∆t(kSm) and 1 Rtm t−1∆t−1(kSm). Since the only
word w ∈ Wm with k(w) ≥ m is w = | · · · |, there is an isomorphism
1 Rt ∼ ∼m t−1∆t−1(kSm) = 1mM ⊗Part−1 ∆t−1(kSm) = 1m∆t−1(kSm)
Hence, as right Sym-modules, the spaces 1m∆t(kSm) and 1mRtt−1∆t−1(kSm) are
isomorphic. Additionally, it is an easy observation that any vector in these subspaces
is a highest weight vector. It follows that there is a homomorphism ∆t(kSm) →
Rtt−1∆t−1(kSm) which is an isomorphism on the 1m-weight spaces.
116
To see that ι is injective, consider a basis element fσ of ∆t(kSm). By the action
of Part through (4.1.15), it follows that
ι(fσ) = (fσ)× · ( |︸·︷·︷·∑ ︸
| )
m
= (fσ)[S ×] · ( |︸·︷·︷·︸| )
S⊆C(fσ)∑ m
= fσ + (fσ)[S ×] · ( |︸·︷·︷·︸| )
S⊆C(fσ)
S=6 m∅
Each term in the final summation apart from fσ contains a partition diagram with
phantoms. With (4.3.1), it now follows that ϕ ◦ ι = id∆t(kSm). So ι is injective.
Observe that, by Lemma 4.3.1, Rtt−1∆t−1(kSm) has a vector space filtration 0 =
N−1 ⊆ N0 ⊆ N1 ⊆ · · · ⊆ Rt〈t−1 where 〉
N ×i := k fσ ∈ Bm | f has at most i phantoms. (4.3.2)
There are projections p` : R
t
t−1∆t−1(kSm)→ N` defined on basis elements by p`(fσ) =
fσ if fσ ∈ N` and p`(fσ) = 0 otherwise. Letting N ` = N`/N`−1 and p` be the
composition of p` with this quotient, the next gr⊕aded decomposition is immediate.
Rtt−1∆t−1(kS ∼m) = N ` (4.3.3)
`∈N
Let Q := Rtt−1∆t−1(kS tm)/ι(∆t(kSm)) and also let π : Rt−1∆t−1(kSm) → Q be
the quotient map.
Lemma 4.3.5. The quotient Q has basis given by the images of fσ ∈ B×m where f
has at least one phantom. Denote this set by BQm.
Proof. By Lemma 4.3.1, the images of all fσ ∈ B×m certainly span the quotient.
Additionally, since p0 : R
t
t−1∆t−1(kSm) → N0 is defined by killing all diagrams with
phantoms, it follows that all those fσ ∈ B×m with π(fσ) ∈ BQm lie in the kernel of p0.
117
Also, the composition p0 ◦ ι : ∆t(kSm)→ N0 is an isomorphism as it sends a basis to
a bas⊕is. Since the k-span of th∑ose fσ where f has phantoms isomorphically projects
onto `≥1N `, it follows that `≥0 p` is an isomorphism.∑ ( )
p + p ⊕
k −−0−−−`≥−1−→`ι(∆t( Sm)) + ker(p0) ∼= N0 ⊕ N `
`≥1
By (4.3.3), we conclude there is an internal decomposition Rtt−1∆t−1(kSm) ∼=
ι(∆t(kSm))⊕ ker(p0). The result now follows.
Although Lemma 4.3.5 allows us to identify BQm with B
×
m, we choose to keep the
two distinct. However, whenever speaking of some π(fσ) ∈ BQm, we allow ourselves
to access the associated phantom partition diagram fσ ∈ B×m
Now enters the next linear map with the aid of Lemma 4.3.5. Given π(fσ) ∈ BQm,
define f̃ σ̃ ∈ Bm+1 to be the unique (phantomless) partition diagram obtained by
collecting all phantoms of fσ into a single tree, connected to the bottom left corner
of fσ. The new diagram has m + 1 components attached to the bottom row. This
assignment produces a linear map
ψ : Q→ ∆t(kSm+1), π(fσ) 7→ f̃ σ̃ (4.3.4)
Example 4.3.6. Consider the summand of 15R∆t−1(kS2) corresponding to the word
w = |× ||× from the previous example. Then ψ is defined on this subspace as follows.
There are decoratory gray lines above which is the normally ordered f̃ and below
which is σ̃. ( ( )) ( ( ))
× ◦×
◦ ×◦ × ◦
ψ π = , ψ π =
( ( σ )) σ◦ ( ( σ )) σ◦× × × ×
ψ π = , ψ π =
σ σ σ σ
118
( ( )) ( ( ))
× × × ×
ψ π = , ψ π =
σ σ σ σ
Lemma 4.3.7. For all n ∈ N, the dimensions of 1nQ and 1n∆t(kSm+1) over k are
equal.
Proof. Notice first that for π(fσ) ∈ BQm, it is true that π(fσ) ∈ 1nQ if and only if the
word along the top row of fσ is of length n. So, it follows that ψ is a weight-space
preserving linear map.
With the above consideration in mind, it is enough to provide a two-sided inverse
to ψ. Given fσ ∈ Bm+1, let f̂ σ̂ ∈ B×m be the unique phantom partition diagram
obtained by erasing the connected component of fσ attached to the bottom left
corner of fσ, replacing those erased boundary points along the top row by phantoms.
Now define ψ̂ : ∆t(kSm+1) → Q on a basis by fσ 7→ π(f̂ σ̂) for fσ ∈ Bm+1. It is
easily seen that for any π(fσ) ∈ BQm, (ψ̂ ◦ ψ)(π(fσ)) = π(fσ). Similarly, for any
fσ ∈ Bm+1, (ψ ◦ ψ̂)(fσ) = fσ.
Before getting into the next proposition, here is a summary of the previous
handful of results. By working with explicit bases, proposition 4.3.4 and Lemma 4.3.5
allow us to view ∆t(kSm) and Q as complimentary linear subspaces of Rtt−1∆t−1(kSm).
Then Lemma 4.3.7 shows that ψ is a weight-space preserving isomorphism
∆ (kS ) ∼t m+1 = Q by constructing an inverse ψ̂ which lifts through Rtt−1∆t−1(kSm),
making the following diagram commute.
119
The next proposition shows that ψ can be replaced by a proper isomorphism of Part
modules.
Theorem 4.3.8. For m ∈ N, there is a short exact sequence of (Part, Sym)-
bimodules
0→ ι π∆t(kS tm)→− Rt−1∆t−1(kSm)→− ∆tE(kSm)→ 0
Proof. Throughout this lemma, we remind ourselves that π(fσ) = fσ + im(ι) for
any π(fσ) ∈ BQm. Thanks to Lemma 4.3.7, it is enough to exhibit a surjection
∆t(kSm) → Q as Part-modules. First, it is easily seen that (4.3.4) restricts to an
isomorphism of left kSm+1-modules 1m+1Q ∼= 1m+1kSm+1.
i i
1 ψ : 1 Q→ kS , ··× ·· + im(ι) →7 ··m+1 m+1 m+1 ··
σ σ
Frobenius reciprocity now gives a homomorphism of Part-modules Ψ : ∆t(kSm+1)→
Q which restricts to an isomorphism between the 1m+1 weight spaces.
Before proving surjectivity of Ψ, here is some notation. Let f be a phantom
partition diagram with ` letters | along the bottom row. If we want to erase the
connected component attached to the ith letter | and replace the erased boundary
points by ×, we will denote this by placing a symbol × at the bottom of the ith letter
×. Illustratively, the diagram
f
· ·×· · (4.3.5)i
denotes the phandom partition diagram obtained from f by erasing the component
labeled ‘i’, and replacing all boundary points by ×.
The proof of surjectivity of Ψ is inductively for each weight space. Fix some
n ∈ Z>m and define subspaces 1nQ[p] ⊆ 1nQ for m ≤ p < n as below, recalling
120
Remark 4.3.2. 〈 〉
1nQ[p] := k fσ ∈ 1nQ ∩B×m | ζ(fσ) has at most p connected components
This provides a vector space filtration
0 ⊆ 1nQ[m] ⊆ 1nQ[m+ 1] ⊆ · · · ⊆ 1nQ[n− 1] = 1nQ.
Now enters the inductive argument. Consider first 1nQ[m]. For any basis element
fσ ∈ 1nQ[m] ∩ B×m, ζ(fσ) consists entirely of upwards trees and trunks. Recycling
the construction in (4.3.4) and using (4.1.15) shows that
Ψ(f̃ σ̃) = f̃∑Ψ(σ̃) ( )i
= f̃ [S×] ·Ψ ·· ··
S⊆∑C(f̃) ( σ )×i
= f̃ [S×] · ·· ·· + im(ι)
S⊆C(f̃) σ
f̃
= ··× + im(ι)i ··
σ
f
= ·· ·· + im(ι)
σ
= fσ + im(ι)
Suppose now that for some m < p ≤ n − 1, it is known that 1nQ[p − 1] ⊆ im(Ψ).
Choose one of our basis elements fσ ∈ 1 ×nQ[p] ∩ Bm. Without loss of genereality,
suppose ζ(fσ) contains exactly p connected components. Let C∪(f̃) ⊂ C(f̃) be the
set of components of f̃ which are branches or leaves. A similar calculation as the base
case shows
Ψ(f̃ σ̃) = f̃Ψ(σ̃)
121
∑ ( )×i
= f̃ [S×] · ·· ·· + im(ι)
S⊆C(f̃) σ∑ f̃ [S×]
= ··× ·· + im(ι)i
S⊂C∪(f̃) σ
f̃ [∅×] ∑ f̃ [S×]
= ··× ·· + ··× ·· + im(ι)i i
σ S⊂C∪(f̃) σ
∑S 6=∅f̃ f̃ [S×]
= ··× + × + im(ι)i ·· ·· i ··
σ S⊂C∪(f̃) σ
S
f ∑=6 ∅ f̃ [S×]
= ·· ·· + ··× + im(ι)i ··
σ S⊂C∪(f̃) σ
∑ S 6=∅f̃ [S×]
= fσ + ··× ·· + im(ι)i
S⊂C∪(f̃) σ
S 6=∅
Every term appearing in the final summation is a diagram with less than p connected
components. So the summation is in the image of Ψ by the inductive hypothesis and
hence there is some g ∈ ∆t(kSm+1) for which∑ f̃ [S×]
Ψ(g) = ··× ·· + im(ι)i
S⊂C∪(f̃) σ
S 6=∅
Finally, it follows that fσ + im(ι) = Ψ(f̃σ̃ − g). The case that p = n − 1 completes
the proof.
Corollary 4.3.9. There is a short exact sequence of functors from Sym -Modfd to
Part -Modlfd.
0→ ∆ tt → Rt−1 ◦∆t−1 → ∆t ◦ E → 0
122
Proof. The functors here are given by tensoring with the bimodules appearing in
Theorem 4.3.8.
Theorem 4.3.10. For λ ∈ P, there is a short exact sequence of (Part, Sym)-
bimodules ⊕ ( )
0→ ∆t(λ)→ Rtt−1∆t−1(λ)→ ∆t λ+ a → 0
a∈add(λ)
Proof. Immediate from Corollary 4.3.9 and the fact that the bimodules appearing in
Theorem 4.3.8 are flat right Sym-modules, seeing as Sym is semisimple.
Recall that a module N has an standard flag if there is a filtration 0 = N0 ⊂
N1 ⊂ · · ·N` = N with Ni/Ni− ∼1 = ∆(λi) for all i = 1, . . . , ` and some partitions
λ1, . . . , λ`.
Corollary 4.3.11. If N is a left Part−1-module with a standard flag, then R
t
t−1N is
a Part-module with a standard flag.
Proof. This follows from Lemma 4.2.1 and Theorem 4.3.10.
4.4 The Comes-Ostrik conjecture
We are now in a position to prove the Comes-Ostrik conjecture. Assume that
t ∈ Z>0. Here is a general (yet specific) lemma to the case at hand. Recall that an
essentially finite algebra is a locally unital algebra A so that each 1iA and A1i is
finite-dimensional.
Lemma 4.4.1. Let A and B be essentially finite algebras and suppose there is an
equivalence F : A-Modfd → B-Modfd. If there is an exact functor F̃ : A-Modfd →
B-Modfd so that F̃L ∼= FL for all irreducibles L ∈ A-Modfd, then F̃ is an equivalence
too.
123
Proof. We first note that F̃ is fully faithful. To see faithfulness, take any nonzero A-
module homomorphism f : V → W so that F̃ (f) = 0. Let L ⊆ W be an irreducible
submodule in the image of f and let V ′ = f−1(L). There is a surjection g : V ′  L
and an injection h : L ↪→ W . The restriction f |V ′ has a factorization fV ′ = h ◦ g.
Either F̃ (h) = 0 or F̃ (g) = 0. But since F̃ is exact and F̃L ∼= FL, this would be
impossible. So F̃ has to be faithful. Consequently, it is also full since Hom-spaces in
these two categories are of the same (finite) dimension due to F being an equivalence.
Now let PL be the projective cover of the irreducible module L ∈ A-Modfd.
Similarly, let PF̃L be the projective cover of F̃L ∈ B-Modfd. We claim that
F̃P ∼L = PF̃L. Since F̃ is fully faithful, Hom (F̃P , F̃L) ∼B L = HomA(PL, L) 6= 0. So
F̃PL is indecomposable (by exactness) and has F̃L ∼= FL as an irreducible quotient.
Consequently, there is a surjection PFL  F̃PL. But in the Grothendieck group
of B, F̃P and FP ∼L L = PFL are equal. So the surjection PFL  F̃PL must be an
isomorphism as both modules have the same dimension.
It now follows that F̃ is essentially surjective when restricted to the subcategories
of projective A- and B-modules. Hence it is an equivalence since it is fully faithful
too. Knowing that F̃ restricts to an equivalence A-Proj → B-Proj, we deduce the
full equivalence A-Mod→ B-Mod (see [BD17, Cor. 2.5]).
The rest of this section has a bit of bookkeeping, so here is some setup. Let
pr0,t denote the projection of Part-Modlfd onto the principal block containing the
irreducible module Lt(∅), as in (3.8.10). Also let (Part-Modlfd)0 be said principal
block. Taking the partition κt = (t), the indecomposable projectives in (Part-Modlfd)0
(n)
are given by Pt(n) := Pt(κt ) as described in Theorem 3.10.5 and ordered by
(0)
Corollary 3.10.7. So κt = ∅
(1) (2)
, κt = (t + 1), κt = (t + 1, 1), and so on. Similarly,
(n)
let ∆t(n) := ∆t(κt ). Also let L (n) = L (κ
(n)
t t ) be the nth irreducible module in the
124
principal block. Recall that ∆t(n) is indecomposable with two composition factors:
irreducible socle Lt(n+ 1) and irreducible head Lt(n) for n ≥ 0.
Lemma 4.4.2. For any n ∈ N,
(pr0,t ◦Rtt−1)(∆t−1(n)) ∼= ∆t(n).
Proof. This is a case analysis using the combinatorial rule provided by
(0)
Theorem 4.3.10. If n = 0 then κt−1 = ∅ and the ∆-factors of Rtt−1∆t−1(n) are
∆t(∅) and ∆t((1)). Since t > 0, only ∆t(∅) = ∆t(0) is in the principal block. So
(pr ◦Rt ∼0,t t−1)(∆t−1(0)) = ∆t(0).
(n)
If n = 1, then κ tt−1 = (t) and the ∆-factors of Rt−1∆t−1(1) are ∆t((t + 1)) and
∆t((t, 1)). Only ∆t((t+ 1)) = ∆t(1) is in the principal block.
(n)
If n ≥ 1, then κ = (t, 1n−1t−1 ) and the ∆-factors of Rtt−1∆t−1(n) are ∆t((t +
1, 1n−1)), ∆t((t, 1
n−1, 1)), and ∆t((t, 1
n)). Only ∆t((t + 1, 1
n−1)) = ∆t(n) is in the
principal block.
Lemma 4.4.3. For any n ∈ N,
(pr t ∼0,t ◦Rt−1)(Lt−1(n)) = Lt(n).
Proof. Suppose for the sake of contradiction that (pr0,t ◦Rt ∼t−1)(Lt−1(n)) 6= Lt(n) for
some n ∈ N. Let ` ∈ N be minimal so that (pr t ∼0,t ◦Rt−1)(Lt−1(`)) 6= Lt(`). From
Lemma 4.2.1 and Lemma 4.4.2, there is an exact sequence
0→ (pr t0,t ◦Rt−1)(Lt−1(`+ 1))→ ∆t(`)→ (pr0,t ◦Rtt−1)(Lt−1(`))→ 0.
Since ∆t(`) has length 2, either (pr0,t ◦Rtt−1)(Lt−1(`)) = 0 or (pr ◦Rt0,t t−1)(L ∼t−1(`)) =
∆t(`). In the first case, it follows that (pr0,t ◦Rtt−1)(L ∼t−1(` + 1)) = ∆t(`). But there
is also an exact sequence
0→ (pr ◦Rt t0,t t−1)(Lt−1(`+ 2))→ ∆t(`+ 1)→ (pr0,t ◦Rt−1)(Lt−1(`+ 1))→ 0.
125
These last two observations show there is a surjection ∆t(`+ 1)→ ∆t(`), impossible.
In the case where (pr t0,t ◦Rt−1)(Lt−1(`)) ∼= ∆t(`), one has (pr ◦Rt0,t t−1)(Lt−1(` +
1)) = 0. Repeating the above argument would give a surjection ∆t(`+2)→ ∆t(`+1),
another contradiction. So it must be that (pr0,t ◦Rt ∼t−1)(Lt−1(n)) = Lt(n) for all n ∈
N.
Theorem 4.4.4. For t ≥ 1, the composition
pr ◦Rt0,t t−1 : (Part−1-Modlfd)0 → (Part-Modlfd)0
is an equivalence.
∼
Proof. By Theorem 3.10.5, there is an equivalence (Part−1-Modlfd)0 −→ (Part-Modlfd)0
and both these categories are essentially finite, in that they are equivalent to
categories of finite-dimensional modules over essentially-finite algebras (see [BS,
Cor. 2.20]). Now apply Lemma 4.4.3 and Lemma 4.4.1.
126
CHAPTER V
THE ABELIAN ENVELOPE
This chapter provides an alternate perspective of the abelian envelope of Rep(St)
involving tilting modules and Ringel duality. After briefly reviewing some basics
about abelian envelopes and splitting objects, we use the restriction functor Rt−1
from chapter IV to show that tilting modules for the partition category can be
identified with splitting objects. This allows us to connect the Benson-Etingof-Ostrik
construction of abelian envelopes in [BEO23] to tilting theory and Ringel duality
studied in [BS].
5.1 Review of abelian envelopes
Throughout this section let C be a locally-finite Karoubian rigid monoidal
category with EndC (1) ∼= k. Generally, C is not abelian and so is not a full-fledged
tensor category in the sense of [EGNO15]. However, one can ask to find an abelian
envelope of C if it exists. Such an abelian envelope is the data of a tensor category
D and a monoidal functor F : C → D so that for any other tensor category D ′,
composition with F induces an equivalence between the category of faithful monoidal
functors C → D ′ and the category of exact monoidal functors D → D ′. That is, for
each G : C → D ′ there exists a unique functor (up to isomorphism) G ′ : D → D ′
making the following diagram commute up to isomorphism:
Any two abelian envelopes of C are equivalent, so we often speak of the abelian
envelope of C .
127
Remark 5.1.1. In the case of Deligne’s category Rep(St), Comes and Ostrik were the
first to construct the abelian envelope by examining the heart of a certain t-structure
on the homotopy category of Rep(St) [CO14]. Later, the abelian envelope of Rep(GLt)
was constructed by Entova, Hinich, and Serganova by clever use of inverse and direct
limits of representations of the general linear supergroup GL(m|n) with m − n = t
[EHS18]. More recently, Harman and Snowden build abelian envelopes for a class of
Oligomorphic groups as a kind of completed group algebra [HS22].
The Benson-Etingof-Ostrik construction (and also Coulembier’s approach in
[Cou21]) of abelian envelopes involves the use of splitting objects. With C as above,
an object S ∈ O(C ) is splitting if for each morphism f : X → Y in C , the morphism
f⊗ idS : X⊗S → Y ⊗S is split. That is, f⊗ idS is the direct sum of a zero morphism
and an isomorphism.
It is proven in [BEO23] that the splitting objects of C form a thick tensor ideal;
the full subcategory consisting of splitting objects of C is Karoubian and closed
under taking tensor products with arbitrary objects of C . Let (Si)i∈I be a family of
irredundant representatives for the isomorphism classes of indecomposable splitting
objects and suppose the following finiteness property : for any splitting object S, there
are finitely many i ∈ I for which HomC (Si, S) is nonzero.⊕Under this assumption,
Benson, Etingof, and Ostrik build the coalgebra C := i,j∈I HomC (Si, Sj)
∗ and
a functor F : C → C-Comodfd from C to the category of finite-dimensional C-
comodules.
After proving that C-Comodfd has a monoidal structure, Benson, Etingof, and
Ostrik show that under certain conditions, C-Comodfd is the abelian envelope of
C . The first condition is that C must be of finite type: there exists a splitting
object S ∈ O(C ) so that every indecomposable splitting object Si (i ∈ I) appears
128
as a summand of X ⊗ S for some X ∈ O(C ). The second condition is that C is
separated, meaning that F is faithful. Lastly, C must be complete, meaning that F(C )
is equivalent to Kar(F(C )). Their main theorem on abelian envelopes is summarized
below. More details can be found in [BEO23].
Theorem 5.1.2 (Benson-Etingof-Ostrik). Let C be a monoidal category with the
finiteness property (as well as the other properties listed at the beginning of this
section). Also suppose C is of finite type. Then C admits a fully faithful monoidal
functor E : C → D into a (multi-)tensor category D with enough projectives if and
only if C is separated and complete. Moreover, in this case there exists a tensor
embedding E ′ : C-Comod → D so that E ∼= E ′fd ◦ F and C-Comodfd is the abelian
envelope of C .
5.2 Ringel duality and the abelian envelope
Since many of the projective modules for Part are self-dual (Theorem 3.10.5),
they have a standard flag and (finite) costandard flag. Hence, they are tilting
modules1. A classification of indecomposable tilting modules up to isomorphism is
provided by Brundan and Stroppel [BS, Thm.4.18]. In the setting for Part-Modlfd,
the classifcation states that there is exactly one indecomposable tilting module T (λ)
for each partition λ ∈ P , up to isomorphism. Specifically, T (λ) is characterized
uniquely by the property that it has ∆(λ) as the bottom section in any standard flag.
Recalling the definition (3.8.12), the tilting module for Part-Modlfd corresponding to
the partition λ ∈ P is:P (κ(n+1)t ) λ = κ(n) for some κ ∈ Pt and n ∈ N
Tt(λ) =  (5.2.1)Pt(λ) otherwise.
1In a general upper-finite highest weight category, tilting modules potentially have infinite
‘descending’ costandard flags.
129
Note that whenever λ is not of the form κ(n) for some κ ∈ Pt and n ∈ N, Pt(λ) = L(λ)
is irreducible.
Lemma 5.2.1. If T ∈ Par tt−1-Modlfd is tilting, then Rt−1T is tilting too.
Proof. This follows from (4.2.2) and Corollary 4.3.11, noting that standard flags turn
into costandard flags upon applying Chevalley duality.
In the next lemma, we make use of the restriction functorsRt−1 : Par−1 -Modlfd →
Part -Modlfd as in Remark 4.2.4. It is an easy consequence of the fact that F
t
t−1 is
monoidal that
(| ⊕ ×) ? F t ∼ tt−1(−) = Ft−1 ◦ (| ?−).
More generally,
(| ⊕ ×−1 ⊕ · · · ⊕ ×t−1) ? F t−1(−) ∼= F t−1 ◦ (| ?−).
Bringing back the functor Dt : Part-Mod→ Part-Mod of (3.9.3), it follows that there
is a natural isomorphism
Dt ◦Rt ∼ t−1 = R−1 ◦ (D−1 ⊕ ︸Id⊕ ·︷·︷· ⊕ Id︸), (5.2.2)
t+1
where Id denotes the identity functor on Part−1-Mod. Letting D
n
t = ︸Dt ◦ ·︷·︷· ◦D︸t,
( ) nwe have ⊕n
Dn ◦Rt ∼ t
n
◦ ` ⊕( )(t+1)(n−`)`t t−1 = R−1 (D−1) (5.2.3)
`=0
Lemma 5.2.2. An indecomposable X ∈ Part-Proj is splitting if and only if it is
tilting.
Proof. We start by showing that every tilting is splitting. Since every projective
appears as a summand of a direct sum of finitely many Qt(n) := Part1n, it is enough
to check that tiltings split morphisms f : Qt(n) → Qt(m) for any n,m ∈ N. By
130
the combinatorial rule provided in Theorem 4.3.10, the module V := Rt−1∆−1(∅) has
a section ∆(∅). Since V is tilting by Lemma 5.2.1, it follows from (5.2.1) that V
contains the tilting module Tt(∅) as a summand. Consider the following commuting
diagram, using Lemma 2.6.1:
The right-most vertical arrow, given by the natural isomorphism (5.2.3), is split as
Par−1-Mod is semisimple. Hence, it follows that V is a splitting object and so is
Tt(∅), being a summand of a splitting object. Moreover, since splitting objects form
a thick tensor ideal, all Tt(λ) for λ ∈ P are splitting since they appear as a summand
of DmV = Q (m) ?ft t V for a suitable m ∈ N.
It remains to see that those indecomposable projectives which are not tilting are
also not splitting. First consider Pt(∅) = ∆(∅). This is the unit with respect to ?f.
Since there is a non-split map f : P (∅)→ P (∅(1)t t ), it is immediate that 1Pt(∅) ?ff = f
is not split. So Pt(∅) is not a splitting object. Consider now Pt(κ(0)) for any κ ∈ Pt
and look at the morphism 1 fP (κ(0)) ?f : Pt(κ(0)) ?fP (∅) → P (κ(0)) ?ft t P (1)t t(∅ ). By
Lemma 3.10.1, Dmt Pt(κ
(0)) = Qt(m) ?fPt(κ(0)) will contain a summand of Pt(∅) for an
appropriate m and we get a commuting diagram below, where the first two vertical
arrows are non-split. Consequently, Pt(κ
(0)) cannot be a splitting object.
131
Lemma 5.2.3. The category Part-Proj has the finiteness property.
Proof. By Lemma 5.2.2, any splitting object S if a finite direct sum of indecomposable
tilting modules. It follows from Theorem 3.10.5 that there are only finitely many Tt(λ)
with HomPart(T (λ), S) being nonzero.
( ⊕⊕ Consider now T := ) opλ∈P Tt(λ) and the algebra B := EndPart(T ) =op
λ,µ∈P HomPart(Tt(λ), Tt(µ) . In the language of [BS], T is a tilting generator.
Equipping B with the profinite topology, Brundan and Stroppel build the coalgebra
Coend(T ) := {f ∈ B∗ | f vanishes on some two-sided ideal of finite codimension}.
There is an identification Comodfd- Coend(T ) = B-Modfd. Brundan and Stroppel also
construct the Ringel duality functor G : Part-Mod → Comodfd- Coend(T ), defined
on any N ∈ Part-M{od below. }
G(N) := f ∈ Hom (N, T )∗ | f vanishes on a submodulePart of HomPart (N,T ) with finite codimension
We can now show that B-Modfd, equipped with the functor G, is the abelian envelope
of Part, and hence, also of Rep(St).
Theorem 5.2.4. The Ringel dual B-Modfd of Part-Modlfd is the abelian envelope of
Rep(St).
Proof. As usual, identify Rep(St) with Part-Proj by means of the Yoneda equivalence.
We just need to check the conditions provided in Theorem 5.1.2. The first is that
Rep(St) is of finite type. We claim the module Tt(∅) is a generator for the splitting
ideal. From Lemma 3.10.1 and Lemma 3.10.3, any splitting (=tilting) object in a
nontrivial block of Part-Modlfd is a summand of D
nTt(∅) ∼= Qt(n) ?fTt(∅) for some
n ∈ N. For those tiltings Tt(λ) in a trivial block, the same is true and is easily
deduced from Theorem 3.9.1.
132
The other conditions we have to check is that Rep(St) is both separated and
complete. Let G : Part-Modlfd → B-Modfd be the Ringel duality functor and let
T = G(Rep(St)) be the image of this functor. Completeness amounts to showing
that T is equivalent to its Karoubi envelope, and separatendess requires us to show
that F is faithful. This, however, is true by [BS, Thm.4.27]. So Rep(St) is separated
and complete.
133
REFERENCES CITED
[Bru17] J. Brundan, Representations of the oriented skein category;
arXiv:1712.08953v1, 2017.
[Bru18] J. Brundan, On the definition of Heisenberg category, Algebr. Comb. 1
(2018), no. 4, pp. 523–544. MR4195648
[BD17] J. Brundan and N. Davidson, Categorical actions and crystals, Contemp.
Math. 683 (2017), pp. 105–147. MR3611712
[BEO23] D. Benson, P. Etingof, V. Ostrik New incompressible symmetric tensor
categories in positive characteristic, Duke Math. J. 172 (2023), no. 1, pp.
105–200. MR4533718
[BEEO20] J. Brundan, I. Entova-Aizenbud, P. Etingof and V. Ostrik,
Semisimplification of the category of tilting modules for GLn, Adv. Math. 375
(2020), 107331, 37 pp.. MR4135417
[BSW20] J. Brundan, A. Savage and B. Webster, Heisenberg and Kac-Moody
categorification, Selecta Math. 26 (2020), no. 5, paper no. 74, 62 pp..
MR4169511
[BS] J. Brundan and C. Stroppel, Semi-infinite highest weight categories, to appear
in Mem. Amer. Math. Soc..
[BV22] J. Brundan and M. Vargas, A new approach to the representation theory of
the partition category, J. Algebra 601 (2022), pp. 198–279. MR4398411
[Com20] J. Comes, Jellyfish partition categories, Alg. Rep. Theory 23 (2020), no. 2,
pp. 327–347. MR4097319
[CO11] J. Comes and V. Ostrik, On blocks of Deligne’s category Rep(St), Adv.
Math. 226 (2011), no. 2, pp. 1331–1377. MR2737787
[CO14] J. Comes and V. Ostrik, On Deligne’s category Repab(Sd), Alg. & Num.
Theory 8 (2014), no. 2, pp. 473–496. MR3212864
[Cou21] K. Coulembier, Monoidal Abelian envelopes, Compositio Mathematica 157
(2021), no. 7, pp. 1584 - 1609. MR4277111
[CEOP23] K. Coulembier, P. Etingof, V. Ostrik, B. Pauwels, Monoidal abelian
envelopes with a quotient property, J. Reine Angew. Math. 794 (2023), pp.
179-214. MR4529414
134
[Cre21] S. Creedon, The center of the partition algebra, J. Algebra 570 (2021), pp.
215–266. MR4186641
[CD23] S. Creedon, M. De Visscher, Defining an affine partition algebra, Algebr.
Represent. Theor. (2023).
[Del07] P. Deligne, “La catégorie des représentations du groupe symétrique St,
lorsque t n’est pas un entier naturel”, in: Algebraic Groups and Homogeneous
Spaces, Tata Inst. Fund. Res. Stud. Math. (2007), pp. 209–273. MR2348906
[EL16] B. Elias and A. Lauda, Trace decategorification of the Hecke category, J.
Algebra 449 (2016), pp. 615–634. MR3448186
[EA16] I. Entova-Aizenbud, Deligne categories and reduced Kronecker coefficients, J.
Algebr. Comb. 44 (2016), no. 2, pp. 345-–362. MR3533558
[EHS18] I. Entova-Aizenbud, V. Hinich, V. Serganova, Deligne categories and the
limit of categories Rep(GL(m|n)), Int. Math. Research Notices 15 (2018), no.
15, pp. 4602–4666. MR4130848
[Eny13] J. Enyang, Jucys–Murphy elements and a presentation for partition
algebras, J. Algebr. Comb. 37 (2013), no. 3, pp. 401–454. MR3035512
[Eny13’] J. Enyang, A seminormal form for partition algebras, J. Combin. Theory
Ser. A 120 (2013), no. 7, pp. 1737–1785. MR3092697
[EGNO15] P. Etingof, S. Gelaki, D. Niksych, V. Ostrik, Tensor categories, Math.
Surveys Monogr., vol. 205 (2015), AMS. MR3242743
[HR05] T. Halverson and A. Ram, Partition algebras, European J. Combin. 26
(2005), no. 6, pp. 869–921. MR2143201
[HS22] N. Harman and A. Snowden, Oligomorphic groups and tensor categories;
arXiv:2204.04526 2022.
[Kho14] M. Khovanov, Heisenberg algebra and a graphical calculus, Fund. Math. 225
(2014), no. 1, pp. 169–210. MR3205569
[KS15] M. Khovanov and R. Sazdanovic, Categorifications of the polynomial ring,
Fund. Math. 230 (2015), no. 3, pp. 251–280. MR3351473
[LSR21] S. Likeng and A. Savage (appendix with C. Ryba), Embedding Deligne’s
category Rep(St) in the Heisenberg category, Quantum Topol. 12 (2021), no. 2,
pp. 211–242. MR4261657
[Lit56] D. Littlewood, The Kronecker product of symmetric group representations, J.
Lond. Math. Soc. 31 (1956), pp. 89–93. MR0074421
135
[Mar91] P. Martin, Potts Models and Related Problems in Statistical Mechanics,
Series on Advances in Statistical Mechanics, vol. 5, World Scientific Publishing,
Teaneck, NJ, 1991. MR1103994
[Mar96] P. Martin, The structure of the partition algebras, J. Algebra 183 (1996),
no. 2, pp. 319–358. MR1399030
[Mar00] P. Martin, The partition algebra and the Potts model transfer matrix
spectrum in high dimensions, J. Phys. A 33 (2000), no. 19, pp. 3669–3695.
MR1768036
[MS22] Y. Moussaaid and A. Savage, Categorification of the elliptic Hall algebra,
Doc. Math. 27 (2022), pp. 1225—1273. MR4466984
[OV05] A. Okounkov and A. M. Vershik, A new approach to the representation
theory of the symmetric groups, J. Math. Sci. 131 (2005), pp. 5471–5494.
[SS15] S. Sam and A. Snowden, Stability patterns in representation theory, Forum
Math. Sigma 3 (2015), paper no. 11, 108 pp.. MR3376738
[SS22] S. Sam and A. Snowden, The representation theory of Brauer categories I:
triangular categories, Appl. Categ. Structures 30 (2022), no. 6, pp. 1203–1256.
MR4519632
136