Deligne categories and representation stability in positive characteristic

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Deligne Categories and Representation Stability

in Positive Characteristic

by

Nate Harman

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

AUG

01 2017

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June 2017

@

Massachusetts Institute of Technology 2017.

Author

All rights reserved.

Signature redacted

Department of Mathematics

May 2, 2017

Certified by...

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Pavel Etingof

Professor of Mathematics

Thesis Supervisor

Accepted by .

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William Minicozzi

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Deligne Categories and Representation Stability

in Positive Characteristic

by

Nate Harman

Submitted to the Department of Mathematics on May 2, 2017, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

We study the asymptotic behavior of the representation theory of symmetric groups S, in positive characteristic as n grows to oc, with the goal of understanding and generalizing the Deligne categories Rep(St) as well as the theory of FI-modules and representation stability in the positive characteristic setting. We also give q-analogs of some of our results in the context of unipotent representations of finite general linear groups in non-defining characteristic.

Thesis Supervisor: Pavel Etingof Title: Professor of Mathematics

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Acknowledgments

First of all I'd like to thank my advisor Pavel Etingof for all his guidance and support these past few years, and for helping me grow as a mathematician. I'd also like to thank all of my math teachers and mentors over the years for cultivating my love of mathematics, in particular I'd like to thank Allison Kyte, P.J. Karafiol, sarah-marie belcastro, David Kelly, Frank Morgan, and Jenia Tevelev. Next, I'd like to thank the MIT math department staff for all that they do to make the math department welcoming and easy to navigate for graduate students. Finally, I'd like to thank my friends and family for their love and support over the years, without them

I never could have done this.

This work was supported in part by the NSF Graduate Research Fellowship, NSF

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Chapter 1 Introduction and setting

Many of the groups and algebras of classical interest in representation theory live in natural families indexed by a natural number n. For example there are general linear groups GL(n), orthogonal groups O(n), unitary groups U(n), symmetric groups

S., the list goes on.

Within each of these families something remarkable happens. If we look at their complex representations as n tends to infinity, then many aspects of the representa-tion theory stabilize and become independent of n. We will refer to such behavior as stability phenomena in representation theory. Let's start with a couple classical examples of this behavior:

Example: Representations of GL,(C)

Consider the polynomial representation theory of GL, (C). Irreducible polynomial representations GL,(C) are naturally indexed by partitions A with at most n parts. The corresponding irreducible V(A) is given by applying the Schur functor SA to the defining representation V = Cn of GL,(C).

A classical example of a stability phenomenon is given by Schur-Weyl duality

which tells us that for all n > k

Vok

efV(A)

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where the multiplicities fA are independent of n (in fact they count standard Young tableaux of shape A). More generally, stability of Schur functors says that if A and it are fixed partitions of size j and k respectively then for all n > jk

SA(V(pt)) - (@p,,V(V)

zA-jk

where the multiplicities p",, (called plethysm coefficients) are independent of n. In this case though, we know these plethysm coefficients stabilize one n is large enough but they are otherwise not very well understood, and there is no known combinatorial formula for them.

Example: Representations of S,

Next let's look at the case of symmetric groups, which will be the setting for most of our results. The irreducible complex representations of S, are usually labeled by partitions A of size n, with SA denoting the irreducible Specht module.

In the setting of stability phenomena it is convenient to reindex slightly by for-getting the first part of the partition. If A = (A1, A2, ... , Ak) is an arbitrary

parti-tion, and n is an integer such that n - JAl > A1 define the padded partition A[n] :=

(n - JA|, A1, A2... , Ak) to be the partition of size n obtained by adding a single part

to A to make it have size n.

Now the irreducible representations of S, are given by Specht modules SA'l" where

A is a partition such that n- AI > A1. If we let V = C" be the standard representation

of S, permuting the coordinates, then analogously to the result for general linear groups for n >> 0

V*k @

0

gkSA[]

IAI<k

with the multiplicities gA,k independent of n with an explicit combinatorial interpre-tation. Another classical example in this setting is a result of Murnaghan from the late 30's [36] that says that for any three partitions A, p, v there are coefficients , independent of n such that

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gAIV = dim(S"[n] Spn] 0 Sv[nI)s.

for all sufficiently large n. The space of such invariants for any fixed n is not well understood, but nevertheless we know they stabilize as n gets large. These coefficients are called the reduced Kronecker coeffcients and equivalently count the multiplicity

of Sv[n] in SA[n] 0 SpI.

Modern incarnations of stability

While numerous instances of stability phenomena in representation theory have appeared in the literature for decades, there has very recently been an increased interest in this sort of behavior for two primary reasons:

1. In 2007 Deligne defined symmetric tensor categories Rep(GLt), Rep(Ot), and

Rep(St), for complex numbers t. These categories are satisfy certain interesting universal properties, and their behavior is closely related to the stable behavior of representations of general linear, orthogonal, and symmetric group represen-tations.

2. In 2010 Church and Farb introduced the notion of representation stability (in

[5]), as a generalization of homological stability from topology. They gave a

number of examples arising in geometry and topology of sequences of vector spaces, where the nth vector space Vn carries an action of Sn, such that the multiplicity of SA[n] in V, is eventually independent of n.

These will serve as the setting for this work, and we will discuss them both in more detail now. We'll note that in both cases, the theory outside of the semisimple characteristic zero setting was largely conjectural or all together undeveloped. One of our primary goals will be understanding what happens to the stability phenomena for symmetric groups when we pass to the modular setting, and what that tells us about these theories.

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1.1 Deligne categories

Deligne defined remarkable categories Rep(GLt), and Rep(St) for t E C that are closely related to this stabilization phenomena for complex representations of general linear and symmetric groups (he also defined analogous categories for orthogonal and symplectic groups, but we won't discuss those). Understanding and generalizing these categories is one of the primary motivations of this work, we will briefly review the theory of these categories now.

1.1.1 Deligne's setting: rigid symmetric tensor categories

First let's discuss the setting in which Deligne was working when he defined these categories, we will be closely following the exposition of Ostrik in [381 (also see [101). Recall that a symmetric monoidal category is a category C equipped with a functor

0 : C x C -+ C, associativity and commutativity isomorphisms, and a unit object 1

satisfying certain axioms (See [201 for more details).

Suppose k is an algebraically closed1 field. A rigid symmetric tensor category over

k is a symmetric monoidal category C such that:

1. C is essentially small and enriched over finite dimensional vector spaces over k.

2. The functor 0 is k-linear and exact in each variable.

3. The natural map k -+ Endc(1) is an isomorphism.

4. Every object X E C is rigid meaning there is a dual object X* E C with evaluation map X* 0 X -+ 1 and a coevaluation map 1 -+ X 0 X* satisfying

certain axioms (again, see [20] for more details).

Sometimes we may drop the fourth condition and just talk about symmetric tensor categories. If, in addition to these four conditions, C is an abelian category where all objects have finite length we say that C is a pre-Tannakian category.

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Example: The category of finite dimensional vector spaces over k with the usual

tensor product and symmetric structure given by v 0 w -+ w 0 v is a pre-tannakian

category. We'll note that coevaluation map 1 -* VOV* can be viewed as the inclusion

of the scalar matrices inside End(V) e V 0V*. In particular, there would be no such natural map if we tried to take V to be infinite dimensional.

Example: The next example to keep in mind is C = Repk(G), the category of finite dimensional representations of a group G (or really, an affine group scheme over k). Here the tensor product and symmetric structure are inherited from category of vector spaces, with G acting diagonally on the tensor factors. The unit object is just the trivial representation. If V is a representation of G, then V* naturally carries a G

action such that the evaluation and coevaluation maps are G-equivariant.

These examples have natural analogs in the setting of super vector spaces and algebraic supergroups, and in some sense these these examples are getting at why one might study pre-Tannakian categories. The definition of pre-Tannakian categories is meant to be a purely categorical formulation of the type of categories that can arise as categories of finite dimensional representations of groups.

Indeed, with the additional data of a "forgetful" functor to finite dimensional (su-per) vector spaces preserving the symmetric tensor structure, Tannakian reconstruc-tion says that C is equivalent to such a category of finite dimensional representareconstruc-tions of a (super) group.

A major part of Deligne's motivation in defining Rep(GLt) and Rep(St) was to

explore in some sense the landscape of rigid symmetric tensor categories and pre-Tannakian categories which are not coming from representations of some (super) group.

Commutative Frobenius algebras: A commutative Frobenius algebra: in

sym-metric monoidal category is an object X along with morphisms

P:XOX -X r7:1-+X 6:X-+XOX 6:X-+1

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we will just give a motivating example.

Suppose G is a finite group acting on a finite set S. k[S], the space of formal k-linear combinations of S with the natural action of G, has a natural structure of a commutative Frobenius algebra defined as follows:

o If a, b are elements of S then [ sends a 0 b to a if a = b, and to 0 otherwise,

extended bilinearly.

o 1 sends 1 E k - 1 (the trivial representation) to the sum of all elements of S. o J sends an element a

E

S to a9a, and extended linearly to formal combinations.

o E takes a formal sum EaES caa to EaES Ca, the sum of its coefficients.

As in this example, a commutative Frobenius algebra in a symmetric tensor category over k is necessarily self dual (and hence rigid) with evaluation and coevaluation given

by e o [ and 6 o rj respectively.

Categorical trace and dimension: One thing that the structure of a rigid symmet-ric tensor category gives us is an internal categosymmet-rical notion of the trace of an

endomor-phism. Explicitly, if

f

E Endc(X) we define the categorical trace Tr(f) C k as follows.

First take the coevaluation map 1 -9 X 0 X*, compose it with the map

f

0 Idx. from X 0 X* to itself, then compose with the symmetry map X 0 X* -+ X* 0 X to

switch the factors, and finally compose it with the evaluation map X* 0 X -+ 1 to

obtain an element of End(1) a k.

If C is the category of finite dimensional vector spaces over k this just recovers

the usual notion of the trace of a linear map V -+ V. Moreover, this is a categorical

invariant in the sense that the categorical trace is preserved by symmetric tensor functors between symmetric tensor categories over k. An important case is if we take

f

to be the identity endomorphism of X, then the trace gives us the categorical

dimension of X.

Remark: It is important to note that the categorical dimension is valued in k, and in particular if k is of characteristic p then the categorical dimension of a vector

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space is the reduction mod p of the usual dimension. With Etingof and Ostrik in [21] we defined two modified versions of the categorical dimension for pre-Tannakian categories in characteristic p that take values in the p-adic integers Zp rather than the ground field, which recover the usual dimension of a vector space.

1.1.2 Deligne categories and their properties

Defining the Deligne categories: First, a short definition that is a moral truth

but technical lie: if t E C then Rep (GLt) is the symmetric tensor category over C freely generated by a rigid object X of categorical dimension t, similarly Repo(St) is the symmetric tensor category over C freely generated by a commutative Frobenius algebra X of categorical dimension t. The Deligne categories Rep(GLt) and Rep(St) are defined to be the Karoubian envelopes2 of these categories.

This definition is very concise and makes it clear that functors out of these cate-gories should satisfy certain universal properties (which we will formulate soon). It's somewhat unclear though what exactly it means to freely generate a category, if such categories should even exist, and to what extent they should be unique. Deligne showed that indeed we can make sense of such freely generated categories, at least up to equivalence of symmetric tensor categories, and he gave concrete diagrammatic descriptions for them.

Such diagrammatic descriptions can be found in [111 and

[7].

Ultimately though these diagrammatic descriptions won't be useful for our purposes, so we won't go into them in detail but we'll briefly sketch the construction for Rep (St).

9 Objects are formal symbols [X®O] for nonnegative integers n.

* Morphisms from [X®n] to [X*m] are given by linear combinations of set parti-tions of the set [n, m] = {1, 2, ... , n} U 1', 2 ... , m'}.

2

Taking the Karoubian envelope of a k linear category C is a mild categorical construction where one first formally adjoins images of idempotents in endomorphism algebras in C, and then adds in all finite direct sums of such objects. If C is a rigid symmetric tensor category, its Karoubian envelope naturally inherits the same structure.

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" Given a set partition on [n + m] and a set partition on [m + i] this induces a set partition on [n, m, = {1, 2, *. , n} U {1', 2',. . ., m'} U {1", 2", ... ."} by

identifying the last m elements of [n + m] with the first m elements of [m + 1]. Composition is given by taking the induced partition on [n + f] (forgetting the middle m points and relabeling) and weighting it by a factor of t', where k is the number of parts of the set partition on [n + m + f] completely contained in the middle m points.

" The monoidal structure is given by [Xn] 0 [X®"] - [Xon+",) on objects, and

sends set partitions on [n + m] and [n'+ m'] to the natural set partition on their disjoint union, which we then identify with [(n + n') + (m + m')].

"

The symmetric braiding [X®n] 0 [X®"n] _> [X m] 0 [XO'] is given by the set

partition of [n + m, n

+

m] into sets of size two pairing a with (a + m)' if a < n,

and with (a - n)' if a > n.

In particular we'll note that endomorphisms of [X*n] are given by the partition algebra Parn(t) at the parameter t. These are a family of algebras that are known to be Schur-Weyl dual to symmetric groups in an appropriate sense. The category Rep (St) is in some sense just a repackaging of these algebras into a single symmetric monoidal category.

Universal properties: As one would expect from something freely generated, these

categories satisfy certain universal properties among k-linear symmetric tensor cate-gories. Explicitly:

Proposition 1.1.1 ([111 props. 8.3 and 10.3) Let C be a symmetric tensor cat-egory over k. Then for every rigid object (resp. Frobenius algebra) V in C of cate-gorical dimension t there is a unique (up to natural isomorphism) symmetric

functor

Fv : Rep₀(GLt) -4 C (resp. Repo(St) -- C) sending X -- V.

Hence these categories are universal among k-linear symmetric tensor categories for having an object or Frobenius algebra of categorical dimension t. If we additionally

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assume that the target category C is idempotent complete and additive, then the categories Rep(GLt) and Rep(St) satisfy the same universal property.

Interpolation: If t = n E N then the universal property of Deligne categories ensures that we have a symmetric tensor functor from Rep(GL,) to Rep(GLn), the category of finite dimensional algebraic representations of GL"(C), sending X to V, the defining n-dimensional representation of GL, (C). Similarly, we get a functor from Rep(Sn) to Rep(Sn) sending X to the standard representation of S,., on C" permuting the coordinates.

One might hope that these maps are equivalences of categories, but this is not the case. It turns out that these functors are essentially surjective and full, but they are not faithful and send some nonzero objects to zero. For example, if we look at the defining object X of dimension n then X has infinitely many nonzero exterior powers in Rep(GLn) (or Rep(Sn)), and all but the first n of them get killed by this functor. In fact we can describe the kernel of this map (i.e. the morphisms that get sent to zero) explicitly. We say that a morphism

f

E Homc(A, B) in a symmetric tensor

category C is negligible if Tr(g o

f)

= 0 for all g C Homc(B, A). Such morphisms form a tensor ideal of C in the sense that if you compose a negligible morphism with another morphism you get a negligible morphism and if you take a tensor product a negligible morphism with another morphism you get a negligible morphism. In particular there is a natural way to quotient C by its ideal of negligible morphisms to obtain a symmetric tensor category C", the semisimplification of C.

Proposition 1.1.2 ([111 thms 6.2 and 10.4) A morphism in Rep(GLn) gets sent

to zero by the natural functor Rep(GLn) to Rep(GLn) if and only if it is negligible, and this defines an equivalence between the semisimplification of Rep(GLn) and Rep(GLn). This proposition remains true if we replace GLn by Sn everywhere.

So we see that, up to this semisimplification procedure, these categories interpolate the categories of representations of general linear and symmetric groups in some sense.

Interpolation redux: It turns out there is another sense in which these categories

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which is more closely related to the type of stability phenomena we are interested in. Define Rep(GLt) d to be the Karoubian envelope of those mixed tensor powers

Xr (9X*®S with r+ s < d, and similarly let Rep(GLn) d be the category of

represen-tations of GL,(C) isomorphic to a direct summand of a direct sum of represenrepresen-tations of the form V' 9 V*O® with r + s < d.

Proposition 1.1.3 ([111 Prop 6.4 and 10.6) If n > 4d then the functor Rep(GL,)

to Rep(GLn) sending X to V restricts to an equivalence of categories from Rep(GLn) d to Rep(GLn) d. The same holds if we replace GLn by Sn.

In the case of symmetric groups, this category Rep(Sn):d is exactly the category of Sr-representations with every irreducible summand of the form S[n] with

JAl

K d.

It will be this relationship with the asymptotic behavior of representations that we will be most interested in understanding and generalizing.

Positive characteristic: So far we have been talking about Deligne categories over

the complex numbers. In fact Deligne's construction gives symmetric tensor categories

Rep(GLt) and Rep(St) over arbitrary commutative rings R where t E R is an arbitrary

element, moreover these still satisfy the same universal property.

However we'll note that neither Proposition 1.1.2 nor Proposition 1.1.3 holds once we stop working over a field of characteristic zero, and these characteristic p Deligne categories don't see very much of the behavior of modular representations of sym-metric groups. One of our goals will be to construct characteristic p versions of these categories that capture more of the representation theory.

1.2 Representation stability and Fl-modules

Let F1 denote the category where the objects are finite sets, and morphisms are injections. An FI-module over a commutative ring k is a covariant functor V from

FI to the category of modules over k.

Since the set of FI-endomorphisms of the set [n] = {1, 2, ... , n} is the symmetric

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(the image of [n] under the functor) of S, for each n along with with compatibility maps from V, to Vm for every injection from [n] to [m]. An FI-module is said to be

finitely-generated in degree at most d if all the V are finitely-generated as k-modules

and V, is spanned by the images of Vd under all injections from [d] to [n] for all n > d. Fl-modules were introduced by Church, Ellenberg, and Farb in [3], where it was shown that over a field of characteristic zero the sequence of symmetric group representations defined by a finitely-generated FI-module exhibits the phenomenon known as representation stability as defined by Church and Farb in [5].

As a motivating example, let M be an orientable manifold of finite type and let

Conf,(M) denote the space of configurations of n distinct ordered points in M. For

any inclusion [n] -+ [m] we get a natural map from Confm(M) to Conf"(M) given

by forgetting points not in the image and reordering the others, these maps define a

contravariant functor from FI to the category smooth manifolds. Taking cohomology gives functors FI -+ k-mod defined by [n] -+ H'(Conf,(M), k). These FI-modules

are known to be finitely generated

([3]

Theorem 6.2.1 for k = Q, and [4] Theorem E

for arbitrary noetherian rings k) and are one of the main examples motivating the definition of FI-modules.

Recall that over a field of characteristic zero every finite-dimensional representa-tion of S,, decomposes as a direct sum of irreducible Specht modules S/, indexed by partitions p of size n. If A = (A, A2,... , Af) is a partition, then for n > |Al

+

A

let A[n] = (n - IAl, A,, A₂, ., A). In other words, A[n] is the partition obtained by

taking A and adding a large first part to make it have size n. The stability result of Church, Ellenberg, and Farb can be stated explicitly as follows.

Proposition 1.2.1 ([3] Proposition 3.3.3) Let V be a finitely-generated FI-module

over a field of characteristic zero. There exist non-negative integer constants cA in-dependent of n and nonzero for only finitely many partitions such that for all n

>

0

V = ®c@ SA[]

A

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Specht module associated to the partition A[n].

In positive characteristic, of course representations of symmetric groups do not in general decompose into a direct sum of irreducible representations so we should expect that for FI-modules over a field k of characteristic p > 0 things are more

complicated, and indeed that is the case.

As an example, consider the natural FI-module V sending a finite set S to k[S], the space of formal linear combinations of elements of S. In this example we have that V, a kn with the action of S, permuting the coordinates in the usual way. This is a direct sum of two irreducible representations if p

{

n, and is an indecomposable

representation with three irreducible composition factors whenever p

I

n (and n > 2). Even in this basic example, we see that things are more complicated in positive characteristic. One of our primary goals will be to understand to what extent we can fix this failure of stability for categorical properties of representations like the composition length and decomposition into indecomposibles.

An important corollary to Proposition 1.2.1 is that the characters of these repre-sentations become polynomial in n in a certain sense. Explicitly:

Proposition 1.2.2 ([3] Theorem 1.5) For a finitely generated FI-module V over a field of characteristic zero the characters Xvn of the Sn representations V, are even-tually equal to a fixed polynomial function Pv (X1, X2.... ) in variables Xi (g) recording

number of cycles of length i in g G Sn.

In particular, if we evaluate the character at the identity element of Sn this implies the dimension of a finitely generated Fl-module is eventually given by a polynomial in n.

While some more refined categorical properties of representations were known to

fail, it was conjectured by Farb in his 2014 ICM address ([22] Problem 6.2) that

we should still see this type of eventual polynomiality when we look at the Brauer characters of finitely generated FI-modules in positive characteristic. Another goal of ours will be to formulate a relaxed version of Proposition 1.2.1 that holds in positive characteristic, and implies this conjecture on Brauer characters.

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1.2.1 Structural results for FI-modules

For the most part we will just be interested in understanding the underlying sequence of symmetric group representations of a finitely generated FI-module from a representation theoretic perspective. The sequences of representations coming from a finitely generated FI-module are far from arbitrary though, and we will need a few structural results about the category of FI-modules to understand how these sequences are constrained.

Freely generated FI-modules: Suppose W is a representation of Sm. We can

define an FI-module M(W), freely generated by W by the formula

M(W)s:= S -+ k[HomFI({m], S)] ®k[Sm] W.

Explicitly M(W)n is the representation Inds- xs (W M 1) of Sn obtained by first

extending W to an Sm x Sn-m representation by letting the second factor act trivially, and then inducing up to Sn. This construction is functorial in W, and in fact is a left adjoint to the functor from FI-modules to Sm representations sending an Fl-module

V to Vm.

Over a field of characteristic zero, there are no nontrivial extensions of freely generated FI-modules, and a direct sum of freely generated FI-modules is called an FI -module. Being a direct sum of such modules is a bit restrictive in positive characteristic, so instead we will say that an FI-module V is h-filtered if it admits a filtration

0=V CV1C...CVd=V

of FI-submodules such that the graded pieces Vi/V2- are of the form M(Wi) for some Wi a representation of Sm.

The following proposition is an important structural result for FI-modules due to Nagpal:

Proposition 1.2.3 ([37I Theorem A) For an arbitrary finitely-generated FI-module V, there exist -filtered FI-modules JI, J2

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V -+ J1,

#

: j + J2

7 . N-1 . N-1 _ jN such that

for

all n

>>

0

0 -+* V, -+ Jn -+ Jn2- -- n

is an exact sequence of Sn representations.

One can form a quotient of the category of finitely generated FI-modules by those supported in finitely many degrees, this has the effect of identifying two FI-modules which are isomorphic in sufficiently large degree. Proposition 1.2.3 can be rephrased as saying that in this quotient every finitely generated FI-module admits a finite resolution by i-filtered FI-modules.

Noetherianity: Probably the most important result in the theory of FI-modules and representation stability is the Noetherianity theorem of Church, Ellenberg, Farb, and Nagpal, which says:

Proposition 1.2.4 ([4] Theorem A) If V is a finitely generated FI-module over

a Noetherian ring k, and W is a sub-Fl-module of V, then W is finitely generated.

This implies that finitely generated FI-modules form an abelian category, and therefore one can pass finite generation between the pages of a spectral sequence of FI-modules, which is an essential step in basically every topological application of the theory of FI-modules.

If we let M(m) := M(k[Sm]), this is called the free FI-module generated in degree m. One easily reformulates the definition of finite generation saying that an FI-module V is finitely generated in degree at most d if and only if there is a surjection from a finite direct sum of free FI-modules M(mi) to V with mi < d for all i.

An important consequence of noetherianity is that the kernel of such a surjection is itself finitely generated, and therefore any finitely generated FI-module is also

finitely presented, meaning that it is a cokernel of a map between two FI-modules

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Chapter 2 Background material and exposition

While ultimately many of our final results will pertain to Deligne categories or

FI-modules, we will be drawing on tools and results from a number of different areas of math. In the interest of making this self contained, this section contains an overview of many of the tools and results we will use.

For the most part the results here are standard and we will leave the proofs to the references, although in some cases we will prove slightly non-standard formulations of results that will be suited to our needs.

2.1 Modular representations of symmetric groups

Many of our results pertain to representations of symmetric groups over fields of positive characteristic. In this section we will review some of the theory of modular representations of symmetric groups, particularly focusing on the aspects of the theory we will need later on. As a standard reference on this material we will refer to book of James on the subject [30].

2.1.1 Some classes of representations

Permutation Modules: For A =

(A,

A2, ... , Ak) a partition of n we define the

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or equivalently we could write this as

M (A)

Inds x

where 1 denotes the trivial representation.

From a combinatorial perspective it will be convenient to think of this as the linearization of the natural S, action on the collection of ordered set-partitions of [n] of type A. That is, ordered k-tuples (A1, A2,..., Ak) of subsets of [n] such that [n] = A, U A2U - -U Awith |Ail = Ai.

We'll note that the definition of a permutation module makes perfect sense if A is a composition (i.e. a list of positive integers, not necessarily weakly decreasing) rather than a permutation. We'll note though that the underlying action of S" on set partitions only depends on the underlying partition, so we don't get anything new this way and we might as well just consider partitions.

Specht Modules: For an ordered set partition A

=

(A1, A2,..., Ak), a transverse

set partition to A is an ordered set partition B = (B1, B2, ... Bj) of [n] into non-empty

subsets satisfying

1. JBi nA J < I for all iJ

2. Bi n Aj

= 0

-> Bi n Aj+1 = o

For any ordered set partition B let SB C S, be the subgroup of S, that sends each Bi to itself. For any pair (A, B) of transverse set partitions with A of type A define the vector EA,B E M(A) as

EA,B := sgn(-)u(A)

CESB

where sgn(u) denotes the sign of the permutation. We define the Specht module SA C M(A) to be the space spanned by the vectors EAB for all choices of A, B.

Using this explicit definition one can show that over any ring k this is a free k-module with rank given combinatorially by the so called hook-length formula. How-ever for our purposes it will be more convenient to use an alternative less explicit

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characterization of the Specht module SA. Namely, S" can be characterized as being the set of all vectors V'

c

M(A) such that q(V) = 0 for all Sn-equivariant maps 0 from

M(A) to M(p) with p > A in the dominance order.

We'll note that similarly we could characterize the dual Specht module S,:

Homk(SA, k) as the quotient of M(A) by the span of the images of all Sn-maps from M(p) to M(A) with p > A in the dominance order.

Irreducible representations: If our base ring k is a field of characteristic 0, then

all maps between permutation modules split, so the Specht and dual Specht modules coincide. Moreover, they form a complete list of isomorphism classes of irreducible representations of S,. Over a field of characteristic p, the situation is somewhat different as we will describe now.

Define S" be the submodule of the permutation module M(A) is spanned by the images of all Sn equivariant maps from M(p) to M(A) with p > A in the dominance order, or alternatively S" can be characterized as the orthogonal space to S\ with respect to the natural bilinear form on M(A) where the ordered set partitions form an orthonormal basis. In particular, SA n SAI is the radical of this form restricted to

s A.

We say that a partition is p-regular if it does not contain p parts of the same size. The following theorem gives a classification of the irreducible representations of S5 over a field of characteristic p.

Theorem 2.1.1 ( [30] section

4)

1. The quotient D' := SA / (SA

n

S A-) is either irreducible or zero. 2. Dx is nonzero iff A is p-regular.

3. If we let A run over the collection of p-regular partitions, the D A form a complete list of the isomorphism classes of irreducible representations of Sn.

We could also define the irreducible representations in terms of p-restricted parti-tions (i.e. those conjugate to a p-regular one) by taking:

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Da :=

soc(SA)

for p-restricted A (see 25]). These are related to our other description of irreducibles

by the formula:

DA = DA' ® sgn

which follows from a similar formula for Specht and dual Specht modules. We'll note that for a fixed partition A the padded partition A[n] is never p-restricted for n

>

0,

so for the most part the first description will better suited for our purposes. However at some point we will want to use a version of Schur-Weyl duality (described in section

2.3.2), for which this second description is better behaved.

Young modules: We defined Specht modules as certain submodules of permutation

modules. In characteristic zero they are of course direct summands, but in general they do not split. It will at times be useful for us to look at the indecomposable summands of permutation modules. The following theorem (see [34] section 4.6, for example) summarizes what happens with these summands:

Theorem 2.1.2

1. There is a unique indecomposable direct summand of M(A) containing the Specht module SA, it is called the Young module Y(A).

2. Y(A) is self dual and hence can also be characterized as the unique indecom-posable direct summand of M(A) such that the quotient map form M(A) to the

dual Specht module SA factors through projection onto Y(A).

3. Any other indecomposable direct summand of M(A) is isomorphic to Y(p) for some p > A in the dominance order.

Note that the regular representation is isomorphic to the permutation module

M(1, 1,1,..., 1). In particular this means that indecomposable projective

represen-tations are Young modules. We'll note though that there are more Young modules than projectives, and in general the projective cover of DA is not Y(A).

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2.1.2 Relationships between representations

Now that we have defined these four classes of representations we'd like to elab-orate a bit more about the relationship between them, and how the characteristic p theory relates to the characteristic zero theory. We'll note that in characteristic zero our definitions of the Specht, dual Specht, irreducible, and Young modules coincide. Reductions modulo p: One important construction in modular representation theory is reduction mod p from characteristic zero. Given a representation of a finite group G on a vector space V over Q (or more generally a number field) and a full rank lattice L c V preserved by G, if k is a field of characteristic p then L 0 k is a representation of G over k that we'll say is a reduction mod p of V.

Note that the choice of such lattice is not unique and it is possible for V to admit two nonisomorphic reductions mod p. In fact, the Specht and dual Specht are both reductions mod p of the corresponding Specht module over Q, but in general they are not isomorphic to one another.

We will note however that any two reductions mod p of the same representation over Q will have the same irreducible composition factors (but possibly in different orders) and their Brauer characters will agree with the usual character of the repre-sentation over Q. In particular this holds for the Brauer characters of Specht (and dual Specht) modules.

Kostka and Littlewood-Richardson filtrations: Recall from characteristic zero

that

M(Wp) =

$

K,,SA and Ind7x _(SA Z S') =

A

-where the constants KA,, and c", are the Kostka numbers and Littlewood-Richardson coefficients respectively. In characteristic p we rarely expect to be able to write something as a direct sum of Specht modules, and indeed these formulas no longer hold on the nose.

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Since both the permutation and Specht modules are reductions mod p it follows for free (by say looking at Brauer characters) that these formulas still hold at the level of Grothendieck groups, but in fact we can say something stronger.

Theorem 2.1.3 ([301 Corollary 17.14)

1. Over an arbitrary field k the permutation module M(p) admits a filtration

0= VO C V1 c ... C V = M(P)

such that the graded pieces V'/V'-1 are isomorphic to Specht modules S t, with

SA occurring with multiplicity equal to the Kostka number K,.

2. For arbitrary partitions A I- k, p F- n - m and over an arbitrary field k, the induced representation Indi.sn (SA 0 SP) has a filtration

O=VO CV c...CVd=Indslxs _(SAZ SA)

such that the graded pieces V/V- 1 are isomorphic to Specht modules SVi, with

S" occurring with multiplicity equal to the Littlewood-Richardson coefficient c'. 3. Both of the above theorems remain true if we replace each Specht module by the

corresponding dual Specht module in all places.

Decomposition numbers and p-Kostka numbers: As mentioned before, in

char-acteristic zero the Specht, irreducible, and Young modules all coincide. We'd like to talk a bit more about the relationship between these modules, unfortunately there are many aspects of this theory that are still not well understood.

First let's consider the multiplicity [D,", SA] of an irreducible module D" in a com-position series for the Specht module SA. These numbers are decomcom-position numbers which together form the decomposition matrix for the symmetric group in question. In general there is no known formula for these numbers, and even the question of when these multiplicities are non-zero is not fully understood. Nevertheless, we can say a few things about these as summarized by the following theorem.

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Theorem 2.1.4 [D",

SA]

= 0 unless p > A in the dominance order. Moreover if A

is p-regular (so DA is defined), then [D A, SA] = 1.

When we introduced Young modules we saw that the decomposition of permu-tation modules into Young has a similar upper-triangularity property, although in that case Y(A) makes sense for all A and Y(A) always appears in M(A) exactly once as a direct summand. Let [Y(p) : M(A)] denotes the multiplicity of Y(p) in M(A) as a direct summand, these numbers are the so called p-Kostka numbers. Like the decomposition numbers these numbers are not well understood in general.

It turns out that the filtrations of permutation modules by Specht modules from Proposition 2.1.3 are compatible with the decomposition into Young modules. In particular each Young module has a filtration by Specht modules (and one by dual Specht modules), and one can ask for the multiplicity of a Specht module in such a filtration. Again, these multiplicities are not well understood

2.1.3 More on permutation modules

Of the four classes of representations we discussed the permutation modules were

the easiest to define and are the easiest to work with. The goal of this section will be to give a more or less complete description over any commutative ground ring k of the category of permutation representations and direct sums thereof.

Maps between permutation modules: Suppose G is a finite group that acts

transitively on two finite sets X and Y. For any G-orbit 0 of the diagonal action of G on X x Y we can define a G-equivariant linear map 00 : k[X] - k[Y] by the formula

(xey)eO

and it is an easy exercise to show that for any commutative ring k these maps form a basis for HomG(k[X], k[Y]), the space of G-equivariant k-linear maps from k[X] to

k[Y].

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with X and Y being the collections of ordered set partitions of [n] of type A and 1y respectively. The following easy lemma describes the Sn orbits on the product of these two sets.

Lemma 2.1.5 Suppose (A1, A2,... Ak) and (A', A's,. . .A' ) are ordered set partitions

of type A, and (B1, B2,... Bj) and (B', B',... B,) are ordered set partitions of type

p. ((A1, A2,... Ak), (B1, B2,... B)) and ((A', A',.. .A'), (BI, BI, ... Bk)) are in the same Sn orbit if and only if

IAi

n Bj| = |A' n BI for all i, j.

In particular to understand all maps between permutation modules we just have to understand the combinatorics of the possible values of Tij =

IAi

n Bj I for ordered

set partitions of type A and p. This ends up being quite easy, here are a few ways to index them:

1. So-called tabloids of shape p and type A, that is a labeling of the boxes of a

Young diagram of shape p with A, boxes labeled 1, A2 boxes labeled 2, etc.

considered up to equivalence of permuting the entries in each row. Here the ri, will count the number of boxes in row i labeled by

j.

2. Tabloids of shape A and type p, like above but with ri, counting the number of boxes in row

j

labeled by i.

3. k x f matrices of non-negative integers with row sum A and column sum A. Here

rj just corresponds to the i,

j

entry of the matrix.

Indexing these orbits in terms of tabloids is slightly more standard in the combi-natorics literature, but the more symmetric version in terms of matrices is often more convenient. In any case for our purposes we will just refer to a tabloid T as the data of the numbers Ti,j, subject to the combinatorial constraints above.

Combining things we have the following characterization of maps between permu-tation modules over arbitrary rings k.

Corollary 2.1.6 Homs, (M(A), M(p)) is a free k module with a basis h' indexed by

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h'(( A1,

A2,.... Ak)) =E

(B1,

B2, ...

Be)

AinBjJ=-ri,j

Composition of morphisms: Now that we have written down bases for the spaces

of maps between permutation modules we want to understand how to compose these maps. More explicitly, suppose h' E Homs,(M(A), M(p)), hr' E Homs.(M(p), M(v)), and h" E Homs,(M(A), M(v)) are as defined in above. We want to compute the coefficient of hT" in the expansion of hr' o h' in this basis.

This has a simple combinatorial interpretation, suppose (A1, A2,.. .Ak) and

(B1, B2, ... Be) are ordered set partitions of type A and v, such that Ai

n

BjI =

r ', or all i, j. The coefficient we care about is just the number of set partitions (C1, C2,. .. , Cm) of type p such that IAi

n

Cjl=

Tj and

ICi

n

BI1

= r for all i,

j.

This can be seen by noting that the coefficient of hT" in this expansion is equal to the coefficient of (B1, B2, ... Be) in h' o hT(A1, A2, ... Ak) in the basis of set partitions,

which we can compute directly.

This interpretation will be enough for many purposes, however at some point we will need to prove certain polynomiality properties of these composition coefficients, and for that we will need to push this a bit further. We note that the sizes of the pairwise intersections of the A's, B's, and C's do not in general determine the size of the triple intersections, so the idea is to break our count up further by the size of the triple intersections.

Lemma 2.1.7 Let S denote the set of (k x f x m)-tuples of nonnegative integers ar,,,t

such that

t r

then the coefficient of hr" in the expansion of hr' o h' is equal to

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Proof: S is just indexing the possible sizes of A,

n

B, n Ct given the sizes of the

pairwise intersections. Once we fix these sizes the multinomial coefficients are just counting all the ways to partition each A, n Bt into pieces of the appropriate sizes. E Remark: While in this case it was easy to do things all at once, we'll note that we could also obtain this recursively by looking at one triple intersection at a time. Suppose i is minimal such that A1 n Bi is nonempty, if we fix the data of smallest

j

such that A1 Bi n Cj is nonempty as well as the size a1,, =

IA

1 n Bi

n

C,' of this

triple intersection, then once we choose this triple intersection inside A1 n Bi (giving a factor of

(

))

we are left with a triple intersection problem of the same type but on a smaller set, which can be counted recursively. Summing over all possibilities of this minimal triple intersection then gives the full count.

This calculation of the structure constants for composition gives us a complete description the category of permutation modules of Sa, just as a category. We'll now briefly discuss some additional structures we have on these categories in slightly less detail.

Duality: Permutation modules come equipped naturally with S-invariant bilinear forms and are hence self dual. One could ask if we look at one of our basis maps h' E Homs (M(A), M(p)), what is its adjoint map inside Homs (M(p), M(A))?

This has a nice and simple answer: If we view T as a k x

e

matrix with row sums A and column sums p, then the adjoint map to h' is just the basis map corresponding to the transpose of T.

Restriction: Given a permutation module M(A) for S,, we could ask what its re-striction to Sk X Sn-k is. A set partition on [n] induces set partitions on the first k elements and on the last n - k, so unsurprisingly we can express this in terms of permutation modules again. Explicitly we get

Res(M(A)) =

®

M(p) M M(v)

|a|i =k

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allowed to have empty parts and not be ordered by size with addition defined by

(A1, A2,..., Ak) -+-(F/1, F/2,7.. -, k) = (A 1+ F11, A2 + F/2,7..., A iD), and p, v denote the partitions we get by ignoring empty rows and rearranging them in decreasing order.

We could also ask what the restriction of a map h E Homs (M(A), M(M)) is in

terms of various maps h' O hr". While straightforward to compute this is somewhat messy and not particularly enlightening to work out, so we will omit it.

Induction: Similarly we could take two permutation modules M(p), M(v) for Sk and Snk and ask what we get if we induce M(p) Z M(v) from Sk x Sn-k up to Sn.

This is even easier since the permutation modules are themselves defined in terms of induction. We have that

Ind(M(p) 0 M(A)) = M(p U A)

where p U A denotes the partition we get by shuffling together the parts of P with the parts of A and reordering them in decreasing order by size. This time what happens to maps h' M hT" is easier to describe, we just get a single h where the tabloid T

is obtained by shuffling the rows of the two tabloids T' and T" together in the same manner.

Tensor products: Given two permutation modules M(A), M(y) for S, over a ring

k we might want to understand the representation M(A) Ok M(P) with the diagonal

action (we will drop the subscript on the tensor product from now on). As it happens this is naturally isomorphic to the module given by first restricting M(p) from S, to

SA X SA X ... X SAk and then inducing back up to S,. So in some sense we should

already understand this module, but nevertheless we can be a bit more explicit. Note that if G is a finite set which acts on two finite sets X and Y then k[XI 9

k[Y] r k[X x Y] as representations of G. In our case when G is So, and X and Y are

the collections of ordered set partitions of [n] of type A and p, we already calculated the orbits S, in Lemma 2.1.5. In particular we can write

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where the direct sum runs over tabloids T of shape p and type A, and M(r) is the corresponding permutation module we get when we think of the collection of Tij's as

a partition of n.

2.2 _{Unipotent representations of GLn(Fq)}

There is a common theme in algebraic combinatorics that if you replace the notion of a finite set with a finite dimensional vector spaces over a finite field Fq, then many combinatorial identities admit nice "q-analogs", formulas in a variable q which simplify to classical combinatorial formulas when q is specialized to 1.

For example, the q-integer [n)l := qn-l+qn-2+- - -+q+ = q1 counts the number

_q-

₁

of one dimensional subspaces in an n-dimensional vector space over Fq analogously to how the usual integer n counts the number of one element subsets of an n element set. Moreover, if we plug in q = 1 into this formula we just recover n, the thing we were q-deforming.

With this philosophy in mind one might hope that representation theoretic results about Sn might have q-analogs in the representation theory of GLn (Fq). This is indeed the case, there is a class of representations of GLn(Fq) called unipotent representations which behaves in many ways like the representation theory of Sn, we will review that theory now.

Let Bn(Fq) C GLn(Fq) denote the subgroup of upper triangular matrices. The

set .Tn(Fq)= GLn(Fq)/Bn(Fq) naturally parameterizes flags

0 = V c V1 c V2 c ... c V = lFq

where V is a d-dimensional subspace of Fn. An easy calculation tells us that

1Fn(F)I

is given by the q-factorial [n]q! := [n]q -[n - 1]q ... [1]q, a natural q-analog of the factorial.

If we linearize the natural action of GLn(Fq) on TFn(Fq) we get a representation of GLn(Fq) over any ring k. We'll call this linearized action on flags the regular

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unipotent representation. Informally, a unipotent representation of GLn(Fq) is any

representation that appears as a subquotient of a direct sum of copies of the regular unipotent representation.

More formally let B E kGLn(Fq) denote the sum of all elements in Bn(Fq), and let

I denote the annihilator of kGLn(Fq) [B] (the left ideal generated by B in kGLn(Fq)). A unipotent representation of GLn(Fq) over k is a module over kGLn(Fq)/I.

2.2.1 Some classes of unipotent representations

Permutation modules: Consider for example the action of GLn(Fq) on Gr(n, m, Fq),

the set of m-dimensional subspaces of Fn. Linearizing this gives us a ["] := [mq!

(a q-binomial coefficient) dimensional representation which arises naturally as a quo-tient of the regular unipotent representation via the map (Vo, V1, ... , V) -+ Vm.

More generally suppose A = (A, 2,... , Aj) is a composition of n. We define the

permutation module M(A) to be the linearization of the action of GLn(Fq) on partial flags of type A

O~VCV 1CA 1 A2 .. C VA1+A 2 ...+At = Fn

0 =VO C V1 C V\1+A2 C -q MM+-+

where as before each V is a subspace with dimension equal to its subscript.

If we look at the actions of GLn(Fq) on partial flags of types A and A', where A

and A' are compositions with the same underlying partition, then unlike the analogous situation for symmetric groups these are not in general isomorphic as finite GLn(Fq)-sets. However we have the following theorem that tells us they become isomorphic after linearizing.

Theorem 2.2.1 ([31] Theorem 14.7) Suppose k is an algebraically closed field of

characteristic not dividing q. Then if A and A' are compositions with the same un-derlying partition, then M(A) 'a M(A') as representations.

As such we will often just restrict ourselves to the case when A is a partition, and for many of our purposes we won't lose much by doing so.

Deligne categories and representation stability in positive characteristic

Deligne Categories and Representation Stability

in Positive Characteristic

by

Nate Harman

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

AUG

01

2017

LIBRARIES

ARCHIVES

June 2017

@

Massachusetts Institute of Technology 2017.

Author

All rights reserved.

Signature redacted

Department of Mathematics

May 2, 2017

Certified by...

Signature redacted

Pavel Etingof

Professor of Mathematics

Thesis Supervisor

Accepted by .

Signature redacted

William Minicozzi

Deligne Categories and Representation Stability

in Positive Characteristic

by

Nate Harman

Abstract

Acknowledgments

Contents

Chapter 1

Introduction and setting

efV(A)

0

1.1

Deligne categories

1.1.1

Deligne's setting: rigid symmetric tensor categories

E

f

f

f

1.1.2

Deligne categories and their properties

[7].

"

+

functor

f

f)

JAl

1.2

Representation stability and Fl-modules

([3]

+

>

{

I

1.2.1

Structural results for FI-modules

#

for

>>

Chapter 2

Background material and exposition

2.1

Modular representations of symmetric groups

2.1.1

Some classes of representations

(A,

M (A)

=

_{Unipotent representations of GLn(Fq)}

_q-