The problem of decomposition numbers of finite classical

The problem of decomposition numbers of finiteclassical groups

Emily Norton

April 24, 2021

Emily Norton New dec numbers April 24, 2021 1 / 25

Modular representation theory

Let G be a finite group and ` a prime dividing |G |.Representation theory of G in characteristic 0 is semisimple (pretty easy).

Representation theory of G in characteristic ` is not semisimple (very hard).

Fix (K,O,k) an `-modular system:

K a finite extension of Q`O ⊂ K ring of integers

k = K/O, char k = `

This allows us to “reduce representations mod `:”

characteristic 0 integral lattice characteristic `

ρ ∈ IrrKG -mod Λρ ∈ OG -mod Λρ ⊗ k ∈ kG -mod


Decomposition numbers and decomposition matrices

Fix a prime ` and an `-modular system (K,O,k). Let ρ ∈ IrrKG , φ ∈ IrrkG .

Definition

The number of times φ appears as a composition factor of ρ is called the decompositionnumber dρ,φ.

Definition

The decomposition matrix of G is the matrix with rows labeled by {ρ ∈ IrrKG}, columnslabeled by {φ ∈ IrrkG}, with entries dρ,φ.

Example G = S3 and ` = 3. The decomposition matrix is:

(3) (2, 1)( )(3) 1 0(2, 1) 1 1

(1, 1, 1) 0 1

Despite powerful methods from Lie theory, the decomposition matrix of Sn for an arbitraryprime ` ≤ n is not known. Problem: how to compute `-Kazhdan-Lusztig polynomials?


Decomposition matrices of finite groups of Lie type I

q a power of a prime, G ⊂ GLn(Fq) connected reductive algebraic group, F : G→ GFrobenius, G(q) := GF is a finite group of Lie type e.g. GLn(q), Sp2n(q), SO2n+1(q).Let ` be a prime s.t. ` - q.

Problem

Determine the decomposition matrix of G(q) in characteristic `.

There is a distinguished subset of IrrKG(q) called unipotent representations, definedusing geometry related to the flag variety of G (Deligne-Lusztig varieties).

#{IrrKG(q)} → ∞ as q →∞ but #{unip reps of KG(q)} is finite, indep of q, dependsonly on the Weyl group W of G .

Definition

The unipotent decomposition matrix of G(q) in characteristic ` is the submatrix of thedecomposition matrix of G(q) given by D = (dρ,φ) s.t. ρ is unipotent, φ ∈ IrrkG(q) s.t.dρ,φ 6= 0 for some unip ρ.

Expectation: the decomposition matrix of G(q) can be recovered from D.


Decomposition matrices of finite groups of Lie type II

Revised problem

Determine the unipotent decomposition matrix D of G(q) in char `.

Example: G(q) = GL3(q), {unip reps of GL3(q)} 1:1↔ {partitions of 3}. Take` | Φ3(q) = q2 + q + 1. Then:

D =

(3) (2, 1) (1, 1, 1)( )(3) 1 0 0(2, 1) 1 1 0

(1, 1, 1) 0 1 1

Properties of the unipotent decomposition matrix D:

the matrix D is square and invertible (Geck-Hiss),

expected by experts but open: for ` large enough, D does not depend on q butonly on the order of q mod `,

D is lower unitriangular if q is a power of a good prime for G (conj: Geck; proof:Brunat-Dudas-Taylor) labels for {φ ∈ IrrkG(q) | dρ,φ 6= 0 some unip ρ}.


The order polynomial of G (q) and cyclotomic polynomialsLet W be the Weyl group of G(q) and assume F acts trivially on W . If G = GLn thenW = Sn, if G = Sp2n or SO2n+1 then W = Bn. R ⊂W set of reflections, m = rkG .

The order of G(q) is a polynomial in q:

|G(q)| = q|R|m∏i=1

(qdi − 1)

where di are the degrees of the generators of S(h)W .

Recall:qa − 1 =

∏d|a

Φd(q),

where Φd(q) is the d ’th cyclotomic polynomial. Thus ` - q and ` | |G(q)| implies` | Φd(q) for some d dividing some di .

The more times Φd(q) divides |G(q)|, the more difficult to determine D in char ` for` | Φd(q).

D is known when:

Φd(q) divides |G(q)| exactly once (1’s on diagonal, some 1’s on subdiagonal, 0’selse),

d = 1 (identity matrix).


Methods for computing decomposition matrices of FGLT

1 Algebraic - Harish-Chandra induction/restriction to produce new characters fromold, connection to Hecke algebras (1990’s)

2 Geometric - Deligne-Lusztig varieties produce projective characters via cohomology,give info about cuspidals (reps that can’t obtained by HC induction) (2010’s)

3 Combinatorial/Lie theoretic - Kac-Moody categorification, mod d combinatorics ofpartitions (type A) or bipartitions (types B/C/D) (reduce, reuse, recycle)

Decomposition matrix known in type A + cases reducible to type A

In the case of GLn(q) (or SLn(q), PGLn(q)), the decomposition matrix when` | Φd(q), 2 ≤ d ≤ n, is given by the decomposition matrix of a q-Schur algebra forq a primitive d ’th root of unity (known for ` >> 0 by Ariki,Lascoux-Leclerc-Thibon...). Parabolic affine KL polynomials of type A.

For Sp2n(q) or SO2n+1(q) and ` | Φd(q) with d odd, the decomposition matrix isdetermined from that of GLn(q) (Gruber-Hiss).


Types B and CFrom now on, q is always a power of an odd prime.

G(q) = Sp2n(q) or SO2n+1(q). Then W = Bn (hyperoctahedral group).

Order polynomial:

|G(q)| = qn2n∏

i=1

(q2i − 1)

Example: G(q) = Sp4(q) or SO5(q). Then

|G(q)| = q4Φ1(q)2Φ2(q)2Φ4(q).

Interesting case: ` | Φ2(q). The decomposition matrix of the principal block (the blockcontaining the trivial representation 2.) was found by Okuyama-Waki:

2. .2 B2 12. .12

2. 1 · · · ·.2 1 1 · · ·B2 · · 1 · ·12. 1 · · 1 ·.12 1 1 2 1 1


Types B and C: parametrization of unipotent representations

G = SO2n+1(q) or Sp2n(q). W = Bn.

Unipotent representations of G :

Organized into “Bt2+t-series,” one series for each t ∈ Z≥0 such that t2 + t ≤ n.

In the Bt2+t-series, there is a unipotent representation for each bipartition λ1.λ2 ofn − (t2 + t).

We writeBt2+t : λ1.λ2

for the unipotent representation in the Bt2+t-series labeled by the bipartition λ1.λ2 ofn − (t2 + t).

Example: take n = 6. The unipotent representations of G are parametrized by:

Bipartitions of 6 ( “principal series”), e.g. 4.12,

Bipartitions of 4 (B2-series), e.g. B2 : 21.1,

Bipartitions of 0 (B6-series), there is only one, denoted simply as B6.


What is the unipotent decomposition matrix in types B and C?

G = SO2n+1(q) or Sp2n(q), W = Bn.

` | Φd(q), d even. (The case not given by type A stuff). Assume ` >> 0, so we expectsome “characteristic-free” decomposition matrix only depending on d .

The unipotent decomposition matrix is lower-unitriangular with rows and columnslabelled by bipartitions Bt2+t : λ1.λ2.

Problem

What is the unipotent decomposition matrix in types B and C? Is there a positivecombinatorial formula for the decomposition numbers in terms of the combinatorics ofbipartitions? Or in terms of (parabolic, affine) Kazhdan-Lusztig polynomials?

Some issues:

We have bipartitions of different sizes. How to relate them? Combinatorics of blocksand “cocores” suggests level-rank duality, but it’s not clear how to use this.

In type A, there is a quasihereditary algebra whose decomposition matrix gives D.But in types B and C we don’t know of an algebra that could play this role.

We don’t have many examples.

In order to attack this problem we should use all available methods, both to understandthe representation category better and to produce more examples.


Kac-Moody categorification in representation theory

The third method for studying decomposition numbers of classical finite groups of Lietype: Kac-Moody categorification and the combinatorics of crystals.

Chuang and Rouquier proved Broue’s Abelian Defect Group Conjecture for symmetricgroups in the early 2000s using categorification. Strategy: turn a category ofrepresentations itself into a representation.

Chuang-Rouquier found a way to define an action of the Lie algebra sl2 on a category.The idea: the generators e and f of sl2 act on a module category C by a biadjoint pairof exact endofunctors E and F in such a way that the images of the functors in theGrothendieck group of C satisfy the sl2-relations.

Categorical actions

yield powerful structural results such asI derived equivalences between blocks,I branching rules for induction and restriction.

cast intricate combinatorial shadows.


Definition of g-categorification

g a Lie algebra (finite or affine Dynkin type).C – abelian category, finite-length

Definition (Chuang-Rouquier)

A g-categorification on C is a collection of exact endofunctors {Ei ,Fi} of C, where iranges over the nodes of the Dynkin diagram of g, satisfying:

For each i , Ei and Fi are a biadjoint pair of functors;

The functors Ei and Fi for all i induce an action of g on the (complexified)Grothendieck group [C] via [Ei ] = ei , [Fi ] = fi where ei , fi are the Chevalleygenerators of g;

The classes [S ] in [C] of the simple objects S ∈ C are g-weight vectors;

Strong: Set E =⊕

i Ei , F =⊕

i Fi . There are natural transformations X ∈ End(E)and T ∈ End(E 2) such that in End(E n), Xj := 1j−1X1n−j and Tk := 1k−1T1n−k−1

satisfy defining relations of an affine Hecke algebra.

Consequence: Soc(Ei (S)) and Head(Fi (S)) are simple or 0 for all simple obj SThe recipe, usually: C is a tower of module categories and E and F are Restriction andInduction.


Example: the symmetric groups in positive characteristic

char k = p > 0, C =⊕n≥0

kSn-mod. Res :=⊕n≥0

Resn+1n , Ind :=

⊕n≥0

Indn+1n .

Theorem (Chuang-Rouquier, Lascoux-Leclerc-Thibon)

There is a slp-categorification on C with

Res = E =⊕

i∈Z/pZ

Ei , Ind = F =⊕

i∈Z/pZ

Fi

λ ` n a p-regular partition, Sλ a simple kSn-module. The head of Fi (Sλ) is simple ifFi (Sλ) 6= 0, set it equal to Sfi (λ)

.

Example: finding f1(λ) when λ = (4, 3) and p = 3:

+−

+

+(−+) +

f1

( )=


The sld -crystal

The graph with

Vertices: {p-regular partitions λ}Edges: {λ→ µ | µ = fi (λ) for some i ∈ Z/pZ}

is called the slp-crystal on the set of p-regular partitions.

Generalizations with representation-theoretic meaning:

can replace prime p with an integer d ≥ 2,

the rule for fi and ei works on any partition, not just p-regular ones,

can replace partitions with `-partitions λ1.λ2. . . . .λ`.

The resulting sld -crystals on partitions or `-partitions give branching rules for inductionand restriction for:

the Hecke algebras Hq(Sn) for q a d ’th root of unity,

the q-Schur algebras for q a d ’th root of unity,

groups GLn(q) in characteristic ` when ` | Φd(q),

cyclotomic Hecke algebras at d ’th roots of 1,

cyclotomic rational Cherednik algebras...


sld -action on module categories of finite classical groupsStarting point: Gerber-Hiss-Jacon conjectured (2014) that the sld -crystal when d is odddescribes the branching rule for Harish-Chandra induction and restriction for unipotentrepresentations of finite unitary groups when ` | Φ2d(q). Proved byDudas-Varagnolo-Vasserot (2015) by constructing a categorical KM-action.

Types B/C: char k = ` > 0, ` | Φd(q), d ≥ 2 even.

Set Cn = kSp2n(q)-modunip or kSO2n+1(q)-modunip, then take

C =⊕n≥0

Cn

Resn+1n : Cn+1 → Cn HC restriction,

Indn+1n : Cn → Cn+1 HC induction.

Then Res =⊕n≥0

Resn+1n , Ind =

⊕n≥0

Indn+1n are exact, biadjoint endofunctors of C.

Theorem (Dudas-Varagnolo-Vasserot, 2016)

There is a sld -categorification on C with

Res = E =⊕

i∈Z/dZ

Ei and Ind = F =⊕

i∈Z/dZ

Fi .


Corollary

The Harish-Chandra branching rule for C is given by the sld -crystal on a sum of level 2Fock spaces Fst corresponding to the unipotent reps in the Bt2+t-series:

[C] ∼=⊕

t∈Z≥0

Fst

Charge st ∈ Z2 for level 2 Fock space Fst :

st =

{(t,−1− t + d

2) if t is even,

(−1− t, t + d2

) if t is odd,

Example: Say d = 4 and we want to determine the simple head of F1(SB2:43.231):

f1(B2 : 43.231) = f1

-2 -1 0 1-3 -2 -1

3 4 +2 31 20 +

= -2 -1 0 1-3 -2 -1

3 42 31 20 1


The unipotent decomposition matrix of SO4n+1(q) or Sp4n(q) when d = 2n

First application of DVV’s theorem to decomposition matrices:

Theorem (Dudas-N., ’21)

We find the unipotent decomposition matrix of SO4n+1(q) or Sp4n(q) when ` | Φ2n(q)and ` is sufficiently large.

The principal block is the only block of defect > 1, so we find the unipotentdecomposition matrix of the principal block.

Unipotent representations in the principal block:

certain hook bipartitions of 2n (principal series), such as

· ∅ , · , ∅ ·

bipartitions of 2n − 2 (B2 series) fitting into certain boxes: ·

bipartitions of 2n − 6 (B6 series) fitting into certain boxes: ·


Example of our theorem when 2n = 6

The unipotent decomposition matrix of the principal block of SO13(q) or Sp12(q) when` | Φ6(q). First found by Dudas-Malle [DM ’20], given by our theorem for n = 3.

6. 1.6 151. 1 12.4 1 1 1

B2 : .22 1

412. 1 1.51 1 121.3 1 1 1 11.41 1 1 1 1

B2 : 1.21 1 1B6 1

212.2 1 1 1 1

12.31 1 1 1 1

B2 : 2.12 1 1

313. 1 1

.412 1 1 1

B2 : 12.2 1 1

213.1 1 1 1 1

13.21 1 1 1 1B2 : 21.1 1 1 1 1 1

B2 : 22. 1 1 1

14.12 1 1 1 1

214. 1 1 1

.313 1 1 1 1

16. 2 1 1 1 1

.214 1 1 1 1

.16 1 2 1 1 1


Example of an argument using the crystal to find decomposition numbers

We used all three methods – algebraic, geometric, combinatorial – to prove our theorem.The combinatorics of the sl2n-crystal gave us an argument showing that almost all theentries in the four “cuspidal columns” are 0 (in particular, the columns that have a 2 inthem).

We also used the sl2n-crystal to establish indecomposability of induced PIMs. The(unipotent part of the) characters of PIMs are the columns of the dec matrix.Example: column B2 : 21.1 in the example matrix. By Fong-Srinivasan,PB2:21. = B2 : 21. . Charge s1 = (−2, 4). We have:

F4 (PB2:21.) = F4

(-2 -1-3

.

)= -2 -1

-3 -2. + -2 -1

-3. 4

this is the character of a projective module bec. Fi takes proj. to proj., but we don’tknow yet if it is indecomposable,

f4(B2 : 21.) = B2 : 21.1 =⇒ PB2:21.1 | F4 (PB2:21.),

it is impossible to tell if F4 (PB2:21.) is indecomposable by HC restriction!!!,

however, we check that ei (B2 : 22.) = 0 for all i ∈ Z/6Z, which implies that PB2:22.

is not a summand of an induced projective module,

so we can conclude that in fact, F4 (PB2:21.) = PB2:21.1.


Any patterns in the decomposition matrix for the 2n = 6 example?

Yes! Let’s look at the submatrix of rows/columns labeled by the B2 series:

1. Consider submatrix labeled B2 : λ1.λ2:

B2 : .22 1B2 : 1.21 1 1B2 : 2.12 1 1B2 : 12.2 1 1B2 : 21.1 1 1 1 1 1B2 : 22. 1 1 1


2. Draw poset determined by nonzero dec numbers:

∅ .

.

..

.

. ∅


Project λ1.λ2 onto λ2:

∅

Question: where have we seen this poset before?


Answer:

This is the poset of Schubert cells in the Grassmannian Gr(2, 4).

This is the poset of parabolic category Op of type A1 × A1 ⊂ A3.

The submatrix of the decomposition matrix labeled B2 : λ1.λ2 from 2 slides ago isthe same as the decomposition matrix of Op (multiplicities of simples in Vermas).

Some history: Category Op of type Ak−1 × An−k−1 ⊂ An−1 is equivalent to the categoryof perverse sheaves on Gr(k, n) (Braden, Stroppel). Poset: Young diagrams fitting ink × (n − k) box, under inclusion of diagrams. In our example, k = 2 and n = 4.

Why does this show up in the unip dec matrix of finite groups of Lie type in typesB and C?

Brundan-Stroppel: the highest weight cover of the Hecke algebra of type Bn in the“d =∞” case (parameter q generic) is equivalent to a sum of categories Op of typeAk−1 × Am−k−1 ⊂ Am−1. Explicit construction of a block as the module category of afinite-dimensional algebra called the Khovanov arc algebra, related to the Temperley-Liebalgebra. Brundan-Stroppel gave an explicit combinatorial formula for the decompositionnumbers for that algebra (and thus for the Hecke algebra of type Bn when q is generic).

When d > |λ1|+ |λ2| we can expect similar behavior to d =∞.


Generic submatrices of the decomposition matrix

By the Bt2+t-submatrix we mean the submatrix of the unipotent decomposition matrix ofkSO2n+1(q) or kSp2n(q) whose rows and columns are labeled by Bt2+t : λ1.λ2, whereλ1.λ2 is a bipartition of n − (t2 + t).

Theorem (Dudas-N. ’21, work in progress)

Let d > n− t2− t be even and let ` = char k be any prime such that ` | Φd(q). Then thedecomposition numbers in the Bt2+t-submatrix are given by Brundan-Stroppel’s formula.

Idea of proof: show that when d > n − t2 − t, the dec numbers are controlled bycombinatorics of sld -crystal.

Consequences:

Explicit, closed, diagrammatic formulas for the entries of the Bt2+t-submatrix.

The Bt2+t-submatrix depends only on the order of q mod `, not on `.

The Bt2+t-submatrix is the same as the decomposition matrix of Category O of therational Cherednik algebra Hd,st (n − t2 − t).

All dec numbers in the Bt2+t-submatrix are 0 or 1.

This is the first result identifying large submatrices of the unip dec matrix in blocksof arbitrary complexity since the 1990s.


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The problem of decomposition numbers of finite classical