Generators of Hecke Ring for Gelfand Pair (S(2n),...

Why o why? Introduction Overview Proof of the generation theorem

Generators of Hecke Ring for Gelfand Pair(S(2n), B(n))

Kürsat Aker, Feza Gürsey Institute - M. B. Can, Tulane Uni.

ODTÜ - Bilkent Algebraic Geometry Seminar2010-12-17 Friday


REPRESENTATIONS: WHAT IS THE PURPOSE OF LIFE?

Representation Theory

I is used in the form Fourier series and Fourier Transform insignal processing,

I helps us multiply matrices faster,I let us analyze random walks on groups, e.g. shuffling cards

(the symmetric group on 52 letters), encoding (to producerandom orthogonal matrices),

I makes it possible to evaluate integrals on unitary,orthogonal, symplectic groups.


REPRESENTATIONS: FUN FACT

I The character table of the Monster group (the final piece ofthe puzzle in the classification of finite simple groups) wasconstructed before the group itself!

I Number of elements of the Monster is ∼ 8 · 1053,I Its character table is 194-by-194 ( ∼ 4 · 104 ).


REPRESENTATIONS: AN EXAMPLE

Definition (Perfect Matching)

A (perfect) matching of the set {1, 2, . . . , 2n} is a partition of theset {1, 2, . . . , 2n} into two-element subsets.

Example

I {{1, 2}, {3, 4}, . . . , {2n− 1, 2n}},I {{1, 3}, {2, 7}, {4, 8}, {5, 6}}.

Set of Matchings

LetM be the set of all matching of the set {1, 2, . . . , 2n}.The symmetric group S(2n) acts onM.


REPRESENTATIONS: ANOTHER EXAMPLE

I Let C[M] the complex vector space with basisM.I The symmetric group S(2n) acts on C[M] as well. Denote

this action by X.I LetHn be the set of C-linear operators on C[M] which

commute with the action of the symmetric group S(2n):

T ∈ Hn ⇐⇒ T · X(w) = X(w) · T

for all w ∈ S(2n).


AIM OF THIS TALK

Let B(n) be the stabilizer of the matching

{{1, 2}, {3, 4}, . . . , {2n− 1, 2n}}.

Then,Hn = EndS(2n)1S(2n)B(n) .

Construct a set of generators ofHn.


JUMP TO THE VERY END. . .Can you find a “nice” set of ring generators for the Hecke ringHn := Hecke(S(2n),B(n)) ?

I S(2n), symmetric group on 2n letters,I B(n), centralizer of the element

t := (1, 2)(3, 4) · · · (2n− 1, 2n),I eB(n) := 1

|B(n)|∑

w∈B(n) w,

I Hn := Hecke(S(2n),B(n)) := eB(n)CS(2n)eB(n).

Theorem (A. - Can, 2010)

The ringHn is generated by the elements

Hi :=∑

w has exactly i circuits

w for i = 1, . . . ,n.


GRAPHS ATTACHED TO ELEMENTS w ∈ S(2n)

w = (1 2 3 4 5 6 7 8 9 10 11 121 7 2 8 6 4 10 3 9 5 12 11

)

8

5

9

12

10

8

6

4

3

2

111

7

1

7

2

610

3

9

5

12

4

11

Figure: Coset Type (3, 2, 1) with 3 circuits


THIS IS A DIRECT ANALOGUE OF. . .

Theorem (Farahat-Higman, 1959)

Let Zn be the the center of integral symmetric group ring, ZS(n).Then, the ring Zn is generated by the elements

Zi :=∑

w has exactly i cycles

w for i = 1, . . . ,n.

I (1, 3)(5, 2, 6, 4) ∈ S(6) is of cycle type (4, 2) and has 2 cycles,I (1, 3)(5, 2, 6, 4) ∈ S(7) is of cycle type (4, 2, 1) and has 3

cycles, since (1,3)(5,2,6,4)=(1,3)(5,2,6,4)(7).


WHY THE CENTER?

I Complex Representations of a finite group GI Complex Characters of GI Center of the group ring CG := {

∑g agg}


TWO INCARNATIONS

(C(G) := {f : G→ C}, ?) (CG, ·)Cl(G) = Class/Central Functions on G Z(CG)Character χ

∑g χ(g)g

1C conjugacy class sum of CCl(G) = ⊕χ∈Irr(G)Cχ Z(ZG) = ⊕CZC

Character Table of G = Change of Basis Matrix from IrreducibleCharacter Basis to Class Sum Basis


MULTIPLICATION IN THE CENTER OF ZG

Given two conjugacy classes/class sums Cλ,Cµ in Z(ZG), theproduct CλCµ has the following expansion:

CλCµ =∑

ν

aνλµCν ,

where, Cν runs over all possible conjugacy classes.Fix any element z ∈ Cν , then

aνλµ := {(x, y)|xy = z, x ∈ Cλ, y ∈ Cµ}.


WHAT ABOUT THE CENTER OF ZG AS A RING?



Zi :=∑


w for i = 1, . . . ,n.

RemarkDimension/Rank of Zn as a Z-module is the number ofpartitions of integer n. Number of ring generators is just n.

Idea of the proof: Let n→∞ and get rid of accidentalrelations.


FARAHAT-HIGMAN AND LATER DEVELOPMENTS

I Farahat-HigmanA new proof of Nakayama’s conjecture for modularrepresentations of S(n)

I Jucys 1974: Zi = en−i(J1, J2, . . . , Jn) for n = 1, 2, . . . ,n.

Jk :=∑

transpositions in S(k)−∑

transpositions in S(k− 1).

I Murphy 1984:Reworked Young’s Orthonormal and SeminormalConstruction of irreducible representations of S(n) usingYJM elements


YJM ELEMENTS

I Vershik-Okounkov 1995:Gelfand-Zetlin algebra GZ(n) := 〈J1, . . . , Jn〉 ⊂ CS(n) playsthe role of Cartan subalgebra in a semisimple complex Liealgebra g.

I Post 1995:I Asymptotic Representation Theory and Free ProbabilityI Weingarten formulas for Unitary and Orthogonal Groups


CENTER→ HECKE: GELFAND PAIRS

Definition (Gelfand Pairs)

A finite group G with a subgroup K is called a Gelfand pair ifany irreducible representation of G appears at most once in theinduced representation 1G

K(= C[G/K]). Equivalently, EndG(1GK)

is commutative. We denote this algebra by Hecke(G,K).

Example (Gelfand Pairs)

I (G× G,∆G): In this case, Hecke(G× G,∆G) ∼= Z(ZG).I (S(2n),B(n)):

1S(2n)B(n)

∼= ⊕λ`nS2λ


SETUP

I G(n) = S(2n), or S(n)× S(n),I K(n) = B(n), or ∆S(n),I Hn, the corresponding Hecke ring, (the centers),I G(∞) = S(2∞), or S(∞)× S(∞),I K(∞) = B(∞), or ∆S(∞).


MULTIPLICATION IN A HECKE RING

Given two double coset sums Kλ,Kµ in a Hecke ring, theproduct KλKµ has the following expansion:

KλKµ =∑

ν

bνλµKν ,

where, Kν runs over all possible double cosets.Fix any element z ∈ Kν , then

bνλµ := {(x, y)|xy = z, x ∈ Kλ, y ∈ Kµ}.


IDEA OF THE PROOF (FARAHAT-HIGMAN):

I Construct a universal ring U to dominate allHn,I Show that U is a free polynomial ring in some elements Ui

for i ∈ N.I Then, the images of Ui will generate each Hecke ringHn.


CONTROL THE DOUBLE COSETS!

I Lemmax and y are in the same K(∞)-double coset if and only ifthey have the same reduced type, λ.

I DefinitionFor any partition λ, denote by Kλ the correspondingK(∞)-double coset in G(∞).Set Kλ(n) := Kλ ∩ G(n).

I LemmaI Kλ(n) is non-empty if and only if |λ|+ `(λ) ≤ n.I In that case, Kλ(n) is a G(n)-double coset.


PRODUCTS Kλ(n)Kµ(n)

Denote the structure coefficients of the algebraHn with respectto the basis Kµ(n) by bν

λµ(n):

Kλ(n)Kµ(n) =∑

ν

bνλµ(n)Kν(n).

Let B be the subring of Q[t] so that f ∈ B iff f (n) ∈ Z for n ∈ Z.The structure constants bν

λµ(n) obey the tricohotomy:

bνλµ(n) =

0 , if |ν| > |λ|+ |µ|,positive integer indep. of n , if |ν| = |λ|+ |µ|,f νλµ(n) , if |ν| ≤ |λ|+ |µ|,

where f νλµ ∈ B.


AN IMPORTANT CALCULATION

TheoremAssume |ν| = |µ|+ r for partition ν. Then,

Kλ(n)K(r)(n) =∑

ρ

b(r+|ρ|)∪λ−ρλ(r) K(r+|ρ|)∪λ−ρ(n) + lower order terms,

where the sum runs over those partitions ρ ⊆ λ so that `(ρ) ≤ r + 1.In this case, the coefficient bν

λ(r) is given by

bνλ(r) =

(mr+|ρ|(λ) + 1)(r + |ρ|+ 1)r!∏i≥0 mi(ρ)!

, (1)

where m0(ρ) := r + 1− `(ρ).


THE UNIVERSAL RING U

I As a B-module: U := ⊕λBKλ.I Multiplication:

KλKµ :=∑

|ν|≤|λ|+|µ|

f νλµKν .

I Surjection onto allHn for all n:

∑fµKµ 7→

∑fµ(n)Kµ(n).

I Setting the degree of the generator Kλ as |λ|, the universalring U is filtered.


UNIVERSAL RING U IS A FREE POLYNOMIAL RING!

Theorem (Farahat-Higman)

The universal ring U is isomorphic to the polynomial ringB[U1,U2, . . .], where for i = 1, 2, . . .,

Ui :=∑|λ|=i

Kµ.


GENERATION THEOREMS



Zi :=∑


w for i = 1, . . . ,n.

I Proof:The universal ring U surjects onto Zn for all n.Recall that ∑

fµKµ 7→∑

fµ(n)Kµ(n).

If |λ| ≥ n, then Kλ 7→ 0. So Un,Un+1, . . . 7→ 0.But, Ui 7→ Zn−i for i = 0, . . . ,n− 1.


GENERATION THEOREMS, CONTINUED. . .

Theorem (A. - Can, 2010)

The ringHn is generated by the elements

Hi :=∑

w has exactly i circuits

w for i = 1, . . . ,n.

I Proof:Similarly, Un,Un+1, . . . 7→ 0.But, Ui 7→ Hn−i for i = 0, . . . ,n− 1.


IN TERMS OF YJM ELEMENTS

Recall that Jucys proved that

Zi = en−i(J1, J2, . . . , Jn)

for n = 1, 2, . . . ,n.Recently, Zinn-Justin and Matsumoto proved that

Hi = en−i(J1, J3, . . . , J2n−1)eB(n)

for n = 1, 2, . . . ,n.

Generators of Hecke Ring for Gelfand Pair (S(2n),...

Documents

Transcript of Generators of Hecke Ring for Gelfand Pair (S(2n),...