Blocks of generic and cyclotomic Hecke algebras Maria Chlouveraki EPFL

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Blocks of generic and cyclotomic Hecke algebras

Maria Chlouveraki EPFL Braids in Paris September 19, 2008

Maria Chlouveraki (EPFL) Blocks of Hecke algebras September 18, 2008 1 / 12

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Symmetric algebras

Ois a Noetherian and integrally closed domain.

Ais anO-algebra, free and finitely generated as anO-module. K is a field such that the algebraKA:=K⊗OAis split semisimple.

Definition

We say that a linear mapt :A→ Ois a symmetrizing form onAif t is a trace function,i.e., t(ab) =t(ba) for alla,b∈A.

The morphism ˆt :A→Hom_O(A,O), a7→(x7→t(ax)) is an isomorphism ofA-modules-A.

Example: IfO=Zand A=Z[G](G a finite group), we can define the following symmetrizing form (“canonical”) on A

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Ais anO-algebra, free and finitely generated as anO-module. K is a field such that the algebraKA:=K⊗OAis split semisimple.

Definition

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Ais anO-algebra, free and finitely generated as anO-module.

K is a field such that the algebraKA:=K⊗OAis split semisimple.

Definition

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Definition

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Definition

We say that a linear mapt :A→ Ois a symmetrizing form onAif

t is a trace function,i.e., t(ab) =t(ba) for alla,b∈A.

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Definition

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Definition

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Symmetric algebras

Definition

t:Z[G]→Z, X

g∈G

a_gg 7→a₁.

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Theorem (Geck) - Definition

1 We have

t = X

χ∈Irr(KA)

1 sχ

χ,

wheres_χ is theSchur element ofχ with respect tot.

2 The Schur element sχ belongs to the integral closureOK ofOinK.

3 LetLbe a field such thatLAis split. Ifθ:OK →Lis a ring homomorphism, then LAis semisimple if and only ifθ(sχ)6= 0 for all χ∈Irr(KA)

4 TheblocksofAare the partsB ofIrr(KA) minimal for the property: X

χ∈B

χ(a) sχ

∈ O, ∀a∈A.

Example: IfO=Z, A=Z[G] (G a finite group) and t is the canonical form on A, we have s_χ=|G|/χ(1).

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Theorem (Geck) - Definition

1 We have

t = X

χ∈Irr(KA)

1 sχ

χ,

χ∈B

χ(a) sχ

∈ O, ∀a∈A.

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Theorem (Geck) - Definition

1 We have

t = X

χ∈Irr(KA)

1 sχ

χ,

χ∈B

χ(a) sχ

∈ O, ∀a∈A.

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Theorem (Geck) - Definition

1 We have

t = X

χ∈Irr(KA)

1 sχ

χ,

4 TheblocksofAare the partsB ofIrr(KA) minimal for the property:

X

χ∈B

χ(a) sχ

∈ O, ∀a∈A.

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Theorem (Geck) - Definition

1 We have

t = X

χ∈Irr(KA)

1 sχ

χ,

4 TheblocksofAare the partsB ofIrr(KA) minimal for the property:

X

χ∈B

χ(a) sχ

∈ O, ∀a∈A.

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Generic Hecke algebras

Every complex reflection groupW has a nice“presentation a la Coxeter”:

I G2=<s,t|ststst=tststs,s²=t²= 1>

I G₄=<s,t|sts=tst,s³=t³= 1>

and a field of realizationK: KG₂=Q, KG₄ =Q(ζ3).

Following Bessis’ theorem, thebraid groupassociated toW has a presentation of the form:

I B(G2) =<S,T|STSTST =TSTSTS >

I B(G4) =<S,T|STS=TST >

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Generic Hecke algebras

Every complex reflection groupW has a nice “presentation a la Coxeter”:

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Generic Hecke algebras

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Generic Hecke algebras

I G4=<s,t|sts=tst,s³=t³= 1>

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Generic Hecke algebras

and a field of realizationK:

KG₂=Q, KG₄ =Q(ζ3).

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Generic Hecke algebras

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Generic Hecke algebras

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Generic Hecke algebras

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Generic Hecke algebras

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We choose a set of indeterminates

u= (us,j)_s,_{0≤j≤o(s)−1}

wheres runs over the set of generators ofW ando(s) denotes the order of s (ifs andt are conjugate inW, thenus,j =ut,j for allj).

The associatedgeneric Hecke algebraH(W) is an algebra over the Laurent polynomial ringZ[u,u⁻¹] and has a presentation of the form:

H(G2) =<S,T|STSTST =TSTSTS, (S−u₀)(S−u₁) = 0, (T−w₀)(T −w₁) = 0> .

H(G4) =<S,T|STS=TST, (S−u₀)(S−u₁)(S−u₂) = 0, (T −u₀)(T −u₁)(T −u₂) = 0> .

Remark: The specialization us,j 7→ζ_o(s)^j sendsH(W) toZKW.

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u= (us,j)_s,_{0≤j≤o(s)−1}

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u= (us,j)_s,_{0≤j≤o(s)−1}

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u= (us,j)_s,_{0≤j≤o(s)−1}

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u= (us,j)_s,_{0≤j≤o(s)−1}

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Assumptions

The algebraHis a freeZ[u,u⁻¹]-module of rank|W|. There exists a unique linear form t:H →Z[u,u⁻¹] such that

I t is a symmetrizing form onH.

I Viaus,j 7→ζ_o(s)^j , the formt becomes the canonical form on Z^KW.

I t satisfies some other condition.

Theorem (Malle)

Let v= (v_s,j)_s,j be a set of indeterminates such that, for alls,j, we have v_s^|µ(K)|_,j :=ζ_o(s^−j₎us,j,

where µ(K) is the group of all the roots of unity inK. Then theK(v)-algebra K(v)His split semisimple.

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Assumptions

The algebraHis a freeZ[u,u⁻¹]-module of rank|W|.

There exists a unique linear form t:H →Z[u,u⁻¹] such that

Theorem (Malle)

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Assumptions

There exists a unique linear formt :H →Z[u,u⁻¹] such that

Theorem (Malle)

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Assumptions

Theorem (Malle)

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Assumptions

I Viau_s,j 7→ζ_o(s)^j , the formt becomes the canonical form on ZKW.

Theorem (Malle)

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Assumptions

Theorem (Malle)

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Assumptions

Theorem (Malle)

Letv= (v_s,j)_s,j be a set of indeterminates such that, for alls,j, we have v_s,j^|µ(K)|:=ζ_o(s^−j₎us,j,

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By “Tits’ deformation theorem”, the specialization vs,j 7→1 induces a bijection Irr(K(v)H) ↔ Irr(W)

χ_v 7→ χ

sχ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

Let χ∈Irr(W). The Schur elements_χ_v is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,i.e.,ifM=Q

s,jv_s,j^a^s,j, then gcd(as,j) = 1 andP

s,jas,j = 0.

Ifp is a prime ideal ofZK such that Ψ(1)∈p, thenM is called ap-essential monomial for χ. We say thatM is ap-essential monomial forW, if there exists χ∈Irr(W) such that M isp-essential forχ.

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χ_v 7→ χ

sχ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

Letχ∈Irr(W). The Schur elements_χ_v is an element ofZK[v,v⁻¹] whose irreducible factors (inK[v,v⁻¹]) are of the form: Ψ(M)

where

s,jas,j = 0.

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χ_v 7→ χ

sχ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

s,jas,j = 0.

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χ_v 7→ χ

sχ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

Ψis aK-cyclotomic polynomial in one variable,

M is a primitive monomial of degree 0,i.e.,ifM=Q

s,jas,j = 0.

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χ_v 7→ χ

sχ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

Ψis aK-cyclotomic polynomial in one variable, M is a primitive monomial of degree 0,

i.e.,ifM=Q

s,jas,j = 0.

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χ_v 7→ χ

sχ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

s,jas,j = 0.

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χ_v 7→ χ

s_χ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

s,jas,j = 0.

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χ_v 7→ χ

s_χ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

s,jas,j = 0.

Ifp is a prime ideal ofZK such that Ψ(1)∈p, thenM is called ap-essential monomial for χ.

We say thatM is ap-essential monomial forW, if there exists χ∈Irr(W) such that M isp-essential forχ.

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χ_v 7→ χ

s_χ_v 7→ |W|/χ(1)

Theorem - Definition (C.)

where

s,jas,j = 0.

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Schur elements of G

2

: X

₀²

:= u

0

, X

₁²

:= −u

1

, Y

₀²

:= w

0

, Y

₁²

:= −w

1

. 2-essential in purple, 3-essential in green.

s₁= Φ₄(X₀X₁⁻¹)·Φ₄(Y₀Y₁⁻¹)·Φ₃(X₀Y₀X₁⁻¹Y₁⁻¹)·Φ₆(X₀Y₀X₁⁻¹Y₁⁻¹)

s2= 2·X₁²X₀⁻²·Φ3(X0Y0X₁⁻¹Y₁⁻¹)·Φ6(X0Y1X₁⁻¹Y₀⁻¹)

Φ4(x) =x²+ 1, Φ3(x) =x²+x+ 1, Φ6(x) =x²−x+ 1 Φ4(1) = 2 Φ3(1) = 3 Φ6(1) = 1

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Cyclotomic Hecke algebras

Lety be an indeterminate. Acyclotomic specializationofHis a ZK-algebra morphismφ:ZK[v,v⁻¹]→ZK[y,y⁻¹] of the form:

φ:vs,j7→yⁿ^s,j wherens,j∈Zfor allsandj.

The correspondingcyclotomic Hecke algebraHφis the Z^K[y,y⁻¹]-algebra obtained as the specialization of theHvia the morphismφ.

Example: The “spetsial” Hecke algebra is the algebra obtained via v_s,07→y, v_s,j 7→1 for1≤j≤o(s)−1, for all s.

Proposition (C.)

The algebraK(y)H_φ is split semisimple.

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Cyclotomic Hecke algebras

The correspondingcyclotomic Hecke algebraHφis theZ^K[y,y⁻¹]-algebra obtained as the specialization of theHvia the morphismφ.

Example: The “spetsial” Hecke algebra is the algebra obtained via v_s,07→y, v_s,j 7→1 for1≤j≤o(s)−1, for all s.

Proposition (C.)

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Cyclotomic Hecke algebras

Example: The “spetsial” Hecke algebra is the algebra obtained via v_s_,07→y, v_s,j 7→1 for1≤j≤o(s)−1, for all s.

Proposition (C.)

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Cyclotomic Hecke algebras

Example: The “spetsial” Hecke algebra is the algebra obtained via v_s_,07→y, v_s,j 7→1 for1≤j≤o(s)−1, for all s.

Proposition (C.)

The algebraK(y)Hφ is split semisimple.

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By “Tits’ deformation theorem”, we obtain that the specialization vs,j 7→1 induces the following bijections :

Irr(K(v)H) ↔ Irr(K(y)Hφ) ↔ Irr(W)

χ_v 7→ χ_φ 7→ χ

s_χ_v 7→ s_χ_φ 7→ |W|/χ(1)

Proposition

The Schur element s_χ_φ associated to the irreducible characterχ_φ ofK(y)Hφ is a Laurent polynomial iny of the form

s_χ_φ =ψ_χ,φ·y^a^χ,φ· Y

Φ∈CK

Φ(y)ⁿ^χ,φ,

where ψ_χ,φ ∈ZK,a_χ,φ∈Z,n_χ,φ∈NandC_K is a set of K-cyclotomic polynomials.

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χ_v 7→ χ_φ 7→ χ

s_χ_v 7→ s_χ_φ 7→ |W|/χ(1)

Proposition

The Schur element s_χ_φ associated to the irreducible characterχ_φ ofK(y)Hφ is a Laurent polynomial iny of the form

Φ∈CK

Φ(y)ⁿ^χ,φ,

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χ_v 7→ χ_φ 7→ χ

s_χ_v 7→ s_χ_φ 7→ |W|/χ(1)

Proposition

The Schur element s_χ_φ associated to the irreducible characterχ_φofK(y)Hφ is a Laurent polynomial iny of the form

Φ∈CK

Φ(y)ⁿ^χ,φ,

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Rouquier blocks, p-blocks and p-essential monomials

TheRouquier blocksof the cyclotomic Hecke algebraH_φ are the blocks of the algebra R_K(y)H_φ, where

RK(y) :=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Proposition

Set O:=ZK[y,y⁻¹]. Two irreducible characters χ, ψ∈Irr(K(y)Hφ) belong to the same Rouquier block if and only if there exist a finite sequence

χ₀, χ₁, . . . , χ_n∈Irr(K(y)H_φ) and a finite sequencep₁, . . . ,p_n of prime ideals of ZK such that

χ0=χandχn=ψ,

foralli (1≤i ≤n),χ_i−1andχi belong to the same block ofOp_iOHφ.

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Rouquier blocks, p-blocks and p-essential monomials

TheRouquier blocksof the cyclotomic Hecke algebraH_φ are the blocks of the algebra R_K(y)H_φ, where

RK(y) :=ZK[y,y⁻¹,(yⁿ−1)⁻¹_n≥1]

Proposition

Set O:=ZK[y,y⁻¹]. Two irreducible characters χ, ψ∈Irr(K(y)Hφ) belong to the same Rouquier block if and only if there exist a finite sequence

χ₀, χ₁, . . . , χ_n∈Irr(K(y)H_φ) and a finite sequencep₁, . . . ,p_n of prime ideals of ZK such that

χ0=χandχn=ψ,

foralli (1≤i ≤n),χ_i−1andχi belong to the same block ofOp_iOHφ.

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

φ:A → O is a cyclotomic specialization.

M₁, . . . ,M_k are all thep-essential monomials which are sent to 1 byφ. q₀:=pA.

q_j :=pA+ (M_j−1)Afor allj (1≤j ≤k). Q:={q0,q1, . . . ,qk}.

Theorem (C.)

Two irreducible characters χ, ψ∈Irr(W) are in the same block ofOpOHφ if and only if there exist a finite sequence χ0, χ1, . . . , χn∈Irr(W) and a finite sequence q_j₁, . . . ,q_j_n∈ Q such that

χ₀=χandχ_n=ψ,

for alli (1≤i ≤n),χ_i−1andχ_i belong to the same block ofA_q_jiH.

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

Theorem (C.)

χ₀=χandχ_n=ψ,

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

Theorem (C.)

χ₀=χandχ_n=ψ,

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

M₁, . . . ,M_k are all thep-essential monomials which are sent to 1 byφ.

q₀:=pA.

Theorem (C.)

χ₀=χandχ_n=ψ,

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

q₀:=pA.

Theorem (C.)

χ₀=χandχ_n=ψ,

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

q₀:=pA.

q_j :=pA+ (M_j−1)Afor allj (1≤j ≤k).

Q:={q0,q1, . . . ,qk}.

Theorem (C.)

χ₀=χandχ_n=ψ,

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

q₀:=pA.

Q:={q0,q1, . . . ,qk}.

Theorem (C.)

χ₀=χandχ_n=ψ,

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A:=ZK[v,v⁻¹].

O:=ZK[y,y⁻¹].

q₀:=pA.

Q:={q0,q1, . . . ,qk}.

Theorem (C.)

Two irreducible characters χ, ψ∈Irr(W) are in the same block ofOpOHφ if and only if there exist a finite sequence χ0, χ1, . . . , χn∈Irr(W) and a finite sequence qj1, . . . ,qjn∈ Q such that

χ₀=χandχ_n=ψ,

for alli (1≤i≤n),χ_i−1andχ_i belong to the same block ofA_q_jiH.