Basic sets for cyclotomic Hecke algebras

(1)

Basic sets for cyclotomic Hecke algebras

Maria Chlouveraki (joint work with Nicolas Jacon)

University of Edinburgh

Nikolaus Conference 2009 December 11, 2009

(2)

Iwahori-Hecke algebras

Let

(W,S) be a finite Weyl group,

u,v:W →Zbe class functions on W (write u(w) =:uw), q be an indeterminate.

The Iwahori-Hecke algebra Hof W is generated over A:=Z[q,q⁻¹] by the elements (Tw)w∈W, whose multiplication is determined by the following rules:







TwT_w⁰ =T_ww⁰, if`(ww⁰) =`(w) +`(w⁰). (Ts−q^2u^s)(Ts+q^2v^s) = 0, for all s ∈S.

Maria Chlouveraki (Edinburgh) Basic sets for cyclotomic Hecke algebras December 11, 2009 2 / 11

(3)

Iwahori-Hecke algebras

Let

u,v:W →Zbe class functions on W (write u(w) =:uw), q be an indeterminate.







(4)

Iwahori-Hecke algebras

Let

u,v:W →Zbe class functions on W (writeu(w) =:uw),

q be an indeterminate.







(5)

Iwahori-Hecke algebras

Let

u,v:W →Zbe class functions on W (writeu(w) =:uw), q be an indeterminate.







(6)

Iwahori-Hecke algebras

Let

u,v:W →Zbe class functions on W (writeu(w) =:uw), q be an indeterminate.







T_wT_w⁰ =T_ww⁰, if`(ww⁰) =`(w) +`(w⁰).

(Ts−q^2u^s)(Ts+q^2v^s) = 0, for all s ∈S.

(7)

Let K :=Q(q) be the field of fractions ofA.

Then the algebra KH:=K ⊗_AH

is split semisimple. By Tits’ deformation theorem, we have a canonical bijection:

Irr(KH)↔Irr(W).

Therefore, if Λ is an indexing set for Irr(W), i.e., Irr(W) ={E^λ|λ∈Λ}, then

Irr(KH) ={V^λ |λ∈Λ}.

(8)

Let K :=Q(q) be the field of fractions ofA. Then the algebra KH:=K ⊗_AH

is split semisimple.

By Tits’ deformation theorem, we have a canonical bijection:

Irr(KH)↔Irr(W).

(9)

Irr(KH)↔Irr(W).

(10)

Irr(KH)↔Irr(W).

Therefore, if Λ is an indexing set for Irr(W), i.e., Irr(W) ={E^λ|λ∈Λ},

then

(11)

Irr(KH)↔Irr(W).

(12)

The a-function

The linear formt :H →Agiven by t(Tw) =

1, if w = 1 0, if w 6= 1

is a symmetrizing form for the Iwahori-Hecke algebraH,

i.e., for all h,h⁰ ∈ H,

1 t(hh⁰) =t(h⁰h),

2 the bilinear form (h,h⁰)7→t(hh⁰) is non-degenerate.

(13)

The a-function

1, if w = 1 0, if w 6= 1

is a symmetrizing form for the Iwahori-Hecke algebraH, i.e., for all h,h⁰ ∈ H,

1 t(hh⁰) =t(h⁰h),

(14)

The a-function

1, if w = 1 0, if w 6= 1

1 t(hh⁰) =t(h⁰h),

(15)

The a-function

1, if w = 1 0, if w 6= 1

1 t(hh⁰) =t(h⁰h),

(16)

We have

t= X

V∈Irr(KH)

1 s_V χ_V

where s_V ∈Ais the Schur element of the irreducible characterχ_V.

Definition

Recall that Irr(KH) ={V^λ|λ∈Λ}.Set s_λ:=s_Vλ and a_λ:=−valuation(s_λ).

Example: Ifs_λ=q⁻²+q⁻¹+q³, thena_λ= 2.

(17)

We have

t= X

V∈Irr(KH)

1 s_V χ_V

Definition

(18)

We have

t= X

V∈Irr(KH)

1 s_V χ_V

Definition

(19)

The decomposition matrix

Let

θ:A=Z[q,q⁻¹]→L, q 7→ξ

be a ring homomorphism such that Lis the field of fractions of θ(A).

We have a good parametrization for Irr(KH). What about the simple modules of the specialized algebra LH?

Assume that LH is split. LetR₀(KH) (resp. R₀(LH)) be the Grothendieck group of finitely generatedKH-modules (resp.

LH-modules). It is generated by the classes [U] of the simpleKH-modules (resp. LH-modules)U.

(20)

The decomposition matrix

Let

θ:A=Z[q,q⁻¹]→L, q 7→ξ

(21)

The decomposition matrix

Let

θ:A=Z[q,q⁻¹]→L, q 7→ξ

(22)

We have a well-defined decomposition map d_θ:R₀(KH)→R₀(LH) such that, for all λ∈Λ , we have

dθ([V^λ]) = X

M∈Irr(LH)

[V^λ:M][M].

The matrix

Dθ =

[V^λ:M]

λ∈Λ,M∈Irr(LH)

is called the decomposition matrix associated with θ.

(23)

We have a well-defined decomposition map d_θ:R₀(KH)→R₀(LH) such that, for all λ∈Λ , we have

dθ([V^λ]) = X

M∈Irr(LH)

[V^λ:M][M].

The matrix

Dθ =

[V^λ:M]

λ∈Λ,M∈Irr(LH)

is called the decomposition matrix associated with θ.

(24)

Basic sets

We say thatH admits acanonical basic set B ⊂Λ with respect to θ:A→Lif and only if the following two conditions are satisfied:

1 For all M ∈Irr(LH), there existsλM ∈ B such that

I [V^λ^M :M] = 1, and

I if [V^µ:M]6= 0, then eitherµ=λor aµ>aλ_M.

2 The map

Irr(LH) → B M 7→ λ_M is a bijection.

(25)

Basic sets

1 For all M ∈Irr(LH), there existsλ_M ∈ B such that

I [V^λ^M :M] = 1, and

2 The map

(26)

Basic sets

I [V^λ^M :M] = 1, and

2 The map

(27)

Basic sets

I [V^λ^M :M] = 1, and

I if [V^µ:M]6= 0, then eitherµ=λoraµ>aλ_M.

2 The map

(28)

Basic sets

I [V^λ^M :M] = 1, and

I if [V^µ:M]6= 0, then eitherµ=λoraµ>aλ_M.

2 The map

(29)

IfL is a field of characteristic 0, then the existence of canonical basic sets for all finite Weyl groups has been proved

by Geck and Rouquier, when us =a∈Nandvs = 0 for all s ∈S, by Geck and Jacon, when us ∈Nandvs = 0 for all s ∈S.

Theorem (C.-Jacon)

Let Lbe a field of characteristic 0. The algebraH admits a canonical basic set with respect to any specializationθ:A→L.

Idea of the proof: Let s ∈S. If us ≥vs, then (us −vs,0)→(us,vs), B 7→ B. Ifu_s <v_s, then

(v_s−u_s,0)→(0,v_s −u_s)→(u_s,v_s), B 7→ B^ε={ε(λ)|λ∈ B} where ε: Λ→Λ is a bijection induced on the set of irreducible representations of KHby the action of the cyclic group Z/2Z.

(30)

by Geck and Rouquier, when us =a∈Nandvs = 0 for all s ∈S,

by Geck and Jacon, when us ∈Nandvs = 0 for all s ∈S. Theorem (C.-Jacon)

(31)

by Geck and Rouquier, when us =a∈Nandvs = 0 for all s ∈S, by Geck and Jacon, when us ∈Nandvs = 0 for alls ∈S.

Theorem (C.-Jacon)

(32)

Theorem (C.-Jacon)

(33)

Theorem (C.-Jacon)

Idea of the proof: Let s ∈S. If us ≥vs, then (us −vs,0)→(us,vs),

B 7→ B.

Ifu_s <v_s, then

(34)

Theorem (C.-Jacon)

Idea of the proof: Let s ∈S. If us ≥vs, then (us −vs,0)→(us,vs), B 7→ B.

Ifu_s <v_s, then

(35)

Theorem (C.-Jacon)

Ifu_s <v_s, then

(v_s−u_s,0)→(0,v_s −u_s)→(u_s,v_s),

B 7→ B^ε={ε(λ)|λ∈ B} where ε: Λ→Λ is a bijection induced on the set of irreducible representations of KHby the action of the cyclic group Z/2Z.

(36)

Theorem (C.-Jacon)

Ifu_s <v_s, then

(v_s−u_s,0)→(0,v_s −u_s)→(u_s,v_s), B 7→ B^ε={ε(λ)|λ∈ B}

where ε: Λ→Λ is a bijection induced on the set of irreducible representations of KHby the action of the cyclic group Z/2Z.

(37)

An example: Type A

Let W be the symmetric group S_n andS the set of transpositions (i,i + 1), fori = 1, . . . ,n−1.

All elements of S are conjugate inW. Hence, the generators of the Iwahori-Hecke algebraH of S_n satisfy

(T_s−q^2u)(T_s+q^2v) = 0, ∀s ∈S for some integers u,v.

The indexing set Λ forIrr(W) is the set P(n) :=

(

(λ₁, λ₂, . . . , λ_r)

λ₁ ≥λ₂ ≥ · · · ≥λ_r ≥1 and

r

X

i=1

λ_i =n )

of partitions of n. The bijectionεon Λ induced by the action ofZ/2Zis simply the conjugation of partitions.

(38)

An example: Type A

Let W be the symmetric group S_n andS the set of transpositions (i,i + 1), fori = 1, . . . ,n−1. All elements of S are conjugate inW.

Hence, the generators of the Iwahori-Hecke algebraH of S_n satisfy (T_s−q^2u)(T_s+q^2v) = 0, ∀s ∈S

for some integers u,v.

(

(λ₁, λ₂, . . . , λ_r)

λ₁ ≥λ₂ ≥ · · · ≥λ_r ≥1 and

r

X

i=1

λ_i =n )

(39)

An example: Type A

Let W be the symmetric group S_n andS the set of transpositions (i,i + 1), fori = 1, . . . ,n−1. All elements of S are conjugate inW. Hence, the generators of the Iwahori-Hecke algebraH of S_n satisfy

(

(λ₁, λ₂, . . . , λ_r)

λ₁ ≥λ₂ ≥ · · · ≥λ_r ≥1 and

r

X

i=1

λ_i =n )

(40)

An example: Type A

(

(λ₁, λ₂, . . . , λ_r)

λ₁ ≥λ₂ ≥ · · · ≥λ_r ≥1 and

r

X

i=1

λ_i =n )

of partitions of n.

The bijectionεon Λ induced by the action ofZ/2Zis simply the conjugation of partitions.

(41)

An example: Type A

(

(λ₁, λ₂, . . . , λ_r)

λ₁ ≥λ₂ ≥ · · · ≥λ_r ≥1 and

r

X

i=1

λ_i =n )

of partitions of n. The bijectionεon Λ induced by the action ofZ/2Zis

(42)

Proposition (Dipper-James, C.-Jacon)

Let θ:Z[q,q⁻¹]→Lbe a specialization such that θ(q)^2u−2v is a primitive root of 1 of order e >1.

1 Ifu >v, thenHadmits a canonical basic set B ⊂Λ with respect toθ and we have

B= Reg_e(n)

where the set Reg_e(n) ofe-regular partitions is defined by

λ= (λ1, . . . , λr)∈/Reg_e(n) ⇐⇒ ∃i ∈N, λi =· · ·=λi+e−1 6= 0.

2 Ifu <v, thenHadmits a canonical basic set B ⊂Λ with respect toθ and we have

B= Res_e(n)

where the set Res_e(n) ofe-restricted partitions is defined by λ= (λ₁, . . . , λ_r)∈Res_e(n) ⇐⇒ ∀i ∈N, λ_i−λ_i+1 <e.

(43)

B= Reg_e(n)

B= Res_e(n)

(44)

B= Reg_e(n)

B= Res_e(n)