Decomposition matrices for cyclotomic Hecke algebras

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Decomposition matrices for cyclotomic Hecke algebras

Maria Chlouveraki

University of Edinburgh

XIV International Conference on Representations of Algebras

August 11, 2010

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Complex reflection groups

Acomplex reflection groupW is a finite group of matrices with coefficients in a finite abelian extension K of Qgenerated by pseudo-reflections.

IfK =Q, then W is a Weyl group.

Shephard-Todd classification (1954) The irreducible complex reflection groups are:

the groups of the infinite series G(de,e,r) (withG(d,1,r)∼=Z/dZoS_r);

the exceptional groupsG4,G5, . . . , G37.

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Hecke algebras of complex reflection groups

Let W be a complex reflection group.

The group W has a presentation given by:

generators: S relations:

I braid relations;

I (s−1)(s−ζe_s)· · ·(s−ζ_e^e_s^s⁻¹) = 0. [ζe_s :=exp(2πi/es)]

Example:

G :=G(3,1,2) =hs,t|stst =tsts,s³ = 1,t² = 1i.

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Let q be an indeterminate and let A:=ZK[q,q⁻¹].

The cyclotomic Hecke algebraH_q(W) has a presentation given by:

generators: (T_s)s∈S

relations:

I braid relations;

I (T_s−1q^m^s,0)(T_s−ζ_e_sq^m^s,1)· · ·(T_s−ζ_e^e^s⁻¹

s q^m^s,es⁻¹) = 0.

Example: G =G(3,1,2)

H_q(G) =

* Ts,Tt

TsTtTsTt =TtTsTtTs,

(Ts−q^m^s,0)(Ts−ζ3q^m^s,1)(Ts−ζ₃²q^m^s,2) = 0, (T_t−q^m^t,0)(T_t+q^m^t,1) = 0

+ .

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Schur elements of Hecke algebras

(We make some assumptions.)

The algebra K(q)H_q(W) is semisimple. By Tits’s deformation theorem, we have a bijection:

Irr(K(q)H_q(W)) ↔ Irr(W)

χq 7→ χ.

Moreover, there exists a “canonical” symmetrizing form t :H_q(W)→A, such that

t= X

χ∈Irr(W)

1 s_χχq

where sχ is the Schur element ofH_q(W) associated with χ.

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We have that sχ belongs toA=ZK[q,q⁻¹] and it is a product of cyclotomic polynomials over K.

Definition

We define a_χ to be the smallest non-negative integer such that q^a^χs_χ∈ZK[q].

Example: Ifs_χ=q⁻¹+ 2 +q, then a_χ = 1.

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The decomposition matrix

Let

θ:A→C, q 7→ξ be a ring homomorphism. Set H_ξ:=C⊗_AH_q(W).

Theorem (Geck-Pfeiffer)

The algebra H_ξ is semisimple if and only if θ(s_χ)6= 0 for all χ∈Irr(W).

We have a well-defineddecomposition map

d_θ:R₀(K(q)H_q(W))→R₀(H_ξ) with corresponding decomposition matrix

D_θ= ( [E :M] )_E∈Irr(W_),_M∈Irr(H

ξ).

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Basic sets

Definition (Geck-Rouquier)

We say thatH_q(W) admits acanonical basic set B^can ⊂Irr(W) with respect to θ:A→Cif there exists a bijection

Irr(H_ξ) ↔ B^can M 7→ E_M such that

[E_M :M] = 1, and

if [E :M]6= 0, then either E =EM or aE >aEM.

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IfH_q(W) admits a canonical basic set B^can with respect toθ, then the decomposition matrix D_θ has the following form:

D_θ =







1 0 · · · 0

∗ 1 · · · 0 ... ... . .. ...

∗ ∗ · · · 1

∗ ∗ ∗ ∗







| {z }

Irr(Hξ)









 B^can











Irr(W)

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Theorem

The algebra H_q(W) admits a canonical basic set with respect to any specialization θ:A→C, if

1 W is a Weyl group;

[Geck-Rouquier, Geck, Geck-Jacon, C.-Jacon]

2 W is a complex reflection group of typeG(d,1,r);

[Dipper-James-Murphy, Geck-Rouquier, Ariki, Uglov, Jacon]

3 W is a complex reflection group of typeG(de,e,r) (for a certain choice of paremeters);

[Genet-Jacon]

4 W ∈ {G₄,G5,G8,G9,G10,G12,G16,G20,G22} (for certain choices of paremeters).

[C.-Miyachi]

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In the last case, we have been also able to show that there exists a subset B^opt⊂Irr(W) such that the decomposition matrixD_θ has the following form:

Dθ =







1 0 · · · 0 0 1 · · · 0 ... ... . .. ...

0 0 · · · 1

∗ ∗ ∗ ∗







| {z }

Irr(H_ξ)









 B^opt











Irr(W)