Blocks and Families for Cyclotomic Hecke Algebras by Maria Chlouveraki

By Maria Chlouveraki

The definition of Rouquier for the households of characters brought by means of Lusztig for Weyl teams by way of blocks of the Hecke algebras has made attainable the generalization of this thought to the case of complicated mirrored image teams. the purpose of this e-book is to check the blocks and to figure out the households of characters for all cyclotomic Hecke algebras linked to complicated mirrored image teams.
This quantity bargains a radical examine of symmetric algebras, masking issues akin to block conception, illustration thought and Clifford idea, and will additionally function an advent to the Hecke algebras of advanced mirrored image groups.

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13. 12. Let e be an idempotent of Op A whose image e¯ in (Op /q )A is central. Then e is central. Proof. We have the following equality: Op A = eOp Ae ⊕ eOp A(1 − e) ⊕ (1 − e)Op Ae ⊕ (1 − e)Op A(1 − e). , eOp A(1 − e) ⊆ qOp A and (1 − e)Op Ae ⊆ qOp A. 1 General Results 29 Since e and (1 − e) are idempotents, we get eOp A(1 − e) ⊆ q eOp A(1 − e) and (1 − e)Op Ae ⊆ q (1 − e)Op Ae. However, q is contained in the maximal ideal p of Op . By Nakayama’s lemma, we obtain that eOp A(1 − e) = (1 − e)Op Ae = {0}.

Hence, g ∈ / Gχ¯ for all χ ¯ ¯(χ) ¯ = χ∈ Tr(G¯b , e¯(χ)). ¯ Thus, 2. Note that b = χ∈ ¯e ¯ ¯ B ¯ B/G ¯ b Tr(G, ¯b) = ¯ e¯(Ω), Tr(G, e¯(χ)) ¯ = ¯ χ∈ ¯ B/G ¯ | Ω∩ ¯ B=∅} ¯ {Ω ¯ by the definition of e¯(Ω). Now let G∨ := Hom(G, K × ). We suppose that K = F . The multiplication of the characters of KA by the characters of KG defines an action of the group G∨ on Irr(KA). This action is induced by the operation of G∨ on the algebra A, which is defined in the following way: ¯ g ∈ G. aag for all ξ ∈ G∨ , a ¯ ∈ A, ξ · (¯ aag ) := ξ(g)¯ In particular, G∨ acts on the set of blocks of A.

Then 1 − eF = eE−F ∈ ZA, which means that E − F is on ZA. If E − F = ∅, then E − F contains an element of PE (ZA), thus contradicting the definition of F . Thus F = E and PE (ZA) is a partition of E. Now let us assume that • O is a commutative integral domain with field of fractions F , • K is a field extension of F , • A is an O-algebra, free and finitely generated as an O-module. Suppose that the K-algebra KA := K ⊗O A is semisimple. Then KA is isomorphic, by assumption, to a direct product of simple algebras: KA ∼ = Mχ , χ∈Irr(KA) where Irr(KA) denotes the set of irreducible characters of KA and Mχ is a simple K-algebra.

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