By Maria Chlouveraki
The definition of Rouquier for the households of characters brought via Lusztig for Weyl teams by way of blocks of the Hecke algebras has made attainable the generalization of this thought to the case of advanced mirrored image teams. the purpose of this booklet 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 an intensive research of symmetric algebras, protecting issues equivalent to block conception, illustration conception and Clifford thought, and will additionally function an advent to the Hecke algebras of advanced mirrored image groups.
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Extra resources for Blocks and Families for Cyclotomic Hecke Algebras
Two KA-modules V, V give rise to the same element in R0 (KA), if V and V have the same composition factors, counting multiplicities. It follows that R0 (KA) is free abelian with basis given by the isomorphism classes of simple modules. Finally, let R0+ (KA) be the subset of R0 (KA) consisting of elements [V ], where V is a ﬁnite-dimensional KA-module. 1. Let x be an indeterminate over K and Maps(A, K[x]) the K-algebra of maps from A to K[x] (with pointwise multiplication of maps as algebra multiplication).
Pn ∈ Spec1 (O) such that • χ0 = χ and χn = ψ, • for all j (1 ≤ j ≤ n), χj−1 ∼pj χj . Proof. Let us denote by ∼ the equivalence relation on Irr(KA) deﬁned as the transitive closure of the relation “there exists p ∈ Spec1 (O) such that χ ∼p ψ”. Thus, we have to show that χ ∼ ψ if and only if χ and ψ belong to the same block of A. We will ﬁrst show that the equivalence relation ∼ is ﬁner than the relation “being in the same block of A”. Let B be a block of A. Then B is a subset of Irr(KA) such that χ∈B eχ ∈ A.
Assume that Irr(KA) is a linearly independent subset of HomK (KA, K). Then the map pK is injective. Proof. Let V, V be two KA-modules such that pK ([V ]) = pK ([V ]). Since [V ], [V ] only depend on the composition factors of V, V , we may assume that V, V are semisimple modules. Let n n ai Vi and V = V = i=1 bi V i , i=1 where the Vi are pairwise non-isomorphic simple KA-modules and ai , bi ≥ 0 for all i. We have to show that ai = bi for all i. If, for some i, we have both ai > 0 and bi > 0, then we can write V = Vi ⊕ V˜ and V = Vi ⊕ V˜ .