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 when it comes to blocks of the Hecke algebras has made attainable the generalization of this idea 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 advanced mirrored image teams.
This quantity bargains a radical research of symmetric algebras, protecting subject matters resembling block thought, illustration idea and Clifford thought, and will additionally function an creation to the Hecke algebras of complicated mirrored image groups.

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6. Assume that we have a symmetrizing form t on A. , EndKA (V ) ∼ = K) and let sV be its Schur element with respect to the induced form tK on KA. Then sV ∈ OK . 52 2 On Blocks Proof. Let R ⊆ K be a valuation ring with O ⊆ R. 4, we can assume that V affords a representation ρ : KA → Mn (K) such that ρ(a) ∈ Mn (R) for all a ∈ A. Let B be an O-basis of A and let B be its dual with respect to t. 2). All terms in the sum lie in R and so sV ∈ R. Since this holds for all valuation rings R in K containing O, property (V3) implies that sV ∈ OK .

E∈E ef . 2, there exists In the above context (1 is a sum of blocks), let us denote by Bl(A) the set of all the blocks of A. 3 implies that the category A mod of A-modules is a direct sum of the categories associated with the blocks: A mod → ˜ e∈Bl(A) Ae mod. 1 General Results 23 In particular, every representation of the O-algebra Ae defines (by composition with πe ) a representation of A and we say, abusing the language, that it “belongs to the block e”. Every indecomposable representation of A belongs to one and only one block.

Let (v1 , . . , vn ) be a K-basis of V and B an O-basis for A. Let V˜ be the R-submodule of V spanned by the finite set {vi b | 1 ≤ i ≤ n, b ∈ B}. Then V˜ is invariant under the action of RA and hence a finitely generated RA-module. Since it is contained in a K-vector space, it is also torsion-free. So (V2) implies that V˜ is an RA-lattice (a finitely generated RA-module which is free as an R-module) such that K ⊗R V˜ ∼ = V . Thus, any R-basis of V˜ is also a K-basis of V with the required property.

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