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 inspiration to the case of complicated mirrored image teams. the purpose of this ebook 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, masking subject matters resembling block concept, illustration thought and Clifford concept, and will additionally function an creation to the Hecke algebras of complicated mirrored image groups.
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Extra resources for Blocks and Families for Cyclotomic Hecke Algebras
Reg = cA . If A is a symmetric algebra with a symmetrizing form t, we obtain a symmetrizing form tK on KA by extension of scalars. Every irreducible character χ ∈ Irr(KA) is a trace function on KA and thus we can deﬁne χ∨ ∈ ZKA. 9. For all χ ∈ Irr(KA), we call Schur element of χ with respect to t and denote by sχ the element of K deﬁned by sχ := ωχ (χ∨ ). 10. For all χ ∈ Irr(KA), sχ ∈ OK . 6. 11. Let O := Z, A := Z[G] (G a ﬁnite group) and t the canonical symmetrizing form. If K is an algebraically closed ﬁeld of characteristic 0, then KA is a split semisimple algebra and sχ = |G|/χ(1) for all χ ∈ Irr(KA).
We deﬁne the map pK : R0+ (KA) → Maps(A, K[x]) [V ] → (a → characteristic polynomial of ρV (a)). Considering Maps(A, K[x]) as a semigroup with respect to multiplication, the map pK is a semigroup homomorphism. , the set of all characters χV , where V is a simple KA-module). The following result is known as the “Brauer-Nesbitt lemma” (cf. , Lemma 2). 2. 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 ]).
With only this exception, all the results in this section are well-known and mostly taken from the seventh chapter of . 1 Grothendieck Groups Let O be an integral domain and K a ﬁeld containing O. Let A be an O-algebra free and ﬁnitely generated as an O-module. 4 Representation Theory of Symmetric Algebras 49 Let R0 (KA) be the Grothendieck group of ﬁnite-dimensional KA-modules. Thus, R0 (KA) is generated by expressions [V ], one for each KA-module V (up to isomorphism), with relations [V ] = [V ] + [V ] for each exact sequence 0 → V → V → V → 0 of KA-modules.