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 idea to the case of advanced 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 advanced mirrored image teams.
This quantity deals a radical research of symmetric algebras, overlaying themes corresponding to block concept, illustration idea and Clifford concept, and will additionally function an advent to the Hecke algebras of complicated mirrored image groups.
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Extra info for Blocks and Families for Cyclotomic Hecke Algebras
10 is also true if we replace Spec1 (O) by Spec(O). However, in this case, Op is not always a discrete valuation ring. 10, we can transfer the problem of the determination of the blocks of A to that of the determination of the p-blocks of A, where p runs over the set or prime ideals (of height 1) of O. 11. Let C6 be the cyclic group of order 6 and A := Z[C6 ]. The group C6 (and thus the algebra Q(ζ6 )[C6 ]) has 6 irreducible characters χ1 , . . , χ6 , where • • • • • χ1 is the trivial character, χ4 (g) ∈ R for all g ∈ C6 , χ2 is the conjugate of χ6 , χ3 is the conjugate of χ5 , χ2 (g)χ3 (g) = χ4 (g) for all g ∈ C6 .
10 implies that all the irreducible characters of C6 belong to the same block of A. 3 Blocks and Quotient Blocks Let p be a prime ideal of O. Then Op is a local ring, whose maximal ideal is p := pOp . Let q be a prime ideal of O such that q ⊆ p. Then q := qOp is a prime ideal of Op . The natural surjection πq : Op Op /q extends to a (Op /q )A, which in turn induces a morphism morphism πq : Op A πq : ZOp A → Z(Op /q )A. 13. 12. Let e be an idempotent of Op A whose image e¯ in (Op /q )A is central.
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.
Blocks and Families for Cyclotomic Hecke Algebras by Maria Chlouveraki