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Blocs, isométries parfaites, catégories dérivées. (Blocks, perfect isometries, derived categories). (French) Zbl 0642.20007

Let R be a complete discrete valuation ring of characteristic 0 with residue field F of prime characteristic p. Let G and H be finite groups such that F and the quotient field of R are splitting fields for G and H. Let e and f be idempotents in ZRG and ZRH, respectively, and denote the corresponding groups of virtual characters by X(G,e) and X(H,f). Then any linear map I: X(G,e)\(\to X(H,f)\) is given by a unique virtual character \(\mu\) of \(G\times H\). Now suppose that \(I=I_{\mu}\) is an isometry. Then I and \(\mu\) are called perfect if the following two conditions are satisfied for \(g\in G\), \(h\in H:\) (1) \(\mu\) (g,h) is divisible in R by \(| C_ G(g)|\) and \(| C_ H(h)|\); (2) If \(\mu\) (g,h)\(\neq 0\) then g has order prime to p if and only if h has order prime to p. In this case I induces an isomorphism ZRGe\(\to ZRHf\) and a bijection between blocks preserving defect, numbers of irreducible ordinary and modular characters, heights, and elementary divisors of the Cartan matrix. Examples of perfect isometries appear in the Déligne-Lusztig theory of characters of finite reductive groups, in the Brauer-Dade theory of blocks with cyclic defect groups and in the representation theory of symmetric groups. Moreover, any equivalence between the bounded derived categories of finitely generated RGe-modules and RHf-modules induces a perfect isometry. The author finishes with two conjectures on perfect isometries and derived categories. After the paper appeared, the smallest Suzuki group was shown to be a counterexample for the prime 2 to the author’s conjecture (3.2); this example was suggested by J. Thompson.
Reviewer: B.Külshammer

MSC:

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
16D90 Module categories in associative algebras
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