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Subpair multiplicities in finite groups. (English) Zbl 0596.20009

Let G be a finite group, p a prime and b a p-block of G. A b-subpair is a pair (X,e) consisting of a p-subgroup X of G and a block e of \(C_ G(X)\) such that \(e^ G=b\). A Brauer b-element is a pair (x,\(\epsilon)\) where \((<x>,\epsilon)\) is a b-subpair. The authors define \(m_{G,b}^{(x,\epsilon)}(X,e)\), the ”multiplicity of (X,e) on the b- subsection of (x,\(\epsilon)\)”, as the dimension of a certain subquotient of the center of the block. These multiplicities refine the ”multiplicities of lower defect groups”, introduced by R. Brauer [in Ill. J. Math. 13, 53-73 (1969; Zbl 0167.298)], and are better adapted to the local approach to representation theory given by J. L. Alperin and M. Broué [in Ann. Math., II. Ser. 110, 143-157 (1979; Zbl 0416.20006)]. The authors apply their general results to the groups \(G=GL(n,q)\) and \(G=U(n,q^ 2)\), where \(p\nmid q\) to obtain a reduction theorem for the multiplicities of an arbitrary block to those of the principal block of a suitable subgroup. These latter multiplicities are related to the number of unipotent conjugacy classes and the number of partitions.
Reviewer: B.Külshammer

MSC:

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
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