## 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

### Citations:

Zbl 0167.298; Zbl 0416.20006
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