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Applications of Frobenius algebras to representation theory of Schur algebras. (English) Zbl 0914.20001
Let $$R$$ be a commutative ring with identity, $$G$$ a finite group, and $$RG$$ the group ring of $$G$$ over $$R$$, with basis $$\{u_g\mid g\in G\}$$ in one-to-one correspondence with $$G$$. Let $$\{E_g\mid g\in G\}$$ be a partition of $$G$$, with $$g\in E_g$$ and $$E_g^{-1}=E_{g^{-1}}$$ for all $$g\in G$$, and let $$G_0$$ be a set of representatives of the $$E_g$$ and put $$s_g=\sum_{x\in E_g} u_x$$. If $$S=\bigoplus_{g\in G_0} Rs_g$$ is a subalgebra with unit $$1_S$$ of $$RG$$, $$S$$ is said to be a Schur algebra in $$RG$$.
The authors consider here the relationship between indecomposable modules for $$RG$$ and those for $$S$$, for two types of Schur algebras. The first type is the double coset algebra determined by a subgroup $$H$$ of $$G$$, with $$| H|$$ a unit in $$R$$. It is $$\varepsilon RG\varepsilon$$, where $$\varepsilon=| H|^{-1}\sum_{h\in H} u_h$$. This is actually discussed in the more general context of Hecke algebras. The second setting is that of fixed rings of automorphism groups arising from automorphisms of $$G$$.
Furthermore, a theory of characters and class functions for Schur algebras is presented. In all, the paper contains a great deal of information of interest to those who work in this area.
##### MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20C25 Projective representations and multipliers 20C30 Representations of finite symmetric groups 20C20 Modular representations and characters 16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) 16S34 Group rings
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