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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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