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Linear groups associated with elements of a group algebra. (English) Zbl 0629.20019
Let K be a field, \(M_ m(K)\) the \(m\times m\) matrices with entries in K, and \(S_ m\) the symmetric group of degree m. If c is a function from \(S_ m\) to K, let \(d_ c(A)=\sum_{\sigma \in S_ m}c(\sigma)\prod^{m}_{i=1}A_{i,\sigma (i)}\) and let S(c) be the group of non-singular matrices B satisfying \(d_ c(BX)=d_ c(X)\) for all \(X\in M_ m(K)\). S(c) gives information about the problem of determining conditions for equality of decomposible symmetric tensors. The authors give necessary and sufficient conditions for a matrix to belong to S(c). In addition, they find properties that matrices which belong to S(c) satisfy.
Reviewer: E.Spiegel

MSC:
20G15 Linear algebraic groups over arbitrary fields
15A30 Algebraic systems of matrices
16Dxx Modules, bimodules and ideals in associative algebras
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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References:
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