Group graded algebras and multiplicities bounded by a constant.
Let \(G\) be a finite group and let \(A\) be a \(G\)-graded algebra satisfying a nontrivial polynomial identity over a field of characteristic zero. In the paper under review the authors study the \(G\)-graded cocharacters of \(A\). The irreducible components of the \(n\)-th cocharacter are described by tensor products \(\chi_{\lambda(1)}\otimes\cdots\otimes\chi_{\lambda(s)}\) of \(s=|G|\) irreducible characters of the product of symmetric groups \(S_{n_1}\times\cdots\times S_{n_s}\), \(n_1+\cdots+n_s=n\). The goal of the paper is to characterize the \(G\)-graded algebras \(A\) with the property that the corresponding multiplicities are bounded by a constant. The first description is that \(A\) satisfies a \(G\)-graded polynomial identity which does not hold for the algebra of \(2\times 2\) upper triangular matrices with any \(G\)-grading. The second description is in terms of representation theory of symmetric groups. There exists a constant \(N\) such that the diagrams \([\lambda(i)]\) such that \(\chi_{\lambda(1)}\otimes\cdots\otimes \chi_{\lambda(s)}\) has a nonzero multiplicity in the cocharacter sequence of \(A\) have \(\leq N\) boxes out of the first row and the first column of the diagram. In the special case of an Abelian group \(G\) the results can be translated in the language of representation theory of the wreath product \(G\wr S_n\).
scientific article; zbMATH DE number 6155417
scientific article