Semisimple commutative algebras with positive bases
Let \(G\) be a finite group and let \(\text{Irr}(G)\) denote the set of irreducible complex characters of \(G\). The set of complex characters \(G\) can be given the structure of a finite-dimensional algebra over \(\mathbb{C}\) with \(\text{Irr}(G)\) as a basis. Also the center of the group algebra \(\mathbb{C} G\) has a similar property. To make a concurrent studying model for both algebras the notion of \(C\)-algebra was defined by \textit{Y. Kawada} [Proc. Phys.-Math. Soc. Japan, III. Ser. 24, 97-109 (1942; Zbl 0063.03172)] and that of table-algebra by \textit{Z. Arad} and \textit{H. I. Blau} [J. Algebra 138, No. 1, 137-185 (1991; Zbl 0790.20015)] and \textit{Z. Arad}, \textit{E. Fisman} and \textit{H. I. Blau} [J. Algebra 138, No. 1, 186-194 (1991; Zbl 0790.05097)]. This model of a finite-dimensional algebra has useful applications in finite group theory especially some properties of the product of complex irreducible characters and the products of conjugacy classes are easily obtained using this model. However, this model does not include the Brauer or the principal indecomposable characters. In the paper under review a model of a finite-dimensional algebra over any subfield \(F\) of the field of real numbers \(\mathbb{R}\) is studied such that it includes the Brauer and the principal indecomposable characters. In this model the emphasis is on the characters as eigenvalues of the regular representation of algebra elements so that results can be obtained about the values of characters. In the paper, examples and applications are given to see how this model of algebra is a useful tool in studying characters, ordinary or Brauer, and conjugacy classes of finite groups.
scientific article; zbMATH DE number 1247576
scientific article