Semisimple commutative algebras with positive bases
type: Item
instance of: scholarly article
title: Semisimple commutative algebras with positive bases
zbMATH Open document ID: 0943.20009
author: David Chillag
published in: Journal of Algebra
publication date: 1999-10-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 05E30
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1247576
zbMATH Keywords: Brauer characters
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: centers
zbMATH Keywords: group algebras
zbMATH Keywords: \(C\)-algebras
zbMATH Keywords: table algebras
zbMATH Keywords: conjugacy classes
zbMATH Keywords: principal indecomposable characters
zbMATH Keywords: values of characters
zbMATH Keywords: finite groups
reviewed by: M. R. Darafsheh
MaRDI profile type: Publication
full work available at URL: jabr.1998.7553
OpenAlex ID: W1987863621
cites work: Q3804890
cites work: On table algebras and applications to finite group theory
cites work: Table algebras and applications to products of characters in finite groups
cites work: Q3943082
cites work: Shorter Notes: Character Values of Finite Groups as Eigenvalues of Nonnegative Integer Matrices
cites work: Q3798849
cites work: Nonnegative matrices, Brauer characters, normal subsets, and powers of representation modules
cites work: Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes
cites work: Q5645586
cites work: Bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial
cites work: Q4103728
cites work: Q5843905
cites work: Proof of a conjecture about the exponent of primitive matrices
cites work: Q4833884
DOI: 10.1006/JABR.1998.7553
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description: scientific article; zbMATH DE number 1247576