Group graded algebras and multiplicities bounded by a constant.
type: Item
instance of: scholarly article
title: Group graded algebras and multiplicities bounded by a constant.
zbMATH Open document ID: 1288.16036
author: Alessio Cirrito
author: Antonio Giambruno
published in: Journal of Pure and Applied Algebra
publication date: 2013-04-18T00:00:00Z
review text: 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 \(G\wr S_n\).
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 6155417
zbMATH Keywords: algebras with polynomial identity
zbMATH Keywords: group graded algebras
zbMATH Keywords: cocharacter sequences
zbMATH Keywords: graded codimensions
zbMATH Keywords: cocharacter multiplicities
zbMATH Keywords: representations of symmetric groups
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.06.005
OpenAlex ID: W1973715427
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DOI: 10.1016/J.JPAA.2012.06.005
description: scientific article; zbMATH DE number 6155417