A tower construction for the radical in Brauer's centralizer algebras
type: Item
instance of: scholarly article
title: A tower construction for the radical in Brauer's centralizer algebras
zbMATH Open document ID: 0834.16015
published in: Journal of Algebra
publication date: 1996-03-24T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/31707
review text: The Brauer centralizer algebras \({\mathcal A}_f^{(x)}\) are studied in the case that the multiplication constant \(x\) is a rational integer. If \(F_f\) is the set of all graphs with \(2f\) points and \(f\) lines in which every point has degree 1 (1-factors on \(2f\) points), then \({\mathcal A}_f^{(x)}\) is an \(\mathbb{R}\)-algebra with \(F_f\) as a basis. The set of all diagrams with at least \(k\) horizontal lines in the rows is an ideal of \({\mathcal A}^{(x)}_f\) denoted by \({\mathcal A}^{(x)}_f (k)\). A description of the radical of the quotient \({\mathcal A}^{(x)}_f (k)/{\mathcal A}^{(x)}_f (k+1)\) was obtained earlier by authors in terms of \((m,k)\)-partial 1-factors. The tower construction is a method to study towers of semisimple algebras. The first step in this paper is to modify the tower construction for the purpose of its application in the case when the algebras of the tower are not semisimple. Let \(R\) be a finite dimensional semisimple \(R\)-algebra with matrix ring decomposition \(R=\bigoplus_\mu R_\mu\), \(V_\mu\) is the corresponding irreducible representation of \(R\), \(M=\bigoplus g_\mu V\) is a left \(R\)-module and \(M'\) is a right \(R\)-module with a linear isomorphism \(M \cong M'\) (\(m \rightsquigarrow m'\)) such that \((rm)'=m'r'\). For every special \(\mathbb{R}\)-bilinear map \(\varepsilon:M \times M'\to R\) is defined the algebra \({\mathcal A}(R, M, \varepsilon)\) with \(\mathbb{R}\)-basis \(M'\otimes_RM\) and multiplication depending on \(\varepsilon\). The map \(\varepsilon\) is determined by the sequence of symmetric matrices \((H_\mu)\). The structure of the algebra \({\mathcal A} (R, M, \varepsilon)\) is described by the characteristic sequence \((H_\mu):{\mathcal A}(R, M, \varepsilon) \cong \bigoplus_\mu \Lambda (H_\mu)\), where \(\Lambda (H_\mu)\) is an \(\mathbb{R}\)-algebra with \(R\)-basis \(\mathbb{R}^d \otimes \mathbb{R}^d\) and multiplication defined by \(H_\mu\). The general construction is used in the situation…
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 579549
zbMATH Keywords: Brauer centralizer algebras
zbMATH Keywords: tower construction
zbMATH Keywords: towers of semisimple algebras
author: Philip J. Hanlon
author: David Wales
reviewed by: A. I. Kashu
MaRDI profile type: Publication
OpenAlex ID: W2037841247
DOI: 10.1006/JABR.1994.1089
description: scientific article; zbMATH DE number 579549