On algebras satisfying symmetric identities
type: Item
instance of: scholarly article
title: On algebras satisfying symmetric identities
zbMATH Open document ID: 0809.16022
DOI: 10.1007/BF01196669
published in: Archiv der Mathematik
publication date: 1995-03-29T00:00:00Z
review text: Consider the non-commutative polynomials \[ s_ n = \sum_{\pi \in \text{Sym}(n)} x_{\pi(1)} \dots x_{\pi(n)}\quad \text{and} \quad d_ n = \sum_{\pi \in \text{Sym}(n)} x_{\pi(1)} y_ 1 \dots y_{n - 1} x_{\pi(n)}. \] Let \(R\) be an algebra over a field of characteristic \(p > 0\). We show that if \(s_ n = 0\) (\(d_ n = 0\), resp.) is a polynomial identity on \(R,\) then \(s_{kn} = 0\), \((d_{(n - 1) k^ 2 + 1} = 0\), resp.) is an identity on the \(k \times k\) matrix algebra over \(R\). These results generalize an earlier theorem due to Chang and Zalesskij on the identities of the matrix algebra over a field of positive characteristic. Motivated by Regev's results on algebras (over a characteristic zero field) satisfying a Capelli identity we investigate what can be said about the modular cocharacter sequence of \(R\) if we assume that \(R\) satisfies the identity \(d_ n = 0\), and prove that certain types of irreducible \(\text{Sym}(n)\)-modules can not occur in the modular cocharacter.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 16R30
zbMATH DE Number: 701016
zbMATH Keywords: polynomial identity
zbMATH Keywords: matrix algebra
zbMATH Keywords: Capelli identity
zbMATH Keywords: modular cocharacter sequence
zbMATH Keywords: irreducible \(\text{Sym}(n)\)-modules
author: Matyas Domokos
reviewed by: Matyas Domokos
MaRDI profile type: Publication
cites work: Q5808057
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full work available at URL: https://doi.org/10.1007/bf01196669
OpenAlex ID: W2003013354
description: scientific article; zbMATH DE number 701016