Central polynomials and matrix invariants
type: Item
instance of: scholarly article
title: Central polynomials and matrix invariants
zbMATH Open document ID: 0874.16017
DOI: 10.1007/BF02785544
author: Antonio Giambruno
author: Angela Valenti
published in: Israel Journal of Mathematics
publication date: 1997-09-24T00:00:00Z
review text: Since the early 70's when Formanek and Razmyslov discovered central polynomials for matrices, many new central polynomials have been constructed. In the paper under review the authors consider multilinear polynomials \(F^{\lambda,\mu}\) of degree \(2n^2\), skew-symmetric in two disjoint sets of \(n^2\) variables and indexed by a pair of partitions \(\lambda,\mu\) of \(n^2\). These polynomials appeared first in \textit{A. Regev} [Commun. Algebra 8, 1417-1467 (1980; Zbl 0437.16012)] and are known to be either polynomial or central identities for the \(n\times n\) matrix algebra \(M_n(K)\) over a field \(K\) of characteristic 0. The purpose of the present paper is to describe conditions which guarantee that \(F^{\lambda,\mu}\) is a central polynomial. It turns out that this happens only in the case when \(\lambda\) and \(\mu\) are derived in a suitable way from the partition \(\delta=(1,3,\ldots,2n-3,2n-1)\). The authors give an explicit formula which involves characters of the symmetric group and allows to evaluate \(F^{\lambda,\mu}\) in \(M_n(K)\). As an application they give a new class of central polynomials for \(M_n(K)\). The proof of the main result is based on techniques from invariant theory of matrices and develops further ideas of \textit{E. Formanek} [J. Algebra 109, 93-114 (1987; Zbl 0625.16015)].
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 997620
zbMATH Keywords: central polynomials of matrices
zbMATH Keywords: polynomial identities of matrices
zbMATH Keywords: matrix invariants
zbMATH Keywords: central identities
zbMATH Keywords: characters of symmetric groups
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02785544
OpenAlex ID: W2053836997
description: scientific article; zbMATH DE number 997620