On the irreducibility of varieties of commuting matrices
type: Item
instance of: scholarly article
title: On the irreducibility of varieties of commuting matrices
zbMATH Open document ID: 0955.15005
DOI: 10.1016/S0022-4049(99)00027-4
author: Roberta Basili
published in: Journal of Pure and Applied Algebra
publication date: 2001-02-19T00:00:00Z
review text: The variety of all the pairs \((A,B)\) of \(n\times n\) matrices over an algebraically closed field \(K\) such that \([A,B]=0\) with rank \(B\leq h\), where \(h\in\{0,\dots,n\}\) is studied. It is shown that this variety is irreducible for all \(n\) and \(h\). On the basis of the knowledge of the form of a matrix which commutes with a given one which is in Jordan canonical form, it is proved that also the corresponding variety of symmetric matrices is reducible for all \(n\) and \(h\) if char \(K\neq 2\) and for all \(n\) and \(h=0\) if char \(K=2\) and similarly that the corresponding variety of antisymmetric matrices is irreducible for all \(n\) and \(h\) if char \(K=0\) and for all \(n\) and \(h=0,1\) if char \(K\neq 2\).
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1474719
zbMATH Keywords: commuting matrix
zbMATH Keywords: Lie algebra
zbMATH Keywords: Jordan canonical form
zbMATH Keywords: irreducible
zbMATH Keywords: variety of symmetric matrices
zbMATH Keywords: variety of antisymmetric matrices
MaRDI profile type: Publication
Wikidata QID: Q128089789
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description: scientific article; zbMATH DE number 1474719