On the number of irreducible components of commuting varieties
type: Item
instance of: scholarly article
title: On the number of irreducible components of commuting varieties
zbMATH Open document ID: 0955.15006
DOI: 10.1016/S0022-4049(99)00036-5
author: Roberta Basili
published in: Journal of Pure and Applied Algebra
publication date: 2000-09-04T00:00:00Z
review text: It is shown that the variety of all the pairs \((A,B)\) of \(n\times n\) matrices over an algebraically closed field \(K\) such that \([A,B]=0\), rank \(A\leq k\), rank \(B\leq h\) has \(\min\{h,n-k\}+1\) irreducible components for any \(n\) and \(h,k\in\{0,...,n\}\), \(h\leq k\). It is also proved that the corresponding variety of symmetric matrices is reducible if \(h,k\in\{1,\dots,n-1\}\) and is irreducible if \(h\) is \(0\) and if char \(K\neq 2\) and \(k\) is \(n\). Similarly it is proved that the corresponding variety of antisymmetric matrices is reducible for all \(h,k\in\{2,\dots,n-1\}\) if \(n\) is even, for all \(h,k\in\{2,\dots,n-2\}\) if \(n\) is odd and is irreducible if \(h=0\), char \(K=0\) and \(k\) is \(n\) or \(n-1\).
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 15A27
zbMATH DE Number: 1474720
zbMATH Keywords: commuting matrix
zbMATH Keywords: Lie algebra
zbMATH Keywords: Jordan canonical form
zbMATH Keywords: commuting varieties
zbMATH Keywords: irreducible components
zbMATH Keywords: variety of symmetric matrices
zbMATH Keywords: variety of antisymmetric matrices
MaRDI profile type: Publication
full work available at URL: s0022-4049(99)00036-5
OpenAlex ID: W2021742067
description: scientific article; zbMATH DE number 1474720