Commuting pairs and triples of matrices and related varieties
type: Item
instance of: scholarly article
title: Commuting pairs and triples of matrices and related varieties
zbMATH Open document ID: 0971.15007
DOI: 10.1016/S0024-3795(00)00065-3
author: B. A. Sethuraman
author: Robert Guralnick
published in: Linear Algebra and its Applications
publication date: 2001-10-28T00:00:00Z
review text: Let \(F\) be an algebraically closed field, let \(\mathcal C(d,n)\) be the set of all commuting \(d\)-tuples of elements in the ring \(M_n(F)\), then \(\mathcal C(d,n)\) is an affine variety. The authors prove that the set of all commutative \(d\)-tuples that are contained in some \(n\)-dimensional commutative subalgebra of \(M_n(F)\) of dimension \(n\) is a closed subset of \(\mathcal C(d,n)\). It is well known that \(\mathcal C(2,n)\) is irreducible. It is also known that \(\mathcal C(d,n)\) is reducible for \(d\geq 4\) and \(n\geq 4\), and that \(\mathcal C(3,n)\) is reducible for \(n\geq 32\) but is irreducible for \(n\leq 3\). The authors show that \(\mathcal C(3,4)\) is irreducible. They set up a certain incidence correspondence in a suitable product of projective varieties, and then project onto a component to find that the set of commutative subalgebras of \(M_n(F)\) of dimension \(n\) is a closed set in the Grassmanian variety of \(n\)-dimensional subspaces of \(n^2\) dimensional space. It is shown to be irreducible of dimension \(n^2-n\) for \(n\leq 4\), but reducible of dimension greater than \(n^2-n\) for \(n\geq 7\).
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A27
zbMATH DE Number: 1453258
zbMATH Keywords: commuting matrices
zbMATH Keywords: commuting variety
zbMATH Keywords: commutative subalgebras of matrix algebras
zbMATH Keywords: matrix pairs
zbMATH Keywords: matrix triples
zbMATH Keywords: irreducible
zbMATH Keywords: reducible
MaRDI profile type: Publication
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Wikidata QID: Q126351540
full work available at URL: s0024-3795(00)00065-3
OpenAlex ID: W2075832453
description: scientific article; zbMATH DE number 1453258