Commuting pairs in the centralizers of 2-regular matrices
type: Item
instance of: scholarly article
title: Commuting pairs in the centralizers of 2-regular matrices
zbMATH Open document ID: 0924.15015
author: Michael G. Neubauer
author: B. A. Sethuraman
published in: Journal of Algebra
publication date: 1999-11-07T00:00:00Z
review text: Let \(M_{n}(k)\) denote the set of all \(n\times n\) matrices over an algebraically closed field \(k\), and let \(V:=k^{n}\) be the underlying vector space. The authors define a matrix \(A\in M_{n}(k)\) to be \(l\)-regular if each eigenspace of \(A\) has dimension at most \(l,\) or equivalently, if \(V\) can be generated by \(l\) elements as a \(k[A]\)-module. A matrix \(A\) is \(1\)-regular if and only if its minimal polynomial and characteristic polynomial coincide (such a matrix is often called nonderogatory); in this case the centralizer of \(A\) in \(M_{n}(k)\) is just \(k[A]\). The paper considers the commutative subalgebras of the centralizer of a \(2\)-regular matrix. Among the results obtained it is shown that the variety of commuting pairs in the centralizer of a \(2\)-regular matrix \(C\) is irreducible. Using this, it is then proved that if \(A,B\) and \(C\) are any triple of commuting matrices in \(M_{n}(k)\), and \(A\) and \(B\) commute with a \(2\)-regular matrix, then \(k[A,B,C]\) has dimension at most \(n\) and indeed is contained in a commutative subalgebra of dimension exactly \(n.\) The latter settles a special case of a question first raised by \textit{M. Gerstenhaber} [Ann. Math., II. Ser. 73, 324-348 (1961; Zbl 0168.28201)] who asked whether the commutative subalgebra \(k[A,B,C]\) is of dimension at most \(n\) for any commuting triple.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1275260
zbMATH Keywords: nonderogatory
zbMATH Keywords: centralizer
zbMATH Keywords: commutative subalgebras
zbMATH Keywords: 2-regular matrix
zbMATH Keywords: commuting pairs
zbMATH Keywords: irreducible
zbMATH Keywords: dimension
MaRDI profile type: Publication
full work available at URL: jabr.1998.7703
OpenAlex ID: W2083342598
cites work: Vector bases for two commuting matrices
cites work: Q3254327
cites work: On dominance and varieties of commuting matrices
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cites work: Two-generated commutative matrix subalgebras
cites work: Pairs of Matrices With Property L. II
cites work: Two-generated commutative subalgebras of \(M_ n(F)\)
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DOI: 10.1006/JABR.1998.7703
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description: scientific article; zbMATH DE number 1275260