Two-generated commutative subalgebras of \(M_ n(F)\)
type: Item
instance of: scholarly article
title: Two-generated commutative subalgebras of \(M_ n(F)\)
zbMATH Open document ID: 0801.16031
author: Michael G. Neubauer
author: David Saltman
published in: Journal of Algebra
publication date: 1994-12-01T00:00:00Z
review text: Let \(A\) and \(B\) be a pair of commuting \(n\times n\)-matrices over a field \(K\). It is well known that \(\dim_ K K[A,B]\leq n\). The authors show that \(\dim_ K\text{Cent}_{M_ n(K)}(A,B)\geq n\), and equality holds in one of these inequalities if and only if it holds in the other. Further characterizations of the case where equality holds are also given. The space of column vectors of length \(n\) is a \(K[A,B]\) module \(M\) in the obvious way; write \(M=M_ 1\oplus\cdots\oplus M_ s\), where each \(M_ i\) is indecomposable. The complete characterization theorem now asserts that the following statements are equivalent: 1) \(\dim_ K K[A,B]=n\); 2) For \(1\leq i\leq s\), each \(M_ i\) is cyclic or cocyclic (i.e., its \(K\)-dual is cyclic), and \(\text{Hom}_{K[A,B]} (M_ i,M_ j)=0\) when \(i\neq j\); 3) \(K[A,B]\) is self-centralizing in \(M_ n(K)\) (i.e., \(K[A,B]\) is a maximal commutative subalgebra of \(M_ n(K)\), with respect to inclusion); 4) \((A,B)\) is a smooth point on the variety of pairs of commuting \(n\times n\) matrices over \(K\); and 5) \(\dim_ K\text{Cent}_{M_ n(K)} (A,B)=n\). In regard to the case where equality does not hold, \textit{M. Gerstenhaber} [Ann. Math., II. Ser. 73, 324-348 (1961; Zbl 0168.282)] showed that there is an \(n\)-dimensional commutative subalgebra \(G\) of \(M_ n(K)\) that contains \(K[A,B]\). The authors give a new proof of this result, and show by an example that the action of \(G\) on \(M\) need not be cyclic or cocyclic.
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A27
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16G10
zbMATH DE Number: 562454
zbMATH Keywords: commuting \(n\times n\)-matrices
zbMATH Keywords: maximal commutative subalgebra
zbMATH Keywords: action
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: jabr.1994.1077
OpenAlex ID: W2008046362
DOI: 10.1006/JABR.1994.1077
Recommended article: Commuting pairs in the centralizers of 2-regular matrices
Recommended article: New classes of examples satisfying the three matrix analog of Gerstenhaber's theorem
Recommended article: On the dimension of faithful modules over finite dimensional basic algebras
Recommended article: A note on commuting pairs of matrices
Recommended article: Vector bases for two commuting matrices
Recommended article: Ein Satz von I. Schur über vertauschbare Matrizen
Recommended article: Two-generated commutative matrix subalgebras
Recommended article: Maximal Commutative Subalgebras ofn×nMatrices
Recommended article: Q3781880
Recommended article: Q3478546
description: scientific article; zbMATH DE number 562454