Commuting pairs in the centralizers of 2-regular matrices (Q1283270)

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.