On the irreducibility of varieties of commuting matrices
The variety of all the pairs \((A,B)\) of \(n\times n\) matrices over an algebraically closed field \(K\) such that \([A,B]=0\) with rank \(B\leq h\), where \(h\in\{0,\dots,n\}\) is studied. It is shown that this variety is irreducible for all \(n\) and \(h\). On the basis of the knowledge of the form of a matrix which commutes with a given one which is in Jordan canonical form, it is proved that also the corresponding variety of symmetric matrices is reducible for all \(n\) and \(h\) if char \(K\neq 2\) and for all \(n\) and \(h=0\) if char \(K=2\) and similarly that the corresponding variety of antisymmetric matrices is irreducible for all \(n\) and \(h\) if char \(K=0\) and for all \(n\) and \(h=0,1\) if char \(K\neq 2\).
scientific article; zbMATH DE number 1474719
scientific article