Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit (Q1743142)

Consider the matrix ring \(M_{n}(\mathbb{K})\) of \(n \times n\) matrices over an infinite field \(\mathbb{K}\). A similarity class of nilpotent matrices is characterized by the partition \(P\) of \(n\) which lists the sizes of the Jordan blocks in decreasing order. If \(A \in M_{n}(\mathbb{K})\) is nilpotent of type \(P\) then it is known that the set of nilpotent matrices with which \(A\) commutes forms an irreducible variety \(\mathcal{N}_{A}\) and so the generic matrices in \(\mathcal{N}_{A}\) are all of the same nilpotent type \(Q\). Thus there is a function \(\mathcal{Q}\) on the set of partitions \(P\) of \(n\) given by \(\mathcal{Q}(P) : =Q\). The number \(r_{P}\) of parts in the partition \(\mathcal{Q}(P)\) has a simple characterization due to \textit{R. Basili} [J. Algebra 268, No. 1, 58--80 (2003; Zbl 1032.15012)] but an outstanding problem is to find a description in terms of \(Q\) of the partitions \(P\) such that \(\mathcal{Q}(P) =Q\). So far this problem has only be solved when \(r_{P} =2\) or \(3\) (see [\textit{L. Khatami}, J. Pure Appl. Algebra 218, No. 8, 1496--1516 (2014; Zbl 1283.05272); \textit{T. Košir} and \textit{P. Oblak}, Transform. Groups 14, No. 1, 175--182 (2009; Zbl 1168.15009)]). Several conjectures about \(\mathcal{Q}^{ -1}(Q)\) have been made by \textit{P. Oblak}, most recently for the special case when \(Q\) has two parts \((u ,u -r)\) with \(2 \leq r <u\). It is conjectured that \(| \mathcal{Q}^{ -1}(Q)| =(r -1)(u -r)\) [Linear Multilinear Algebra 60, No. 5, 599--612 (2012; Zbl 1252.15019)]. The main part of this paper is devoted to proving the latter conjecture in a strong form due to R. Zhao (unpublished), which gives precise information about the number of parts in the various partitions \(P\) which give rise to \(Q\). The paper ends with some tentative results dealing with cases for which \(Q\) has three parts.