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

Abstract

The Jordan type of a nilpotent matrix is the partition giving the sizes of its Jordan blocks. We study pairs of partitions $(P,Q)$, where $Q={\mathcal Q}(P)$ is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type $ P$. T. Ko\v{s}ir and P. Oblak have shown that $Q$ has parts that differ pairwise by at least two. Such partitions, which are also known as "super distinct" or "Rogers-Ramanujan", are exactly those that are stable or "self-large" in the sense that ${\mathcal Q}(Q)=Q$. In 2012 P. Oblak formulated a conjecture concerning the cardinality of the set of partitions $P$ such that ${\mathcal Q}(P)$ is a given stable partition $ Q$ with two parts, and proved some special cases. R. Zhao refined this to posit that those partitions $P$ such that ${\mathcal Q}(P)= Q=(u,u-r)$ with $u>r\ge 2$ could be arranged in an $(r-1)$ by $(u-r)$ table ${\mathcal T}(Q)$ where the entry in the $k$-th row and $\ell$-th column has $k+\ell$ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions $P$ for which ${\mathcal Q}(P)=Q$, for an arbitrary stable partition $Q$.