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=Q(P) is the Jordan type of a generic nilpotent matrix A commuting with a nilpotent matrix B of Jordan type P. T. Košir 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 Q(Q)=Q.In 2012 P. Oblak formulated a conjecture concerning the cardinality of Q−1(Q) when Q has two parts, and proved some special cases. R. Zhao refined this to posit that the partitions in Q−1(Q) for Q=(u,u−r) with u>r>1 could be arranged in an (r−1)×(u−r) table T(Q) where the entry in the k-th row and ℓ-th column has k+ℓ parts. We prove this Table Theorem, and then generalize the statement to propose a Box Conjecture for the set of partitions Q−1(Q) for an arbitrary partition Q whose parts differ pairwise by at least two.