Papers › Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit
Nilpotent matrices having a given Jordan type as maximum commuting nilpotent orbit
Anthony Iarrobino, Leila Khatami, Bart Van Steirteghem, Rui Zhao
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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. 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 𝒬(Q)=Q. In 2012 P. Oblak formulated a conjecture concerning the cardinality of the set of partitions P such that 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 𝒬(P)= Q=(u,u-r) with u>r≥2 could be arranged in an (r-1) by (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 P for which 𝒬(P)=Q, for an arbitrary stable partition Q.
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