The smallest part of the generic partition of the nilpotent commutator of a nilpotent matrix

TL;DR

This paper provides an explicit combinatorial formula for the smallest part of the Jordan type of a generic nilpotent matrix commuting with a fixed nilpotent matrix, independent of the field's characteristic, and describes cases with up to three parts.

Contribution

It introduces a combinatorial approach to determine the smallest part of the generic commuting nilpotent matrix's Jordan type, extending understanding of the nilpotent commutator structure.

Findings

Explicit formula for the smallest part of Q(P)

Complete description of Q(P) with up to three parts

Independence of the characteristic of the field

Abstract

Let $k$ be an infinite field. Fix a Jordan nilpotent $n$ by $n$ matrix $B = J_P$ with entries in $k$ and associated Jordan type $P$. Let $Q(P)$ be the Jordan type of a generic nilpotent matrix commuting with $B$. In this paper, we use the combinatorics of a poset associated to the partition $P$, to give an explicit formula for the smallest part of $Q(P)$, which is independent of the characteristic of $k$. This, in particular, leads to a complete description of $Q(P)$ when it has at most three parts.