Bound on the Jordan type of a generic nilpotent matrix commuting with a given matrix

TL;DR

This paper investigates the Jordan types of generic nilpotent matrices commuting with a given nilpotent matrix, providing partial validation for a conjecture about their possible partitions and establishing an order relation.

Contribution

It proves a partial result confirming that the conjectured Jordan type is less than or equal to the actual type in dominance order.

Findings

Confirmed the conjecture's partial aspect in the dominance order.

Established an inequality relation between conjectured and actual Jordan types.

Provided new insights into the structure of commuting nilpotent matrices.

Abstract

It is well-known that a nilpotent n by n matrix B is determined up to conjugacy by a partition of n formed by the sizes of the Jordan blocks of B. We call this partition the Jordan type of B. We obtain partial results on the following problem: for any partition P of n describe the type Q(P) of a generic nilpotent matrix commuting with a given nilpotent matrix of type P. A conjectural description for Q(P) was given by P. Oblak and restated by L. Khatami. In this paper we prove "half" of this conjecture by showing that this conjectural type is less than or equal to Q(P) in the dominance order on partitions.