Identifying Partitions with maximum commuting orbit $Q=(u,u-r)$

TL;DR

This paper demonstrates that two different methods for identifying partitions with maximal commuting orbits in the context of nilpotent matrices are equivalent, unifying previous approaches in the literature.

Contribution

It proves the equivalence of the partition $P_{k,l}(Q)$ defined via the table $ ext{mathcal T}(Q)$ and the partition $P_{k,l}^Q$ defined through the Burge correspondence.

Findings

The partition $P_{k,l}(Q)$ matches $P_{k,l}^Q$ for stable partition $Q$.

Unifies two approaches to maximal nilpotent commutator partitions.

Provides a deeper understanding of the structure of partitions with maximal commuting orbits.

Abstract

The authors here show that the partition $P_{k,l}(Q)$ in the table $\mathcal T(Q)$ of partitions having maximal nilpotent commutator a given stable partition $Q$, defined in [IKVZ2], is identical to the analogous partition $P_{k,l}^Q$ defined by the authors in [BIK] using the Burge correspondence.