On the maximum nilpotent orbit which intersects the centralizer of a matrix

TL;DR

This paper presents an algorithm to determine the largest nilpotent orbit intersecting the centralizer of a matrix, based on matrix ranks and zero entries, confirming a conjecture by Polona Oblak.

Contribution

It introduces a method to identify the maximum nilpotent orbit intersecting a matrix's centralizer, depending on submatrix ranks and zero entries, and proves a related conjecture.

Findings

The maximum nilpotent orbit depends only on submatrix ranks.

The maximum orbit depends on which entries are zero in the centralizer.

An explicit algorithm for determining the maximum orbit is provided.

Abstract

We introduce a method to determine the maximum nilpotent orbit which intersects a variety of nilpotent matrices described by a strictly upper triangular matrix over a polynomial ring. We show that the result only depends on the ranks of its submatrices and we introduce conditions on a subvariety so that it intersects the same orbit. Then we describe a maximal nilpotent subalgebra of the centralizer of any nilpotent matrix; the previous method allows us to show that the maximum nilpotent orbit which intersects that centralizer only depends on which entries are identically 0 in that subalgebra. The aim of the paper is to prove a simple algorithm for the determination of the maximum nilpotent orbit which intersects that centralizer, which was conjectured by Polona Oblak.