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

Roberta Basili

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arXiv · 1710.03588

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

Roberta Basili

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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.

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Roberta Basili. 2017-10-06. On the maximum nilpotent orbit which intersects the centralizer of a matrix. https://arxiv.org/abs/1710.03588

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