The poset of the nilpotent commutator of a nilpotent matrix

Details The poset of the nilpotent commutator of a nilpotent matrix The poset of the nilpotent commutator of a nilpotent matrix

2012-02-27

arxiv.org/abs/1202.6089v1

Mathematics

Commutative Algebra

Scientific paper

Fix an n by n nilpotent matrix B with entries in an infinite field k. Assume that B is in Jordan canonical form with associated Jordan block partition P. It is well known that the nilpotent commutator of B, comprised of those nilpotent matrices commuting with B, is an irreducible algebraic variety. So there is a unique partition Q(P) that is the Jordan partition of the generic element of the nilpotent commutator of B. There have been several results on the partition Q(P) but it has been completely understood only when it has one or two parts. In this paper we study a poset associated to the nilpotent commutator of B, and we give a formula for the smallest part of Q(P). This, in particular, leads to a complete description of Q(P) when it has three parts.

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