Finite Parabolic Conjugation on Varieties of Nilpotent Matrices

TL;DR

This paper investigates the action of parabolic subgroups on nilpotent matrices, providing criteria for finiteness of orbits, explicit orbit representatives, and analyzing the complexity of the associated algebraic structures.

Contribution

It introduces a criterion for finiteness of orbits under parabolic conjugation and describes orbit representatives and degenerations, connecting to representation theory.

Findings

Finite orbit criterion established

Explicit representatives for 2-nilpotent matrices provided

Wild representation type shown for non-finite cases

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$ on the variety of $x$-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of $2$-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the corresponding quiver algebra is of wild representation type.