Conjugation on varieties of nilpotent matrices

TL;DR

This paper studies the conjugation action of parabolic subgroups on nilpotent matrices, providing criteria for finite orbits, explicit representatives, orbit closures, and invariants, advancing understanding of nilpotent matrix varieties.

Contribution

It introduces new criteria for orbit finiteness, explicit orbit representatives, and invariant descriptions for parabolic actions on nilpotent matrices, including normal forms and degenerations.

Findings

Finite orbit criterion for 2-nilpotent matrices

Explicit orbit representatives and closures

Generators of semi-invariant rings

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. 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. Concerning the action on the nilpotent cone, we obtain a generic normal form of the orbits which yields a U-normal form as well, here U is the standard unipotent subgroup. We describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the U-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related.