Finite Parabolic Conjugation on Varieties of Nilpotent Matrices

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\mathrm{GL}}_n(\mathbf{C})$ 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.

The final publication is available at http://link.springer.com/article/10.1007%2Fs10468-014-9464-0#. arXiv admin note: substantial text overlap with arXiv:1205.5197