Unipotent Invariant Matrices

TL;DR

This paper characterizes the fixed points of unipotent operators on matrix spaces, analyzing their determinants and ranks to understand the structure and singularities of associated bilinear forms.

Contribution

It provides a detailed description of the fixed point variety under unipotent actions and computes properties like determinant and rank for generic matrices within this variety.

Findings

Determined the structure of fixed points of unipotent operators on matrices.

Computed the determinant and rank of generic matrices in the fixed variety.

Gained insights into the singular locus of bilinear forms associated with these matrices.

Abstract

We describe the variety of fixed points of a unipotent operator acting on the space of matrices. We compute the determinant and the rank of a generic (symmetric, or anti-symmetric) matrix in the fixed variety, yielding information about the generic singular locus of the corresponding bilinear form.