Anti-commuting varieties

TL;DR

This paper investigates the structure of the anti-commuting variety of matrix pairs, detailing its irreducible components, dimensions, and properties of related quotient spaces, advancing understanding in algebraic geometry and matrix theory.

Contribution

It provides an explicit description of the irreducible components and dimensions of the anti-commuting variety and its semi-nilpotent subset, including their GIT quotients.

Findings

Irreducible components of the anti-commuting variety are explicitly described.

The GIT quotient has pure dimension n.

Semi-nilpotent anti-commuting variety has pure dimension n^2.

Abstract

We study the anti-commuting variety which consists of pairs of anti-commuting $n\times n$ matrices. We provide an explicit description of its irreducible components and their dimensions. The GIT quotient of the anti-commuting variety with respect to the conjugation action of $GL_n$ is shown to be of pure dimension $n$. We also show the semi-nilpotent anti-commuting variety (in which one matrix is required to be nilpotent) is of pure dimension $n^2$ and describe its irreducible components.