# Involutions of the Symmetric Group and Congruence B-orbits of Anti-Symmetric Matrices

## TL;DR

This paper explores the structure of Borel congruence classes of anti-symmetric matrices, establishing a bijection with involutions of the symmetric group and providing formulas for the poset's rank function.

## Contribution

It introduces a novel correspondence between Borel congruence classes of anti-symmetric matrices and symmetric group involutions, along with rank formulas for the poset.

## Key findings

- Bijection between Borel congruence classes and involutions
- Two formulas for the rank function of the poset
- Description of the poset structure of matrix classes

## Abstract

We present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of closures. We show that there exists a bijection between the set of these classes and the set of involutions of the symmetric group. We give two formulas for the rank function of this poset.