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.

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.