Lexicographic shellability of the Bruhat-Chevalley order on fixed-point-free involutions

TL;DR

This paper proves that the Bruhat-Chevalley order on fixed-point-free involutions in the symmetric group is EL-shellable and triangulates a ball, also comparing it to the Deodhar-Srinivasan poset.

Contribution

It establishes EL-shellability of the Bruhat-Chevalley order on fixed-point-free involutions and compares it with the Deodhar-Srinivasan poset.

Findings

Bruhat-Chevalley order on fixed-point-free involutions is EL-shellable

The order complex triangulates a ball

Deodhar-Srinivasan poset is a proper subposet

Abstract

The main purpose of this paper is to prove that the Bruhat-Chevalley ordering of the symmetric group when restricted to the fixed-point-free involutions forms an $EL$-shellable poset whose order complex triangulates a ball. Another purpose of this article is to prove that the Deodhar-Srinivasan poset is a proper, graded subposet of the Bruhat-Chevalley poset on fixed-point-free involutions.