Boolean complexes of involutions

TL;DR

This paper introduces the boolean complex of involutions in Coxeter groups, determines its homotopy type for many cases including all finite groups, and provides recurrence formulas for the number of spheres in the wedge.

Contribution

It extends the concept of boolean complexes to involutions in Coxeter groups and determines their homotopy type using discrete Morse theory.

Findings

Homotopy type is a wedge of spheres of dimension |S|-1 for many Coxeter groups.

Recurrence formulas for counting the spheres in the wedge.

Includes all finite Coxeter groups.

Abstract

Let (W,S) be a Coxeter system. We introduce the boolean complex of involutions of W which is an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By applying discrete Morse theory, we determine the homotopy type of the boolean complex of involutions for a large class of (W,S), including all finite Coxeter groups, finding that the homotopy type is that of a wedge of spheres of dimension |S| - 1. In addition, we find simple recurrence formulas for the number of spheres in the wedge.