A two-sided analogue of the Coxeter complex

TL;DR

This paper introduces a new two-sided boolean complex related to Coxeter systems, extending the Coxeter complex, and explores its topological and combinatorial properties, including shellability and Eulerian polynomials.

Contribution

It constructs a novel two-sided boolean complex for Coxeter systems, generalizing the Coxeter complex and analyzing its topological and combinatorial features.

Findings

The complex is shellable and thin.

For finite Coxeter groups, the complex is a sphere.

A refinement of the $h$-polynomial is given by the two-sided $W$-Eulerian polynomial.

Abstract

For any Coxeter system $(W,S)$ of rank $n$, we introduce an abstract boolean complex (simplicial poset) of dimension $2n-1$ that contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples $(I,w,J)$, where $I$ and $J$ are subsets of the set $S$ of simple generators, and $w$ is a minimal length representative for the parabolic double coset $W_I w W_J$. There is exactly one maximal face for each element of the group $W$. The complex is shellable and thin, which implies the complex is a sphere for the finite Coxeter groups. In this case, a natural refinement of the $h$-polynomial is given by the "two-sided" $W$-Eulerian polynomial, i.e., the generating function for the joint distribution of left and right descents in $W$.