# Construction of double coset system of a Coxeter group and its applications to Bruhat graphs

## TL;DR

This paper develops a combinatorial framework for parabolic double cosets in finite Coxeter groups, extending Coxeter complex concepts and exploring their applications to properties of Bruhat graphs.

## Contribution

It introduces a double coset system generalizing Coxeter complexes and analyzes its structure, with applications to Bruhat graph regularity and Eulerian properties.

## Key findings

- Every parabolic double coset is regular.
- Degree invariance on Bruhat graph lower intervals.
- Noncritical Bruhat intervals satisfy out-Eulerian property.

## Abstract

We develop combinatorics of parabolic double cosets in finite Coxeter groups as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner and Petersen. (1) We construct a double coset system as a generalization of a two-sided analogue of a Coxeter complex and present its order structure with its local dimension function on certain connected components. As applications of double cosets to Bruhat graphs, we also prove: (2) every parabolic double coset is regular, (3) invariance of degree on Bruhat graph on lower intervals as an analogy of the one for Kazhdan-Lusztig polynomials, (4) every noncritical Bruhat interval satisfies out-Eulerian property.