Bruhat intervals and parabolic cosets in arbitrary Coxeter groups

TL;DR

This paper extends the known property of intersections between Bruhat intervals and parabolic cosets, originally proven for Weyl groups, to all Coxeter groups, revealing a broader structural consistency.

Contribution

It demonstrates that the intersection of a Bruhat interval with a parabolic coset forms a Bruhat interval in any Coxeter group, generalizing previous results.

Findings

Intersections are Bruhat intervals in arbitrary Coxeter groups

Unique maximal and minimal elements exist in these intersections

Generalizes previous Weyl group results to all Coxeter groups

Abstract

In [Journal of Pure and Applied Algebra {224} (2020), no 12, 106449], V. Mazorchuk and R. Mr{\dj}en (with some help by A. Hultman) prove that, given a Weyl group, the intersection of a Bruhat interval with a parabolic coset has a unique maximal element and a unique minimal element. We show that such intersections are actually Bruhat intervals also in the case of an arbitrary Coxeter group.