A note on Bruhat order and double coset representatives

TL;DR

This paper establishes a precise relationship between the Bruhat order of minimal double coset representatives and their maximal representatives in finite Coxeter groups, clarifying the order structure within these groups.

Contribution

It proves that the Bruhat order relation between minimal double coset representatives is equivalent to that between their maximal representatives in finite Coxeter groups.

Findings

Bruhat order equivalence for minimal and maximal representatives

Clarification of order structure in Coxeter groups

Theoretical insight into double coset relations

Abstract

Let $W$ be a finite coxeter group, let $W_I$, $W_J$ be standard parabolic subgroups, let $u$, $v$ be minimal double cosets representatives of double cosets in $W_I W / W_J$ and let $u'$, $v'$ be the maximal representatives of those same two cosets. We show that $u < v$ if and only if $u' < v'$ for the Bruhat order.