A note on Bruhat order and double coset representatives

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 $u&#39;$, $v&#39;$ be the maximal representatives of those same two cosets. We show that $u < v$ if and only if $u&#39; < v&#39;$ for the Bruhat order.