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 $u^{\prime }$, $v^{\prime }$ be the maximal representatives of those same two cosets. We show that $u<v$ if and only if $u^{\prime }<v^{\prime }$ for the Bruhat order.

4 pages