Shifted lower Bruhat intervals are EL-shellable

Abstract

Let $W$ be an arbitrary Coxeter group. The shifted Bruhat interval $[w_1,w_2]\,x^{-1}$, the translate of the Bruhat interval $[w_1,w_2]$ by an element $x$, is partially ordered by the Bruhat order of $W$. These posets arise from affine pavings of Richardson varieties, and in general they are neither twisted intervals nor tilted Bruhat intervals. Our main result is that the shifted lower intervals $[e,w]\,x^{-1}$ are EL-shellable for every Coxeter group, via an explicit labeling of each cover by a reflection. Along the way we show that $[e,w]\,x^{-1}$ is a graded poset with a unique maximum given by the Demazure product and a unique minimum given by an opposite Demazure operator that we introduce.