$J$-ascent sets in parabolic quotients of Coxeter groups

Abstract

Let $(W,S)$ be a Coxeter system and let $J\subseteq S$. The right $J$-ascent set $$A_R^J(w)=\{s\in S\mid ws\in {}^{J}W,\ell (ws)=\ell (w)+1\}$$ is a natural refinement of the classical ascent set adapted to the parabolic quotient ${}^{J}W$. We establish a local transition formula describing the behaviour of $J$-ascent sets under right multiplication by a simple reflection. As a consequence, for every $w\in {}^{J}W$ and every $s\in A_R^J(w)$, we prove that $$||A_R^J(w)|-|A_R^J(ws)||\le \max \{1,\text{deg}(s)-1\}.$$

arXiv comment9 pages, 2 figures