A Central Limit Theorem for the Two-Sided Descent Statistic on Coxeter Groups

Abstract

We study the asymptotic behaviour of the statistic $(\operatorname{des}+\operatorname{ides})_W$ which assigns to an element $w$ of a finite Coxeter group $W$ the number of descents of $w$ plus the number of descents of $w^{-1}$. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and builds upon work of Chatterjee-Diaconis, Özdemir and Röttger.