A central limit theorem for the two-sided descent statistic on Coxeter groups

TL;DR

This paper proves a central limit theorem for a specific descent-based statistic on finite Coxeter groups, revealing its asymptotic normality and extending previous research in algebraic combinatorics.

Contribution

It introduces a CLT for the (des+ides) statistic on Coxeter groups, generalizing prior results and answering an open question.

Findings

The (des+ides) statistic follows a normal distribution asymptotically.

The result extends previous work on descent statistics.

It provides a new probabilistic understanding of Coxeter group elements.

Abstract

We study the asymptotic behaviour of the statistic (des+ides) which assigns to an element w of a finite Coxeter group W the number of descents of w plus the number of descents of its inverse. Our main result is a central limit theorem for the probability distributions associated to this statistic. This answers a question of Kahle-Stump and generalises work of Chatterjee-Diaconis, \"Ozdemir and R\"ottger.