Central limit theorems for generalized descents and generalized inversions in finite root systems

TL;DR

This paper establishes central limit theorems for generalized descents and inversions in finite Weyl groups, providing new probabilistic insights into their distributional properties.

Contribution

It introduces Coxeter-theoretic properties of indicator variables and applies the dependency graph method to prove CLTs for generalized descents and inversions.

Findings

Variance formulas for classical types

Central limit theorems for generalized descents

Central limit theorems for generalized inversions

Abstract

We consider generalized inversions and descents in finite Weyl groups. We establish Coxeter-theoretic properties of indicator random variables of positive roots such as the covariance of two such indicator random variables. We then compute the variances of generalized inversions and descents in classical types. We finally use the dependency graph method to prove central limit theorems for general antichains in root posets and in particular for generalized descents, and then for generalized inversions.