Proper elements of Coxeter groups

TL;DR

This paper extends the concept of proper elements to all Coxeter groups, proving that their probability tends to zero in infinite families and enumerating them in exceptional cases, confirming related conjectures.

Contribution

It generalizes proper elements to all Coxeter groups and proves a conjecture about the proportion of Levi spherical Schubert varieties in infinite families.

Findings

Probability of proper elements tends to zero in infinite Coxeter groups

Proper elements are enumerated in exceptional Coxeter groups

Confirms conjecture on Levi spherical Schubert varieties

Abstract

We extend the notion of proper elements to all Coxeter groups. For all infinite families of finite Coxeter groups we prove that the probability a random element is proper goes to zero in the limit. This proves a conjecture of the third author and A. Yong regarding the proportion of Schubert varieties that are Levi spherical for all infinite families of Weyl groups. We also enumerate the proper elements in the exceptional Coxeter groups.