Expected length of a product of random reflections

TL;DR

This paper derives formulas for the expected length and inversions in permutations generated by random reflections, extending to various Coxeter groups, providing exact expressions for these expectations.

Contribution

It introduces a unified approach to compute the expected length of products of random reflections across different Coxeter groups, generalizing previous results.

Findings

Exact formulas for expected inversions in permutations after random transpositions.

Generalized expressions for expected length in Coxeter groups of types A, B, D, and I.

Applicable to permutations generated by random reflections in finite irreducible Coxeter groups.

Abstract

We present a simple formula for the expected number of inversions in a permutation of size $n$ obtained by applying $t$ random (not necessarily adjacent) transpositions to the identity permutation. More general, for any finite irreducible Coxeter group belonging to one of the infinite families (type A, B, D, and I), an exact expression is obtained for the expected length of a product of $t$ random reflections.