Braid moves in commutation classes of the symmetric group

TL;DR

This paper proves that the expected number of braid moves in a specific commutation class of the symmetric group’s long element is one, extending previous results and exploring symmetries and actions on these classes.

Contribution

It introduces a bijective proof that the expected braid moves in a particular commutation class are one and explores homomesy under promotion operators.

Findings

Expected braid moves in the class is one

Provides a refinement on orbits under promotion operators

Extends techniques to more general posets and statistics

Abstract

We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.