On some actions of the 0-Hecke monoids of affine symmetric groups

TL;DR

This paper explores actions of the 0-Hecke monoid on involutions in affine symmetric groups, constructing bijections to weighted matchings, and analyzing the structure of atoms and Bruhat order within this context.

Contribution

It introduces new bijections between involutions and weighted matchings, and characterizes the structure of atoms and Bruhat order for involutions in affine symmetric groups.

Findings

Constructed length-preserving bijections to weighted matchings.

Derived a formula for the generating function counting involutions.

Classified covering relations in the Bruhat order for involutions.

Abstract

There are left and right actions of the 0-Hecke monoid of the affine symmetric group $\tilde{S}_n$ on involutions whose cycles are labeled periodically by nonnegative integers. Using these actions we construct two bijections, which are length-preserving in an appropriate sense, from the set of involutions in $\tilde{S}_n$ to the set of $\mathbb{N}$-weighted matchings in the $n$-element cycle graph. As an application, we compute a formula for the bivariate generating function counting the involutions in $\tilde S_n$ by length and absolute length. The 0-Hecke monoid of $\tilde{S}_n$ also acts on involutions (without any cycle labelling) by Demazure conjugation. The atoms of an involution $z \in \tilde{S}_n$ are the minimal length permutations $w$ which transform the identity to $z$ under this action. We prove that the set of atoms for an involution in $\tilde{S}_n$ is naturally a bounded, graded poset, and give a formula for the set's minimum and maximum elements. Using these properties, we classify the covering relations in the Bruhat order restricted to involutions in $\tilde{S}_n$.