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

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 $\tilde{S}_n$.