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

There are left and right actions of the 0-Hecke monoid of the affine symmetric group S˜n on involutions whose cycles are labelled 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 S˜n to the set of 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 S˜n by length and absolute length. The 0-Hecke monoid of S˜n also acts on involutions (without any cycle labelling) by Demazure conjugation. The atoms of an involution z∈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 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 S˜n.