Actions of the 0-Hecke Monoids of Affine Symmetric Groups

Eric Marberg

Séminaire lotharingien de combinatoire, 80B (2018)

Résumé

There are left and right actions of the 0-Hecke monoid of the affine symmetric group 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 S^~_n to the set of N-weighted matchings in the n-element cycle graph. As an application, we show that the bivariate generating function counting the involutions in S^~_n by length and absolute length is a rescaled Lucas polynomial. The 0-Hecke monoid of S^~_n also acts on involutions (without any cycle labelling) by Demazure conjugation. The atoms of an involution z in 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.