Séminaire Lotharingien de Combinatoire, 80B.65 (2018), 12 pp.

Eric Marberg

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 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.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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