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 ${\tilde{S}}_n$.

34 pages, 2 figures; v2: minor corrections, updated references, added index of symbols; v3: correction to Proposition 8.9(a), a few typos fixed, final version

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Cited by in corpus (6)

• Bumping operators and insertion algorithms for queer supercrystals
• Extending a word property for twisted Coxeter systems
• Involution pipe dreams
• Atoms for signed permutations
• Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions
• Affine transitions for involution Stanley symmetric functions