#####
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.

The following versions are available: