Séminaire Lotharingien de Combinatoire, 80B.65 (2018), 12 pp.
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 labeled periodically by nonnegative
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
also acts on involutions (without any cycle labelling)
by Demazure conjugation. The atoms of an involution z in
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
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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