Positive
m-divisible non-crossing partitions and their Kreweras maps
(169 pages)
Abstract.
We study positive m-divisible non-crossing partitions, as
introduced by Buan, Reiten and Thomas. In all classical types,
we describe combinatorial realisations of these partitions
as certain non-crossing set partitions.
Furthermore, we discuss a translation of a representation-theoretic functor of
Simões and of Buan, Reiten and Thomas into Coxeter combinatorics
(which was also studied by the second author, Thomas
and Williams), called the positive Kreweras map.
Again for all classical types, we realise this positive
Kreweras map as pseudo-rotations on a circle, respectively on an
annulus.
Using our combinatorial models, we enumerate positive m-divisible
non-crossing partitions in classical types that are invariant under a power of the
positive Kreweras map with respect to several parameters.
In order to cope with the exceptional types, we develop a different combinatorial
model (that actually works in any type) describing positive
m-divisible non-crossing partitions that are invariant under a
power of the
positive Kreweras map. We finally show that altogether these results
establish several cyclic sieving phenomena as
defined by Reiner, Stanton and White.
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