On Orbits of Order Ideals of Minuscule Posets

Details On Orbits of Order Ideals of Minuscule Posets On Orbits of Order Ideals of Minuscule Posets

2011-08-26

arxiv.org/abs/1108.5245v1

Mathematics

Combinatorics

23 pages

Scientific paper

An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets, it is shown that the Fon-Der-Flaass action, together with the generating function that counts order ideals by their cardinality, exhibits the cyclic sieving phenomenon as defined by Reiner, Stanton, and White. The proof is uniform, and it is accomplished by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements arranged in Bruhat lattices, which allows for an equivariance between the Fon-Der-Flaass action and an arbitrary Coxeter element to be demonstrated. If $P$ is a minuscule poset, it is shown that the Fon-Der-Flaass action on order ideals of the Cartesian product $P \times [2]$ also exhibits the cyclic sieving phenomenon, only the proof is by appeal to the classification of minuscule posets and is not uniform.

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