On Orbits of Order Ideals of Minuscule Posets

TL;DR

This paper demonstrates that the Fon-Der-Flaass action on order ideals of minuscule posets exhibits the cyclic sieving phenomenon, using a bijection with Weyl group elements and extending results to Cartesian products.

Contribution

It provides a uniform proof of cyclic sieving for minuscule posets via a bijection with Weyl group elements and extends the phenomenon to Cartesian products of these posets.

Findings

Fon-Der-Flaass action exhibits cyclic sieving on minuscule posets

A bijection with Weyl group elements is equivariant under the action

Cyclic sieving also occurs on Cartesian products of minuscule posets

Abstract

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 exhibits the cyclic sieving phenomenon, as defined by Reiner, Stanton, and White. A uniform proof is given by investigation of a bijection due to Stembridge between order ideals of minuscule posets and fully commutative Weyl group elements. This bijection is proven to be equivariant with respect to a conjugate of the Fon-Der-Flaass action and an arbitrary Coxeter element. 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.