On Orbits of Order Ideals of Minuscule Posets II: Homomesy

TL;DR

This paper proves that the average size of order ideals in orbits under Fon-Der-Flaass action is invariant for all minuscule posets, extending known results and revealing deep connections with Lie algebra representations.

Contribution

It extends the homomesy phenomenon to all minuscule posets by linking order ideals to Lie algebra representations and Weyl group actions, providing a uniform proof.

Findings

Homomesy holds for the size of order ideals in all minuscule posets.

The cardinality of order ideals is determined by Weyl group actions on associated weights.

The number of maximal elements in order ideals also exhibits homomesy.

Abstract

The Fon-Der-Flaass action partitions the order ideals of a poset into disjoint orbits. For a product of two chains, Propp and Roby observed --- across orbits --- the mean cardinality of the order ideals within an orbit to be invariant. That this phenomenon, which they christened homomesy, extends to all minuscule posets is shown herein. Given a minuscule poset $P$, there exists a complex simple Lie algebra $\mathfrak{g}$ and a representation $V$ of $\mathfrak{g}$ such that the lattice of order ideals of $P$ coincides with the weight lattice of $V$. For a weight $\mu$ with corresponding order ideal $I$, it is demonstrated that the behavior of the Weyl group simple reflections on $\mu$ not only uniquely determines $\mu$, but also encodes the cardinality of $I$. After recourse to work of Rush and Shi mapping the anatomy of the lattice isomorphism, the upshot is a uniform proof that the cardinality statistic exhibits homomesy. A further application of these ideas shows that the statistic tracking the number of maximal elements in an order ideal is also homomesic, extending another result of Propp and Roby.