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The Fon-Der-Flaass action partitions the order ideals of a poset intodisjoint orbits. For a product of two chains, Propp and Roby observed ---across orbits --- the mean cardinality of the order ideals within an orbit tobe invariant. That this phenomenon, which they christened homomesy, extends toall 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 latticeof 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 behaviorof the Weyl group simple reflections on $\mu$ not only uniquely determines$\mu$, but also encodes the cardinality of $I$. After recourse to work of Rushand Shi mapping the anatomy of the lattice isomorphism, the upshot is a uniformproof that the cardinality statistic exhibits homomesy.  A further application of these ideas shows that the statistic tracking thenumber of maximal elements in an order ideal is also homomesic, extendinganother result of Propp and Roby.

On Orbits of Order Ideals of Minuscule Posets II: Homomesy