  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.
