Let be a finite Weyl group, and let be a simple root of corresponding to a minuscule weight of the associated Lie algebra. The set of positive roots, , of in which appears with nonzero coefficient is equipped with a natural partial order. Remarkably, this partial order makes into a distributive lattice. Also remarkably, the lattice of ideals of is isomorphic to the lattice of weights of the minuscule representation. There are signs that this result is folklore, but I have never seen it stated. I will discuss the strangeness of this result, speculate on possible extensions of it, and lament the lack (to my knowledge) of a conceptual proof.
Roots, weights, and distributive lattices