Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure (“weight poset”). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with $\mathbb Z$-gradingsofsimpleLiealgebras(arXiv:math\mathrm{.}CO1411.7683)\mathrm{.}If$\mathfrak g$isasimpleLiealgebra,thena$\mathbb Z$-gradingof$\mathfrak g$inducesa$\mathbb Z$-gradingofthecorrespondingrootsystem$\Delta$. In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets $\mathrm{\Delta }(1)$. In particular, we provide a bijection between the lower ideals in $\mathrm{\Delta }(1)$ and certain elements of the Weyl group of $\mathfrak{g}$. An inspiring observation is that, to a great extent, the theory of lower ideals in $\mathrm{\Delta }(1)$ is similar to the theory of upper (= ad-nilpotent) ideals in the whole poset of positive roots ${\mathrm{\Delta }}^{+}$.