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 $\mathbb Z$-gradings of simple Lie algebras (arXiv: math.CO 1411.7683). If $\mathfrak g$ is a simple Lie algebra, then a $\mathbb Z$-grading of $\mathfrak g$ induces a $\mathbb Z$-grading of the corresponding root system $Δ$. In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets $Δ(1)$. In particular, we provide a bijection between the lower ideals in $Δ(1)$ 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 $Δ(1)$ is similar to the theory of upper (= ad-nilpotent) ideals in the whole poset of positive roots $Δ^+$.