Properties of weight posets for weight multiplicity free representations

We study weight posets of weight multiplicity free (WMF) representations of reductive Lie algebras. Specifically, we are interested in relations between $\dim\mathcal R$ and the number of edges in the Hasse diagram of the corresponding weight poset $\#\mathcal E(\mathcal R)$. We compute the number of edges and upper covering polynomials for the weight posets of all WMF-representations. We also point out non-trivial isomorphisms between weight posets of different irreducible WMF-representations.
Our main results concern WMF-representations associated with periodic gradings or $\mathbb Z$-gradings of simple Lie algebras. For $\mathbb Z$-gradings, we prove that $0<2\dim\mathcal R-\#\mathcal E(\mathcal R)<h$, where $h$ is the Coxeter number of $\mathfrak g$. For periodic gradings, we prove that $0\le2\dim\mathcal R-\#\mathcal E(\mathcal R)$.