Properties of weight posets for weight multiplicity free representations

We study weight posets of weight multiplicity free (=wmf) representations $R$ of reductive Lie algebras. Specifically, we are interested in relations between $\dim R$ and the number of edges in the Hasse diagram of the corresponding weight poset, # E(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 Z-gradings of simple Lie algebras. For Z-gradings, we prove that 0< 2dim R-# E(R) < h, where $h$ is the Coxeter number of $\mathfrak g$. For periodic gradings, we prove that 0\le 2dim R-# E(R).