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 dimR 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 WMFrepresentations. 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 < 2 dimR −#E(R) < h, where h is the Coxeter number of g. For periodic gradings, we prove that 0 6 2 dimR −#E(R). 2000 Math. Subj. Class. Primary: 05E15; Secondary: 06A07, 17B20.