Properties of weight posets for weight multiplicity free representations

TL;DR

This paper investigates the structure of weight posets in weight multiplicity free representations of reductive Lie algebras, focusing on relations between representation dimensions and poset edges, with specific results for gradings.

Contribution

It computes edges and polynomials of weight posets for all wmf-representations and establishes bounds relating dimension and edges for gradings of simple Lie algebras.

Findings

Proves bounds on 2*dim R - #E(R) for Z-gradings.

Establishes non-trivial isomorphisms between different weight posets.

Provides explicit computations for all wmf-representations.

Abstract

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 $0\le 2dim R-# E(R)$.