Properties of weight posets for weight multiplicity free representations

2008/10/16 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT

paper · pdf · doi:10.48550/arxiv.0810.2919

18 pages

arxiv created 2008/10/16 · openalex publication_date 2008/10/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≤ 2dim R-# E(R).

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