2014/11/27 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #06A07 #17B20 #20F55 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:06A07 #msc:17B20 #msc:20F55
paper · pdf · doi:10.48550/arxiv.1411.7683
28 pages
arxiv created 2014/11/27 · openalex publication_date 2014/11/27 · arxiv updated 2014/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a reductive Lie algebra \mathfrak h and a simple finite-dimensional \mathfrak h-module V, the set of weights of V, P(V), has a natural poset structure. We consider antichains in the weight poset P(V) and a certain operator \mathfrak X acting on antichains. Eventually, we impose stronger constraints on (\mathfrak h,V) and stick to the case in which \mathfrak h and V are associated with a Z-grading of a simple Lie algebra \mathfrak g. Then V is a weight multiplicity free \mathfrak h-module and P(V) can be regarded as a subposet of Δ+, where Δ is the root system of \mathfrak g. Our goal is to demonstrate that antichains in the weight posets associated with Z-gradings of \mathfrak g exhibit many good properties similar to those of Δ+ that are observed earlier in arXiv: math.CO 0711.3353 (=Ref. [14] in the text).