Antichains in weight posets associated with gradings of simple Lie algebras

For a reductive Lie algebra and a simple finite-dimensional -module V, the set of weights of , is equipped with a natural partial order. We consider antichains in the weight poset and a certain operator acting on antichains. Eventually, we impose stronger constraints on and stick to the case in which and for a -grading of a simple Lie algebra . Then V is a weight multiplicity free -module and can be regarded as a subposet of , where is the root system of . Our goal is to demonstrate that the weight posets associated with -gradings exhibit many good properties that are similar to those of that are observed earlier in Panyushev (Eur J Combin 30(2):586-594, 2009).