Antichains in weight posets associated with gradings of simple Lie algebras

Abstract

For a reductive Lie algebra ${\mathfrak {h}}$ and a simple finite-dimensional ${\mathfrak {h}}$-module V, the set of weights of $V,\,\mathcal {P}(V)$, is equipped with a natural partial order. We consider antichains in the weight poset $\mathcal {P}(V)$ 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 ${\mathfrak {h}}={\mathfrak {g}}(0)$ and $V={\mathfrak {g}}(1)$ for a ${\mathbb {Z}}$-grading ${\mathfrak {g}}=\bigoplus _{i\in {\mathbb {Z}}}{\mathfrak {g}}(i)$ of a simple Lie algebra ${\mathfrak {g}}$. Then V is a weight multiplicity free ${\mathfrak {h}}$-module and $\mathcal {P}(V)$ can be regarded as a subposet of $\Delta ^+$, where $\Delta$ is the root system of ${\mathfrak {g}}$. Our goal is to demonstrate that the weight posets associated with ${\mathbb {Z}}$-gradings exhibit many good properties that are similar to those of $\Delta ^+$ that are observed earlier in Panyushev (Eur J Combin 30(2):586–594, 2009).