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 $V$, $P(V)$, 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 ${\Delta }^{+}$, where $\Delta$ 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 ${\Delta }^{+}$ that are observed earlier in arXiv: math.CO 0711.3353 (=Ref. [14] in the text).

28 pages