Antichains in weight posets associated with gradings of simple Lie algebras

TL;DR

This paper studies the structure of antichains in weight posets arising from gradings of simple Lie algebras, revealing properties analogous to those of root systems and contributing to the understanding of Lie algebra representations.

Contribution

It introduces a detailed analysis of antichains in weight posets associated with Z-gradings of simple Lie algebras, extending known properties of root systems to these posets.

Findings

Antichains in weight posets share properties with root systems.

The operator al Xf acts on antichains with notable structure.

Results generalize previous findings on root systems and poset properties.

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).