Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

Abstract

The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure ("weight poset"). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with $\mathbb{Z}$-gradings of simple Lie algebras (arXiv: math.CO 1411.7683). If $\mathfrak{g}$ is a simple Lie algebra, then a $\mathbb{Z}$-grading of $\mathfrak{g}$ induces a $\mathbb{Z}$-grading of the corresponding root system $\Delta$. In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets $\Delta (1)$. In particular, we provide a bijection between the lower ideals in $\Delta (1)$ and certain elements of the Weyl group of $\mathfrak{g}$. An inspiring observation is that, to a great extent, the theory of lower ideals in $\Delta (1)$ is similar to the theory of upper (= ad-nilpotent) ideals in the whole poset of positive roots ${\Delta }^{+}$.

arXiv comment19 pages

References in corpus (2)

• B-stable ideals in the nilradical of a Borel subalgebra
• Exponents for B-stable ideals

Cited by in corpus (1)

• Orbits of antichains in certain root posets