Weak faces and a formula for weights of highest weight modules, via parabolic partial sum property for roots

TL;DR

This paper introduces a parabolic partial sum property for roots in Lie algebras, leading to new descriptions of weights in highest weight modules and classifications of certain weight subsets, with implications for Lie theory.

Contribution

It generalizes the partial sum property to parabolic settings and applies this to refine weight formulas and classify weight subsets in highest weight modules.

Findings

Minimal description of weights for non-integrable modules

Minkowski difference formula for weights

Classification and equivalence of weak faces and 212-closed subsets

Abstract

Let $\mathfrak{g}$ be a finite or an affine type Lie algebra over $\mathbb{C}$ with root system $\Delta$. We show a parabolic generalization of the partial sum property for $\Delta$, which we term the parabolic partial sum property. It allows any root $\beta$ involving (any) fixed subset $S$ of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from $S$, with each partial sum also being a root. We show three applications of this property to weights of highest weight $\mathfrak{g}$-modules: (1)~We provide a minimal description for the weights of all non-integrable simple highest weight $\mathfrak{g}$-modules, refining the weight formulas shown by Khare [J. Algebra} 2016] and Dhillon-Khare [Adv. Math. 2017]. (2)~We provide a Minkowski difference formula for the weights of an arbitrary highest weight $\mathfrak{g}$-module. (3)~We completely classify and show the equivalence of two combinatorial subsets - weak faces and 212-closed subsets - of the weights of all highest weight $\mathfrak{g}$-modules. These two subsets were introduced and studied by Chari-Greenstein [Adv. Math. 2009], with applications to Lie theory including character formulas. We also show ($3'$) a similar equivalence for root systems.