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

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.