  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.
