Weak Faces of Highest Weight Modules and Root Systems

Abstract

Chari and Greenstein (Adv. Math. 220(4), 1193–1221, 2009) introduced certain combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak {g}$ , which were important in studying Kirillov–Reshetikhin modules over $U_{q}(\widehat {\mathfrak {g}})$ and their specializations. Later, Khare (J. Algebra. 455, 32–76, 2016) studied these subsets for large classes of highest weight $\mathfrak {g}$ -modules (in finite type), under the name of weak- $\mathbb {A}$ -faces (for a subgroup $\mathbb {A}$ of $(\mathbb {R},+)$ ), and more generally, ({2};{1,2})-closed subsets. These notions extend and unify the faces of Weyl polytopes as well as the aforementioned combinatorial subsets. In this paper, we consider these “discrete” notions for an arbitrary Kac–Moody Lie algebra $\mathfrak {g}$ , in three prominent settings: (a) the weights of an arbitrary highest weight $\mathfrak {g}$ -module V; (b) the convex hull of the weights of V; (c) the weights of the adjoint representation. For (a) (respectively, (b)) for all highest weight $\mathfrak {g}$ -modules V, we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the sets of weights on the exposed faces (respectively, equal the exposed faces) of the convex hull of weights conv(wtV ). This completes the partial progress of Khare in finite type, and is novel in infinite type. Our proofs are type-free and self-contained. For (c) involving the root system, we similarly achieve complete classifications. For the weights of $\text {ad} \mathfrak {g}$ for any indecomposable Kac–Moody $\mathfrak {g}$ , we show that the weak- $\mathbb {A}$ -faces and ({2};{1,2})-closed subsets agree, and equal the Weyl group translates of the sets of weights in certain “standard parabolic faces” (which also holds for highest weight modules). This was proved by Chari and her coauthors for Δ ⊔{0} in finite type, but is novel for other types.