The geometry, and the (exposed) faces, of $X$ a “Root polytope” or “Weyl polytope” over a complex simple Lie algebra $\mathfrak{g}$, have been studied for many decades for various applications, including by Satake, Borel–Tits, Casselman, and Vinberg among others. This talk focuses on two recent combinatorial analogues to these classical faces, in the discrete setting of weight-sets $X$.

Chari et al [*Adv. Math.* 2009, *J. Pure Appl. Algebra* 2012]
introduced and studied two combinatorial subsets of $X$ a root system or
the weight-set wt $V$ of an integrable simple highest weight
$\mathfrak{g}$-module $V$, for studying Kirillov–Reshetikhin
modules over the specialization at $q=1$ of quantum affine algebras
$U_q(\hat{\mathfrak{g}})$ and for constructing Koszul algebras. Later,
Khare [*J. Algebra* 2016] studied these subsets under the names
“weak-$\mathbb{A}$-faces” (for subgroups $\mathbb{A}\subseteq
(\mathbb{R},+)$) and “$212$-closed subsets”.
For two subsets $Y\subseteq X$ in a vector space, $Y$ is said to be
$212$-closed in $X$, if $y_1+y_2=x_2+x_2$ for $y_i\in Y$ and $x_i\in X$
implies $x_1,x_2\in Y$.

In finite type, Chari et al classified these discrete faces for $X$ root
systems and wt $V$ for all integrable $V$, and Khare for all
(non-integrable) simple $V$. In the talk, we extend and completely solve
this problem for *all* highest weight modules $V$ over *any*
Kac–Moody Lie algebra $\mathfrak{g}$.
We classify, and show the equality of, the weak faces and
$212$-closed subsets in the three prominent settings of $X$:
(a) wt $V$ $\forall V$,
(b) the hull of wt $V$ $\forall V$,
(c) wt $\mathfrak{g}$ (consisting of roots and 0).
Moreover, in the case of (a) (resp. of (b)), such subsets are precisely
the weights falling on the exposed faces (resp. the exposed faces) of the
hulls of wt $V$.

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Last updated: 24 Mar 2023