Grossberg-Karshon twisted cubes and basepoint-free divisors

TL;DR

This paper provides criteria for when Grossberg-Karshon twisted cubes are true polytopes, linking their geometric properties to representation theory and toric degenerations, and offers new conditions for untwistedness.

Contribution

It introduces several equivalent criteria for the untwistedness of twisted cubes, connecting geometric, combinatorial, and representation-theoretic aspects, including basepoint-free divisors and convexity conditions.

Findings

Untwisted twisted cubes are true polytopes with positive character formulas.

Basepoint-freeness of certain divisors characterizes untwistedness.

Convexity and positivity conditions suffice for untwistedness.

Abstract

Let $G$ be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible $G$-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e. closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e. each lattice point appears precisely once in the formula, with coefficient $+1$. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $\lambda$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $\lambda$ and $\underline{w}$.