GROSSBERG-KARSHON TWISTED CUBES AND BASEPOINT-FREE DIVISORS
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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 $\underline{w}$.