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 λ an integral weight and w a choice of word decomposition of a Weyl group element, we give two simple necessary conditions Date: November 15, 2018.

We describe, under certain conditions, the Newton-Okounkov body of a Bott-Samelson variety as a lattice polytope defined by an explicit list of inequalities. The valuation that we use to define the Newton-Okounkov body is different from that used previously in the literature. The polytope that arises is a special case of the Grossberg-Karshon twisted cubes studied by Grossberg and Karshon in connection to character formulae for irreducible G-representations and

The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation νmax coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton-Okounkov bodies of Bott-Samelson varieties with respect to a different valuation ν min in terms of Grossberg-Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton-Okounkov bodies coincide.

We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question: ''Describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety. As an application, we understand G. Mikhalkin's correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves.

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