Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes

TL;DR

This paper characterizes the Newton-Okounkov bodies of Bott-Samelson varieties as explicit lattice polytopes, using a novel valuation, linking these to Grossberg-Karshon twisted cubes and representation theory.

Contribution

It introduces a new valuation for defining Newton-Okounkov bodies of Bott-Samelson varieties and relates these bodies to Grossberg-Karshon twisted cubes, expanding understanding of their geometric and combinatorial structure.

Findings

Explicit inequalities define the Newton-Okounkov body as a lattice polytope.

The valuation used differs from previous approaches, providing new insights.

Connections established between Newton-Okounkov bodies, twisted cubes, and representation theory.

Abstract

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 also studied previously by the authors in relation to certain toric varieties associated to Bott-Samelson varieties.