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

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.

arXiv comment17 pages, 3 figures. References added

References in corpus (5)

• Crystal bases and Newton-Okounkov bodies
• Convex bodies and algebraic equations on affine varieties
• Standard Monomial Theory for Bott-Samelson Varieties of GL(n)
• Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules
• Grossberg-Karshon twisted cubes and basepoint-free divisors

Cited by in corpus (9)

• A comparison of Newton-Okounkov polytopes of Schubert varieties
• Newton-Okounkov polytopes of flag varieties
• On a notion of anticanonical class for families of convex polytopes
• Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases
• Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums
• Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance
• K-orbit closures and Barbasch-Evens-Magyar varieties
• Singular string polytopes and functorial resolutions from Newton-Okounkov bodies
• Bott canonical basis?