Newton-Okounkov bodies for Bott-Samelson varieties and string polytopes for generalized Demazure modules

Abstract

A Newton-Okounkov convex body is a convex body constructed from a projective variety with a valuation on its homogeneous coordinate ring; this generalizes a Newton polytope for a toric variety. This convex body has various kinds of information about the original projective variety; for instance, Kaveh showed that the string polytopes from representation theory are examples of Newton-Okounkov bodies for Schubert varieties. In this paper, we extend the notion of string polytopes for Demazure modules to generalized Demazure modules, and prove that the resulting generalized string polytopes are identical to the Newton-Okounkov bodies for Bott-Samelson varieties with respect to a specific valuation. As an application of this result, we show that these are indeed polytopes.

30 pages

References in corpus (1)

• Convex bodies and algebraic equations on affine varieties

Cited by in corpus (6)

• A comparison of Newton-Okounkov polytopes of Schubert varieties
• Newton-Okounkov bodies of Bott-Samelson varieties and Grossberg-Karshon twisted cubes
• 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
• Singular string polytopes and functorial resolutions from Newton-Okounkov bodies