W4292191549

Representation Theory of the American Mathematical Society, vol. 26

Abstract

Assume that the valuation semigroup $\Gamma (\lambda )$ of an arbitrary partial flag variety corresponding to the line bundle $\mathcal {L_\lambda }$ constructed via a full-rank valuation is finitely generated and saturated. We use Ehrhart theory to prove that the associated Newton–Okounkov body — which happens to be a rational, convex polytope — contains exactly one lattice point in its interior if and only if $\mathcal {L}_\lambda$ is the anticanonical line bundle. Furthermore, we use this unique lattice point to construct the dual polytope of the Newton–Okounkov body and prove that this dual is a lattice polytope using a result by Hibi. This leads to an unexpected, necessary and sufficient condition for the Newton–Okounkov body to be reflexive.

Authors 1

1. Christian Steinert corresponding Aachen Department of Mathematics Chair for Algebra and Representation Theory

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, Chair for Algebra and Representation Theory, RWTH Aachen University, 25042 Aachen, Germany