Reflexivity of Newton–Okounkov bodies of partial flag varieties

Steinert, Christian

doi:10.1090/ert/621

Reflexivity of Newton–Okounkov bodies of partial flag varieties
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by Christian Steinert

Represent. Theory 26 (2022), 859-873

DOI: https://doi.org/10.1090/ert/621

Published electronically: August 16, 2022

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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.

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