Newton-Okounkov bodies of flag varieties and combinatorial mutations

Naoki Fujita, Akihiro Higashitani

Published 2020-03-21Version 1

A Newton-Okounkov body is a convex body constructed from a polarized variety with a higher rank valuation on the function field, which gives a systematic method of constructing toric degenerations of polarized varieties. Its combinatorial properties heavily depend on the choice of a valuation, and it is a fundamental problem to relate Newton-Okounkov bodies associated with different kinds of valuations. In this paper, we address this problem for flag varieties using the framework of combinatorial mutations which was introduced in the context of mirror symmetry for Fano manifolds. By applying iterated combinatorial mutations, we connect specific Newton-Okounkov bodies of flag varieties including string polytopes, Nakashima-Zelevinsky polytopes, and FFLV polytopes.