Newton–Okounkov polytopes of Schubert varieties arising from cluster structures

The theory of Newton–Okounkov bodies is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of projective varieties. In this paper, we study Newton–Okounkov bodies of Schubert varieties from the theory of cluster algebras. We construct Newton–Okounkov bodies using specific valuations which generalize extended g-vectors in cluster theory, and discuss how these bodies are related to string polytopes and Nakashima–Zelevinsky polytopes.