Newton-Okounkov bodies arising from cluster structures

Abstract: A toric degeneration is a flat degeneration from a projective variety to a toric variety, which can be used to apply the theory of toric varieties to other projective varieties. In this talk, we discuss relations among the following three constructions of toric degenerations: representation theory, Newton-Okounkov bodies, and cluster algebras. More precisely, we construct Newton-Okounkov bodies using cluster structures, and realize representation-theoretic and cluster-theoretic toric degenerations from this framework. As an application, we connect two kinds of representation-theoretic polytopes (string polytopes and Nakashima-Zelevinsky polytopes) by tropicalized cluster mutations. We also discuss relations with combinatorial mutations which was introduced in the context of mirror symmetry for Fano varieties. More precisely, we relate dual polytopes of these representation-theoretic polytopes by combinatorial mutations. This talk is based on joint works with Hironori Oya and Akihiro Higashitani.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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