Newton-Okounkov bodies of flag varieties and combinatorial mutations. (English) Zbl 1524.14109
The authors investigate the Newton-Okounkov bodies of certain projective varieties, denoted as \(X\). These bodies are convex bodies associated with a globally generated line bundle on \(X\) and with a higher rank valuation on the function field \(\mathbb{C}(X)\). Newton-Okounkov bodies, introduced by Okounkov and further developed by Lazarsfeld-Mustata and Kaveh-Khovanskii, play a crucial role in constructing toric degenerations of \(X\).
Utilizing the framework of iterated combinatorial mutations for lattice polytopes introduced by Akhtar-Coates-Galkin-Kasprzyk, the authors establish connections between specific Newton-Okounkov bodies of flag varieties. This includes string polytopes, Nakashima-Zelevinsky polytopes, and FFLV polytopes.
Utilizing the framework of iterated combinatorial mutations for lattice polytopes introduced by Akhtar-Coates-Galkin-Kasprzyk, the authors establish connections between specific Newton-Okounkov bodies of flag varieties. This includes string polytopes, Nakashima-Zelevinsky polytopes, and FFLV polytopes.
Reviewer: Irem Portakal (Leipzig)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 05E10 | Combinatorial aspects of representation theory |
| 13F60 | Cluster algebras |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
Keywords:
Newton-Okounkov body; combinatorial mutation; flag variety; cluster algebra; tropicalized mutationReferences:
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