Combinatorial Mutations of Gelfand-Tsetlin Polytopes, Feigin-Fourier-Littelmann-Vinberg Polytopes, and Block Diagonal Matching Field Polytopes

TL;DR

This paper explores the combinatorial relationships among Gelfand-Tsetlin, Feigin-Fourier-Littelmann-Vinberg, and matching field polytopes, revealing mutations and degenerations that connect these structures within the context of Grassmannians.

Contribution

It demonstrates that these polytopes are interconnected through combinatorial mutations and are all Newton-Okounkov bodies, providing new insights into their geometric and algebraic properties.

Findings

Polytopes are among matching field polytopes.

They are related by combinatorial mutations.

All are Newton-Okounkov bodies for Grassmannians.

Abstract

The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that they are related by a sequence of combinatorial mutations that passes only through matching field polytopes. As a result, we obtain a family of matching fields that give rise to toric degenerations for the Grassmannians. Moreover, all polytopes in the family are Newton-Okounkov bodies for the Grassmannians.