Newton-Okounkov polytopes of flag varieties and marked chain-order polytopes

TL;DR

This paper links marked chain-order polytopes to Newton-Okounkov bodies of flag varieties, unifying previous polytope realizations and demonstrating degenerations to toric varieties with applications in representation theory.

Contribution

It provides a new realization of marked chain-order polytopes as Newton-Okounkov bodies of flag varieties, connecting various polytope models in a uniform framework.

Findings

Flag variety degenerates into a toric variety associated with a marked chain-order polytope.

Constructs a basis of irreducible highest weight representations parametrized by lattice points.

Unifies previous realizations of Gelfand-Tsetlin and Feigin-Fourier-Littelmann-Vinberg polytopes.

Abstract

Marked chain-order polytopes are convex polytopes constructed from a marked poset, which give a discrete family relating a marked order polytope with a marked chain polytope. In this paper, we consider the Gelfand-Tsetlin poset of type A, and realize the associated marked chain-order polytopes as Newton-Okounkov bodies of the flag variety. Our realization connects previous realizations of Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann-Vinberg polytopes as Newton-Okounkov bodies in a uniform way. As an application, we prove that the flag variety degenerates into the irreducible normal projective toric variety corresponding to a marked chain-order polytope. We also construct a specific basis of an irreducible highest weight representation which is naturally parametrized by the set of lattice points in a marked chain-order polytope.