Polyptych lattices and marked chain-order polytopes

TL;DR

This paper introduces polyptych lattices as a framework to generate families of toric degenerations related by piecewise-linear transformations, extending concepts from cluster algebras to marked chain-order polytopes.

Contribution

It studies polyptych lattices involving transfer maps for marked chain-order polytopes and computes the Cox ring of the associated projective variety.

Findings

Established a family of toric degenerations to marked chain-order polytopes

Connected polyptych lattices with Gelfand-Tsetlin poset

Computed the Cox ring of the projective variety

Abstract

The theory of polyptych lattices is a framework to obtain a family of toric degenerations whose polytopes are related by piecewise-linear transformations. It can be regarded as a generalization of toric degenerations arising from cluster algebras. In this paper, we study polyptych lattices consisting of transfer maps for marked chain-order polytopes, and obtain a family of toric degenerations of a projective variety to marked chain-order polytopes for the Gelfand-Tsetlin poset. We also compute the Cox ring of this projective variety.