A construction of polyptych lattices via a pair of Gorenstein PL cones
Abstract
The theory of polyptych lattices seeks to incorporate, into a single combinatorial framework, the piecewise-linear bijections (mutations) that appear in many theories that generalize toric geometry, such as Newton-Okounkov bodies, toric degenerations, cluster varieties, and other areas. The link to algebraic geometry comes from the accompanying notion of detropicalizations of polyptych lattices, and their associated compactifications. The main result of this manuscript gives a concrete construction of a strict dual pair (ℰ, ℱ) of polyptych lattices, which we call a Gorenstein PL cone extension. To define our construction, we introduce the notion of Gorenstein PL cones, which are a polyptych analogue of a Gorenstein cone in the classical setting. If the original polyptych lattices are detropicalizable, then the new polyptych lattices ℰ,ℱ are also detropicalizable, via a simple explicit formula. Our Gorenstein PL cone extensions give a rich source of examples of strict dual pairs of polyptych lattices. In particular, any pair of Gorenstein-Fano polytopes Δ,Δ' of dimension n,n' respectively, gives rise to a strict dual pair (ℰ(Δ,Δ'), ℱ(Δ,Δ')) of detropicalizable polyptych lattices of rank n+n'+1. Other examples can be built from cluster data.
Related
- Directionally Fine Families of Cones: Construction Principles, Approximation, and Separation
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- The Generative Mechanism of Dimension: An Axiomatic Construction of Dimensional Conventions (Constructive Mathematics Series, Paper 6; Lean formalization, model-pair independence, numerical evidence)
- Virtually Gorenstein Artin algebras are weakly Gorenstein
- GORENSTEIN FANO POLYTOPES ARISING FROM ORDER POLYTOPES AND CHAIN POLYTOPES
BibTeX
@misc{escobar2026construction,
title = {{A construction of polyptych lattices via a pair of Gorenstein PL cones}},
author = {Laura Escobar and Megumi Harada and Christopher Manon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32549},
eprint = {2609.32549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32549},
}
RIS
TY - UNPB
TI - A construction of polyptych lattices via a pair of Gorenstein PL cones
AU - Laura Escobar
AU - Megumi Harada
AU - Christopher Manon
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32549
UR - https://arxiv.org/abs/2609.32549
AB - The theory of polyptych lattices seeks to incorporate, into a single combinatorial framework, the piecewise-linear bijections (mutations) that appear in many theories that generalize toric geometry, such as Newton-Okounkov bodies, toric degenerations, cluster varieties, and other areas. The link to algebraic geometry comes from the accompanying notion of detropicalizations of polyptych lattices, and their associated compactifications. The main result of this manuscript gives a concrete construction of a strict dual pair (ℰ, ℱ) of polyptych lattices, which we call a Gorenstein PL cone extension. To define our construction, we introduce the notion of Gorenstein PL cones, which are a polyptych analogue of a Gorenstein cone in the classical setting. If the original polyptych lattices are detropicalizable, then the new polyptych lattices ℰ,ℱ are also detropicalizable, via a simple explicit formula. Our Gorenstein PL cone extensions give a rich source of examples of strict dual pairs of polyptych lattices. In particular, any pair of Gorenstein-Fano polytopes Δ,Δ' of dimension n,n' respectively, gives rise to a strict dual pair (ℰ(Δ,Δ'), ℱ(Δ,Δ')) of detropicalizable polyptych lattices of rank n+n'+1. Other examples can be built from cluster data.
ER -