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Polyptych lattices and marked chain-order polytopes

Naoki Fujita, Akihiro Higashitani

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.

Polyptych lattices and marked chain-order polytopes

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.
Paper Structure (11 sections, 23 theorems, 179 equations, 5 figures)

This paper contains 11 sections, 23 theorems, 179 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Basic definitions on polyptych lattices
  3. Relation with marked chain-order polytopes
  4. Marked chain-order polytopes
  5. Marked chain-order polytopes as PL polytopes
  6. Marked posets and spaces of points
  7. Case of Gelfand--Tsetlin posets of type $C$
  8. Spaces of points and strict dual pairs
  9. Detropicalizations and toric degenerations
  10. Cox rings of compactifications
  11. Case of Gelfand--Tsetlin posets of type $A$

Key Result

Theorem 1.1

The map ${\bf B} \rightarrow \mathcal{M}$ induces a valuation $\nu$ on $\mathcal{A}$ to the canonical $\mathbb{Z}_{\geq 0}$-semialgebra $S_{\mathcal{M}}$ associated with $\mathcal{M}$, which makes $\mathcal{A}$ a detropicalization of $\mathcal{M}$ with convex adapted basis $\overline{\bf B}$; see Se

Figures (5)

  • Figure 3.1: The Hasse diagram of $\Pi_1$.
  • Figure 3.2: The marked Hasse diagram of $(\Pi_2, \Pi_2^\ast, \lambda)$.
  • Figure 4.1: The marked Hasse diagram of the Gelfand--Tsetlin poset $(\Pi_C, \Pi^\ast_C, \lambda)$.
  • Figure 4.2: A Hasse diagram satisfying condition $(\spadesuit)$.
  • Figure 5.1: The marked Hasse diagram of the Gelfand--Tsetlin poset $(\Pi_A, \Pi^\ast_A, \lambda)$.

Theorems & Definitions (50)

  • Theorem 1.1: see Theorem \ref{['t:detropicalization_type_C']} and Remark \ref{['r:general_case']}
  • Theorem 1.2: see Theorem \ref{['t:NO_degeneration_type_C']} and Remark \ref{['r:general_case']}
  • Theorem 1.3: see Theorem \ref{['t:Cox_type_C']} and Remark \ref{['r:general_case']}
  • Definition 2.1: see EHM
  • Definition 2.2: EHM
  • Definition 2.3: EHM
  • Definition 2.4: EHM
  • Definition 2.5: EHM
  • Definition 3.1: FFLP
  • Theorem 3.2: see FFLP
  • ...and 40 more