Two poset polytopes are mutation-equivalent

Abstract

The combinatorial mutation ${\mathrm{mut}}_w(P,F)$ for a lattice polytope $P$ was introduced in the context of mirror symmetry for Fano manifolds in [1]. It was also proved in \cite{ACGK} that for a lattice polytope $P\subseteq N_{\mathbb{R}}$ containing the origin in its interior, the polar dual $P^{\ast }\subseteq M_{\mathbb{R}}$ and ${\mathrm{mut}}_w(P,F{)}^{\ast }\subseteq M_{\mathbb{R}}$ have the same Ehrhart quasi-polynomial. To extend this framework, we introduce combinatorial mutation for rational pointed polyhedra in $N_{\mathbb{R}}$ containing the origin in their interiors. Such polyhedra are Minkowski sums of rational polytopes and rational polyhedral pointed cones. On the dual side $M_{\mathbb{R}}$, the construction applies to full-dimensional rational polytopes containing the origin, not necessarily in their interiors. As an application of this extension of the combinatorial mutation, we prove that the chain polytope of a poset $\Pi$ can be obtained by a sequence of combinatorial mutations in $M_{\mathbb{R}}$ from the order polytope of $\Pi$. Namely, the order polytope and the chain polytope of the same poset $\Pi$ are mutation-equivalent.

18 pages, 7 figures

References in corpus (5)

• Mirror Symmetry and Fano Manifolds
• Minkowski Polynomials and Mutations
• Gelfand-Tsetlin polytopes and Feigin-Fourier-Littelmann polytopes as marked poset polytopes
• Deformations of Dimer Models
• Enriched order polytopes and Enriched Hibi rings

Cited by in corpus (3)

• Newton-Okounkov bodies of flag varieties and combinatorial mutations
• Combinatorial mutations of Newton-Okounkov polytopes arising from plabic graphs
• Two enriched poset polytopes