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 [1] that for a lattice polytope $P\subseteq N_{\mathbb{R}}$ containing the origin in its interior, the polar duals $P^{\ast }\subseteq M_{\mathbb{R}}$ and ${\mathrm{mut}}_w(P,F{)}^{\ast }\subseteq M_{\mathbb{R}}$ have the same Ehrhart series. For extending this framework, in this paper, we introduce the combinatorial mutation for the Minkowski sum of rational polytopes and rational polyhedral pointed cones in $N_{\mathbb{R}}$. We can also introduce the combinatorial mutation in the dual side $M_{\mathbb{R}}$, which we can apply for every rational polytope in $M_{\mathbb{R}}$ containing the origin (not necessarily in the interior). 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 the combinatorial mutation 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.

14 pages, 2 figures

References in corpus (1)

• Minkowski Polynomials and Mutations