Restricted Chain-Order Polytopes via Combinatorial Mutations

TL;DR

This paper investigates restricted chain-order polytopes linked to Young diagrams, demonstrating they are interconnected via combinatorial mutations and identifying a broad class of polytopes exhibiting period collapse phenomena.

Contribution

It introduces the concept of restricted chain-order polytopes for Young diagrams and proves their interrelation through combinatorial mutations, linking these to period collapse phenomena.

Findings

All restricted chain-order polytopes for a fixed Young diagram are mutation-equivalent.

These polytopes exhibit the period collapse phenomenon.

The study extends understanding of Ehrhart polynomials in this context.

Abstract

We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the period collapse phenomenon is invariant under combinatorial mutations, we provide a large class of rational polytopes that give rise to period collapse.