Repetitive higher cluster categories of type A_n

TL;DR

This paper introduces a geometric model for the repetitive higher cluster category of type A_n, generalizing previous models and connecting orbit categories with diagonals in polygons.

Contribution

It generalizes the construction of cluster categories to a broader setting using diagonals in polygons, unifying previous models and providing a geometric interpretation.

Findings

Equivalence between the repetitive higher cluster category and a diagonals-based category.

Recovery of known models when p=m=1 and p=1, m>1.

Development of a geometric model for the bounded derived category.

Abstract

We show that the repetitive higher cluster category of type A_n, defined as the orbit category D^b(mod kA_n)/(tau^{-1}[m])^p, is equivalent to a category defined on a subset of diagonals in a regular p(nm+1)-gon. This generalizes the construction of Caldero-Chapoton-Schiffler, which we recover when p=m=1, and the work of Baur-Marsh, treating the case p=1, m>1. Our approach also leads to a geometric model of the bounded derived category D^b(mod kA_n).