Grothendieck groups of repetitive cluster categories

TL;DR

This paper investigates the Grothendieck groups of repetitive cluster categories, extending classical results and revealing new structural patterns related to the repetitive parameter, thereby deepening understanding of their algebraic and categorical properties.

Contribution

It computes key results on Grothendieck groups for specific repetitive cluster categories, extending known theories and uncovering new structural insights.

Findings

Extended classical Grothendieck group computations to repetitive categories

Identified new structural patterns linked to the parameter p

Strengthened the connection between Grothendieck groups, AR theory, and Coxeter transformations

Abstract

In order to study cluster-tilted algebras and their intermediate coverings, Zhu introduced the notion of repetitive cluster categories, defined as the orbit categories $\mathcal D^b(\mathcal H)/\langle(\tau^{-1}\Sigma)^p\rangle$ for $1\leq p\in\mathbb{N}$, where $\mathcal H$ is a hereditary abelian category with tilting objects. In this paper, we compute partial but essential results on the Grothendieck groups of the repetitive cluster categories $\mathcal D^b({\rm mod}KA_n)/\langle(\tau^{-1}\Sigma)^p\rangle$ and $\mathcal D^b({\rm mod} KD_n)/\langle(\tau^{-1}\Sigma)^p\rangle$. Our results extend the known computations for classical cluster categories, reveal new structural patterns arising from the repetitive parameter $p$, and provide further evidence of the close interplay between Grothendieck groups, Auslander-Reiten theory, and Coxeter transformations.