Grothendieck groups of repetitive cluster categories

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.