Grothendieck groups of repetitive cluster categories

2025/01/19 by Huimin Chang, Chang, Huimin, Murphy, Dave +1
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2501.11021

openalex publication_date 2025/01/19 · openalex created_date 2025/01/23 · openalex updated_date 2026/07/28

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)/⟨(τ^-1Σ)^p⟩ for 1≤ p∈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 modKA_n)/⟨(τ^-1Σ)^p⟩ and \mathcal D^b(\rm mod KD_n)/⟨(τ^-1Σ)^p⟩. 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.

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