The Grothendieck groups of <i>m</i> -cluster categories of type <i> D _n </i> (Q6561419)

In this paper, the authors study the Grothendieck groups of \(m\)-cluster categories of type \(\mathcal D_n\), denoted by \(\mathcal C^m_{\mathcal D_n}\), focusing on the case where \(m\) is odd, through the \(m\)-cluster tilting subcategories of \(\mathcal C^m_{\mathcal D_n}\). When \(m=1\), this result generalizes the work of \textit{M. Barot} et al. [J. Pure Appl. Algebra 212, No. 1, 33--46 (2008; Zbl 1148.16005)] on the Grothendieck groups of cluster categories of type \(D_n\). This generalization not only deepens our understanding of the structure of cluster categories but also offers new perspectives for exploring higher-dimensional cluster categories.