Ptolemy diagrams and torsion pairs in m-cluster categories of type <i>D</i> (Q6600725)

In this paper, the authors provide a comprehensive classification of torsion pairs in \(m\)-cluster categories of type \(D\) when \(m\) is odd, denoted by \(\mathcal{C}^m_{\mathcal{D}_n}\), through a bijection to combinatorial structures known as Ptolemy diagrams of type \(D\). This novel classification offers a precise correspondence between torsion pairs and these diagrams, revealing a deeper combinatorial understanding of the category. As an application, the authors further classify \(m\)-rigid subcategories in \(\mathcal{C}^m_{\mathcal{D}_n}\), which in turn leads to Jacquet-Malo's classification of \(m\)-cluster tilting subcategories in \(\mathcal{C}^m_{\mathcal{D}_n}\). Additionally, when \(m=1\), their work generalizes the classification of torsion pairs in cluster categories of type \(D_n\), previously established by \textit{T. Holm} et al. [Adv. Appl. Math. 51, No. 5, 583--605 (2013; Zbl 1301.05021)]. These results not only extend existing theories but also offer new insights into the structure of \(m\)-cluster categories, especially in type \(D_n\) settings.