Cluster tilting modules for mesh algebras (Q820999)

In this article, the authors study cluster tilting modules and their mutations for mesh algebras [\textit{K. Erdmann} and \textit{A. Skowroński}, in: Trends in representation theory of algebras and related topics. Proceedings of the 12th international conference on representations of algebras and workshop (ICRA XII). Zürich: European Mathematical Society (EMS). 201--251 (2008; Zbl 1210.16012)]. Their first main result shows that all mesh algebras of Dynkin type have cluster tilting module. Furthermore, they remark that a different construction of \textit{S. Asai} [Algebr. Represent. Theory 21, No. 3, 635--681 (2018; Zbl 1401.16005)] combined with results of [\textit{W. Yang} et al., Commun. Algebra 45, No. 1, 299--311 (2017; Zbl 1395.16010)] strengthens their result by implying necessity. Their second main result concerns mutation of cluster tilting. On mesh algebra, they obtain that the global dimension of the endomorphism ring of mesh algebra of Dynkin type except \(\mathbb{G}_2\) is 3. The key ingredient is a symmetry of \(\mathrm{Ext}^1\)-spaces equivariant under an automorphism of order smaller or equal to 2, a condition respected for all mesh algebras of Dynkin type except \(\mathbb{G}_2\). However, they remark that if any cluster tilting module of a mesh algebra is reachable from a fixed cluster tilting module by successive mutations in the module category, then the results would also hold for the \(\mathbb{G}_2\) case. It remains an open question whether this property holds. Their mutation is applied in a non-stably 2-Calabi-Yau abelian category with an explicit description of the combinatorics at the linear algebra level, modifying [\textit{C. Geiß} et al., Invent. Math. 165, No. 3, 589--632 (2006; Zbl 1167.16009)] to the skew-symmetrizable setting. Finally, a worked-out example is provided for type \(\mathbb{B}_3\).