Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories

Abstract

We establish a bijection between $d$-simple-minded systems ($d$-SMSs) of $(-d)$-Calabi-Yau cluster category ${\mathcal{C}}_{-d}(H)$ and silting objects of ${\mathcal{D}}^{\mathrm{b}}(H)$ contained in ${\mathcal{D}}^{\le 0}\cap {\mathcal{D}}^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in ${\mathcal{C}}_{-d}(H)$ is the positive Fuss-Catalan number $C_d^{+}(W)$ of the corresponding Weyl group $W$, by applying this bijection and Buan-Reiten-Thomas' and Zhu's results on Fomin-Reading's generalized cluster complexes. Our results are based on a refined version of silting-$t$-structure correspondence.

15 pages, many improvements due to referees, the algebra KQ for Dynkin quiver Q has been generalized to hereditary algebra H of Dynkin type, End(X)=k in the definition of SMC has been generalized to End(X)=a division k-algebra, an appendix about exceptional mutation has been added, to appear in International Mathematics Research Notices

References in corpus (1)

• On triangulated orbit categories

Cited by in corpus (2)

• On simple-minded systems over representation-finite self-injective algebras
• Proper abelian subcategories of triangulated categories and their tilting theory