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

TL;DR

This paper establishes a bijection between certain simple-minded systems in negative Calabi-Yau categories and silting objects, revealing their count matches positive Fuss-Catalan numbers, thus connecting cluster theory and combinatorics.

Contribution

It introduces a new bijection linking $d$-simple-minded systems in negative Calabi-Yau categories with silting objects, and computes their number using Fuss-Catalan numbers.

Findings

Number of $d$-SMSs equals positive Fuss-Catalan number $C_{d}^{+}(W)$

Established a bijection between $d$-SMSs and silting objects in specific derived categories

Connected cluster theory with combinatorial Fuss-Catalan numbers

Abstract

We establish a bijection between $d$-simple-minded systems ($d$-SMSs) of $(-d)$-Calabi-Yau cluster category ${\cal C_{-d}}(H)$ and silting objects of ${\cal D^{\rm b}}(H)$ contained in $\cal D^{\le 0}\cap \cal D^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in ${\cal C_{-d}}(H)$ 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.