On Clusters and Exceptional Sets in Types $\mathbb{A}$ and $\tilde{\mathbb{A}}$

TL;DR

This paper investigates cluster structures and exceptional sets in types A and e4, revealing combinatorial counts and structural similarities using triangulations, Dehn twists, and Catalan numbers.

Contribution

It introduces a classification of clusters in type e4 via Dehn twists and counts exceptional sets in type A and e4 using combinatorial methods.

Findings

Clusters in e4 are grouped into families counted by Catalan numbers.

Structural similarities between annuli diagrams for clusters and exceptional sets are identified.

Explicit counts of exceptional sets for type A quivers are provided.

Abstract

In this paper we first study clusters in type $\tilde{\mathbb{A}}$ by collecting them into a finite number of infinite families given by Dehn twists of their corresponding triangulations, and show that these families are counted by the Catalan numbers. We also highlight the similarities and differences between the annuli diagrams used to study clusters and those used to study exceptional sets in type $\tilde{\mathbb{A}}$. We then focus on exceptional collections (sets) of modules over path algebras of quivers by first showing that the notion of relative projectivity in exceptional sets is well defined. We finish by counting the number of exceptional sets of representations of type $\mathbb{A}$ quivers with straight orientation and using this to count the number of families of exceptional sets of type $\tilde{\mathbb{A}}$ with straight orientation.