Combinatorics of Exceptional Sequences of Type $\tilde{\mathbb{A}}_n$

TL;DR

This paper classifies and parametrizes exceptional sequences of Euclidean quivers of type A_n, using combinatorial and geometric models to show finitely many such families exist.

Contribution

It introduces a combinatorial and geometric framework to classify exceptional sequences of type A_n, establishing a finite parametrization and bijections with arc diagrams.

Findings

Finitely many parametrized families of exceptional sequences are classified.

Bijection established between exceptional collections and arc diagrams.

Arc diagrams provide a geometric realization of the classification.

Abstract

It is known that there are infinitely many exceptional sequences of quiver representations for Euclidean quivers. In this paper we study those of type $\tilde{\mathbb{A}}_n$ and classify them into finitely many parametrized families. We first give a bijection between exceptional collections and a combinatorial object known as strand diagrams. We will then realize these strand diagrams as chord diagrams and then arc diagrams on an annulus. Using arc diagrams, we will define parametrized families of exceptional collections and use arc diagrams to show that there are finitely many such families. We moreover show that these families of exceptional collections are in bijection with equivalence classes of small arc diagrams. Finally, we provide an algebraic explanation of parametrized families using the transjective component of the bounded derived category.