Combinatorics of exceptional sequences in type A

TL;DR

This paper introduces strand diagrams to classify exceptional sequences, c-matrices, and their relations to posets and noncrossing partitions in type A quivers, extending previous classifications.

Contribution

It provides a new combinatorial model for classifying exceptional sequences and related structures in type A quivers, expanding prior work.

Findings

Classified exceptional sequences using strand diagrams.

Established a bijection between exceptional sequences and chains in noncrossing partitions.

Extended classification results from linearly-ordered quivers to type A Dynkin diagrams.

Abstract

Exceptional sequences are certain ordered sequences of quiver representations. We introduce a class of objects called strand diagrams and use this model to classify exceptional sequences of representations of a quiver whose underlying graph is a type $A_n$ Dynkin diagram. We also use variations of this model to classify c-matrices of such quivers, to interpret exceptional sequences as linear extensions of posets, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This work extends a classification of exceptional sequences for the linearly-ordered quiver obtained in an earlier paper by the first and third authors.