A Combinatorial Model for Exceptional Sequences in Type A

TL;DR

This paper introduces a combinatorial model using noncrossing edge-labeled trees to classify and interpret exceptional sequences, c-matrices, and their mutations in type A quivers, providing new insights into their structure.

Contribution

It develops a novel combinatorial framework connecting exceptional sequences, c-matrices, and noncrossing partitions for type A quivers, expanding on previous models.

Findings

Classifies exceptional sequences using noncrossing trees.

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

Interprets c-matrix mutation through noncrossing tree transformations.

Abstract

Exceptional sequences are certain ordered sequences of quiver representations. We use noncrossing edge-labeled trees in a disk with boundary vertices (expanding on T. Araya's work) to classify exceptional sequences of representations of Q, the linearly-ordered quiver with n vertices. We also show how to use variations of this model to classify c-matrices of Q, to interpret exceptional sequences as linear extensions, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. In the case of c-matrices, we also give an interpretation of c-matrix mutation in terms of our noncrossing trees with directed edges.