Hom-Ext Quivers and the Classification of Exceptional Sequences
Overview
Quivers are directed graphs that encode algebraic relationships and play an important role in modern mathematics, including representation theory, cluster algebras, and string theory. A central challenge in this area is classifying exceptional sequences of modules over quiver algebras, which quickly becomes intractable by traditional methods as the size of the quiver grows.
This work develops a computational workflow to explore these sequences for Dynkin-type quivers. By generating large datasets of sequences and their associated Hom-Ext quivers, we detect recurring combinatorial patterns that suggest general principles beyond what is visible in small cases. These patterns provide evidence for new theorems about Hom-Ext quivers and their relationship with exceptional sequences, including a conjectural formula that refines known counting results by decomposing the total count according to Hom-Ext isoclass.
Beyond Hom-Ext quivers, another area of investigation for classifying exceptional sequences is the compatibility graph, whose vertices are indecomposable modules and whose edges encode local sequence rules. We have discovered that the vertex degree in this graph is strongly connected to the number of exceptional sequences beginning with a given module, and there is a lot of unexplored structure that could point to a new formula.
Building on these computational discoveries, the next phase of this research will focus on proving the conjectures that have emerged from the data. This includes verifying the Hom-Ext isoclass formula across all Dynkin types and establishing a geometric model for type D. We also aim to extend this framework to Euclidean quivers.