Hom-Ext Quivers and the Classification of Exceptional Sequences
Overview
Quivers are directed graphs that encode algebraic relationships and play an important role in representation theory. A central challenge in this area is understanding and classifying exceptional sequences of modules over quiver algebras, whose combinatorial complexity grows rapidly with the size of the quiver. This independent study used computational methods to investigate exceptional sequences and their associated Hom-Ext quivers. By generating large datasets of exceptional sequences for Dynkin-type quivers, we identified recurring combinatorial structure in the Hom-Ext quivers and used these patterns to formulate and prove classification and counting results. For quivers of type \(A_n\), this work led to a connection between exceptional collections, Hom-Ext quivers, iterated tilted algebras, and non-crossing spanning trees. In particular, isomorphism classes of iterated tilted algebras of type \(A_n\) can be identified with non-crossing spanning trees on a convex \((n+1)\)-gon up to rotation. This work resulted in the preprint “Enumerating iterated tilted algebras in type \(A\),” with Alexander E. Black and Ray Maresca.
Paper
Enumerating iterated tilted algebras in type \(A\)
Alexander E. Black, Jonathan E. Gordon, Ray Maresca
arXiv preprint, 2026.
Abstract
We show that isoclasses of iterated tilted algebras in type \(A_n\) are in bijection with non-crossing spanning trees up to rotation on a convex \((n+1)\)-gon. This is done by constructing a relationship between iterated tilted algebras up to isomorphism and exceptional sets up to isomorphic Hom-Ext quiver.
Poster
This work was presented at the Joint Mathematics Meetings 2026 in the AMS-PME Poster Session.