Abstract representation theory via coherent Auslander-Reiten diagrams

TL;DR

This paper develops a general framework using Auslander-Reiten diagrams to study quiver representations in stable homotopy theories, revealing universal autoequivalences and connecting classical functors with spectral Picard groups.

Contribution

It introduces an abstract equivalence between quiver representations and mesh categories in stable $ppa$-categories, generalizing classical representation theory tools.

Findings

Establishes an equivalence between $ppa^Q$ and mesh $ppa$-categories.

Constructs universal autoequivalences from Auslander-Reiten quiver symmetries.

Realizes the derived Picard group as a factor of the spectral Picard group for tree quivers.

Abstract

We provide a general method to study representations of quivers over abstract stable homotopy theories (e.g. arbitrary rings, schemes, dg algebras, or ring spectra) in terms of Auslander-Reiten diagrams. For a finite acyclic quiver $Q$ and a stable $\infty$-category $\mathcal{C}$, we prove an abstract equivalence of the representations $\mathcal{C}^Q$ with a certain mesh $\infty$-category $\mathcal{C}^{\mathbb{Z}Q,\, \mathrm{mesh}}$ of representations of the repetitive quiver $\mathbb{Z}Q$, that we build inductively using abstract reflection functors. This allows to produce, from the symmetries of the Auslander-Reiten quiver, universal autoequivalences of representations $\mathcal{C}^Q$ in any stable $\infty$-category $\mathcal{C}$, which are the elements of the spectral Picard group of $Q$. In particular, we get abstract versions of key functors in classical representation theory -- e.g. reflection functors, the Auslander-Reiten translation, the Serre functor, etc. Moreover, for representations of trees this enables us to realize the whole derived Picard group over a field as a factor of the spectral Picard group.