Abstract representation theory via coherent Auslander-Reiten diagrams
Álvaro Sánchez
TL;DR
The paper develops an abstract representation theory for quivers in stable ∞-categories, proving an equivalence between representations of a quiver $Q$ and mesh representations of its repetitive quiver $\mathbb{Z}Q$, built via abstract reflection functors. This framework yields universal autoequivalences organized by the spectral Picard group and provides abstract counterparts to key functors in classical representation theory, including reflection, AR translation, and Serre functors; it also explains how derived/picard-type symmetries specialize to field cases and, for trees, recover known Picard-structure results. The methods generalize Happel’s results beyond fields, making the entire theory robust across arbitrary stable homotopy theories and enabling base-change arguments that connect spectral Picard groups to ordinary Picard groups. Overall, the work offers a unified, homotopy-coherent approach to AR-diagrams, reflection functors, and automorphism groups in a broad categorical setting, with concrete consequences for symmetries and invariants of quiver representations. The results have potential impact on understanding how categorical symmetries propagate through derived and spectral contexts and on computing Picard-type groups for quivers in stable homotopy theory.
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.