A categorification of combinatorial Auslander-Reiten quivers

TL;DR

This work develops a categorification of Oh–Suh’s combinatorial Auslander–Reiten quivers for simply-laced types by embedding the construction in the perfectly valued derived category $\mathrm{pvd}(\Pi_Q)$ of the 2-dimensional Ginzburg dg algebra. It introduces categories $\mathcal{C}([\mathbf{i}])$, their repetition $\mathcal{R}([\mathbf{i}])$ and c-derived $\mathcal{D}([\mathbf{i}])$ associated to commutation classes, generated via spherical twists, and proves that the combinatorial AR quiver is recovered from the Gabriel quiver of $\mathcal{C}([\mathbf{i}])$ while extending to a mesh/triple framework that yields a generalized $\mathfrak{g}$-additive property. The paper develops a robust root-system categorification: $K_0(\mathrm{pvd}(\Pi_Q))$ identifies with the root lattice $\mathsf{Q}$ and the Euler form with the Cartan pairing, and a Serre-type duality is established in the Q-data setting via a generalized twisted Coxeter element. For Q-data, the authors reinterpret inverses of quantum Cartan matrices in terms of Euler forms on the categorified structures, linking representation-theoretic constructions to quantum group data. Overall, this framework provides a categorical bridge between Weyl-group combinatorics, AR-quiver theory, and quantum Cartan phenomena, with potential applications to quantum loop algebras and their derived Hall algebras.

Abstract

We provide a categorification of Oh and Suh's combinatorial Auslander-Reiten quivers in the simply laced case. We work within the perfectly valued derived category $\mathrm{pvd}(Π_Q)$ of the 2-dimensional Ginzburg dg algebra of a Dynkin quiver $Q$. For any commutation class $[i]$ of reduced words in the corresponding Weyl group, we define a subcategory $C([i])$ of $\mathrm{pvd}(Π_Q)$ whose objects are obtained by applying a sequence of spherical twist functors to the simple objects. We describe the Hom-order for $C([i])$ in terms of $[i]$, generalizing a result of Bédard. Furthermore, when $[i]$ is a commutation class for the longest element, we construct a category $D([i])$ generalizing the bounded derived category of $Q$. It is realized as a certain subquotient of $\mathrm{pvd}(Π_Q)$. We demonstrate the existence of particular distinguished triangles in $\mathrm{pvd}(Π_Q)$ with corners in $D([i])$, which allows us to extend the classical mesh-additivity to arbitrary commutation classes. Additionally, we define an analog of the Euler form and prove that its symmetrization yields the corresponding Cartan-Killing form. For commutation classes $[i]$ arising from Q-data, a generalization of Dynkin quivers with a height function introduced by Fujita and Oh, we establish the existence of a partially Serre functor on $D([i])$. Lastly, we apply our results to reinterpret a formula by Fujita and Oh for the inverse of the quantum Cartan matrix.

Figures (1)

• Figure 1: The action of the automorphisms $\vee$ and $\widetilde{\vee}$ on the nonfixed vertices is shown in red. After identifying the orbits via the blue arrows, we get the nonsimply laced Dynkin diagrams.

Theorems & Definitions (141)

• Theorem 1.1: Theorem \ref{['thm:combinatorial AR quiver is obtained from Gabriel quiver']}
• Theorem 1.2: Theorem \ref{['thm:Serre duality']}
• Theorem 1.3: Proposition \ref{['prop:computing the inverse of the quantum cartan matrix']}
• Theorem 2.1: OhSuh19a
• Theorem 2.2: OhSuh19a
• Example 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof