A categorification of combinatorial Auslander–Reiten quivers

Source abstract

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 of the 2‐dimensional Ginzburg dg algebra of a Dynkin quiver . For any commutation class of reduced words in the corresponding Weyl group, we define a subcategory of whose objects are obtained by applying a sequence of spherical twist functors to the simple objects. We describe the ‐order for in terms of , generalizing a result of Bédard. Furthermore, when is a commutation class for the longest element, we construct a category generalizing the bounded derived category of . It is realized as a certain subquotient of . We demonstrate the existence of particular distinguished triangles in with corners in , 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 arising from Q‐data — a generalization of Dynkin quivers with a height function introduced by Fujita and Oh — we establish the existence of a partial Serre functor on . Lastly, we apply our results to reinterpret a formula by Fujita and Oh for the inverse of the quantum Cartan matrix.