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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.06147 |
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| _version_ | 1866914615002136576 |
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| author | Canesin, Ricardo |
| author_facet | Canesin, Ricardo |
| contents | 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 partial 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A categorification of combinatorial Auslander-Reiten quivers Canesin, Ricardo Representation Theory 05E10, 18G80, 20F55 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 partial 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. |
| title | A categorification of combinatorial Auslander-Reiten quivers |
| topic | Representation Theory 05E10, 18G80, 20F55 |
| url | https://arxiv.org/abs/2505.06147 |