Mathematics > Representation Theory
arXiv:2505.06147 (math)
[Submitted on 9 May 2025 (v1), last revised 24 Jun 2026 (this version, v3)]
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 $\mathrm{pvd}(\Pi_Q)$ 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 $\mathrm{pvd}(\Pi_Q)$ 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 $\mathrm{pvd}(\Pi_Q)$. We demonstrate the existence of particular distinguished triangles in $\mathrm{pvd}(\Pi_Q)$ 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.
| Comments: | 51 pages. v3: corrected Remark 6.9. v2: improved Lemma 8.7 and Proposition 8.15, added Remarks 8.16 and 8.17, corrected typos; version in the JLMS |
| Subjects: | Representation Theory (math.RT) |
| MSC classes: | 05E10, 18G80, 20F55 |
| Cite as: | arXiv:2505.06147 [math.RT] |
| (or arXiv:2505.06147v3 [math.RT] for this version) | |
| https://doi.org/10.48550/arXiv.2505.06147 arXiv-issued DOI via DataCite |
|
| Journal reference: | J. London Math. Soc. (2) 113 (2026), no. 5, Paper No. e70579 |
| Related DOI: | https://doi.org/10.1112/jlms.70579
DOI(s) linking to related resources |
Submission history
From: Ricardo Canesin [view email]
[v1]
Fri, 9 May 2025 15:58:08 UTC (70 KB)
[v2]
Wed, 27 May 2026 09:13:41 UTC (69 KB)
[v3]
Wed, 24 Jun 2026 15:31:02 UTC (70 KB)
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