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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.23354 |
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| _version_ | 1866911541769535488 |
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| author | Kleinau, Markus |
| author_facet | Kleinau, Markus |
| contents | Reading constructed a Cambrian lattice $C_Γ$ for each oriented finite type Coxeter diagram $Γ$. We show that the derived category of representations of $C_Γ$ is fractionally Calabi-Yau for any $Γ$, confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type $A$ with linear orientation, better known as Tamari lattices. If $Γ$ is crystallographic, then $C_Γ$ is given by the lattice of torsion classes of any hereditary algebra $Λ$ of type $Γ$. In this case we introduce and study a class of intervals in $C_Γ$ whose combinatorics matches the combinatorics of $2$-cluster tilting objects in the 2-cluster category of $Λ$. This allows us to compute the Calabi-Yau dimension of $C_Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23354 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics Kleinau, Markus Representation Theory 16G20, 05E10, 16E35, 16S90 Reading constructed a Cambrian lattice $C_Γ$ for each oriented finite type Coxeter diagram $Γ$. We show that the derived category of representations of $C_Γ$ is fractionally Calabi-Yau for any $Γ$, confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type $A$ with linear orientation, better known as Tamari lattices. If $Γ$ is crystallographic, then $C_Γ$ is given by the lattice of torsion classes of any hereditary algebra $Λ$ of type $Γ$. In this case we introduce and study a class of intervals in $C_Γ$ whose combinatorics matches the combinatorics of $2$-cluster tilting objects in the 2-cluster category of $Λ$. This allows us to compute the Calabi-Yau dimension of $C_Γ$. |
| title | Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics |
| topic | Representation Theory 16G20, 05E10, 16E35, 16S90 |
| url | https://arxiv.org/abs/2603.23354 |