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Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics

Markus Kleinau

Abstract

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 $C_Γ$.

Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics

Abstract

Reading constructed a Cambrian lattice for each oriented finite type Coxeter diagram . We show that the derived category of representations of is fractionally Calabi-Yau for any , confirming a conjecture of Chapoton. This extends a result of Rognerud for Cambrian lattices of type with linear orientation, better known as Tamari lattices. If is crystallographic, then is given by the lattice of torsion classes of any hereditary algebra of type . In this case we introduce and study a class of intervals in whose combinatorics matches the combinatorics of -cluster tilting objects in the 2-cluster category of . This allows us to compute the Calabi-Yau dimension of .
Paper Structure (16 sections, 65 theorems, 102 equations, 3 figures)

This paper contains 16 sections, 65 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fractionally Calabi-Yau lattices
  3. Cambrian lattices
  4. 2-cluster combinatorics
  5. The Ingalls-Thomas bijections
  6. Subcategories
  7. Support tilting modules
  8. Lattices
  9. The 2-cluster category
  10. Mutable intervals and 2-cluster tilting objects
  11. Interval mutation and almost 2-cluster tilting objects
  12. The Serre functor and representations of incidence algebras
  13. Cambrian lattices are periodic Serre formal
  14. Example: A₃
  15. Tamari lattices and geometric models
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $\Lambda$ be a finite dimensional connected hereditary representation finite algebra and let $I$ be a mutable interval in $\mathop{\mathrm{tors}}\nolimits(\Lambda)$. Then there exists another mutable interval $\mathbb{S} I$ and an integer $K_I$ such that $\mathbb{S} M_I \cong M_{\mathbb{S} I}[K_

Figures (3)

  • Figure 1: A Venn diagram of the wide categories involved in a mutable pair. There are no morphisms or extension going from a category on the left to one further to the right.
  • Figure 2: Construction of the planar dual of a noncrossing tree.
  • Figure 3: A quadrangulation of a 10-gon and its rotation with the corresponding noncrossing trees drawn in dashed lines.

Theorems & Definitions (141)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Definition 2.5
  • ...and 131 more