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A categorification of cluster algebras of type B and C through symmetric quivers

Azzurra Ciliberti

Abstract

We express cluster variables of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$. Then we associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$. Under this correspondence, cluster variables of type $B_n$ (resp. $C_n$) correspond to orthogonal (resp. symplectic) indecomposable representations of $Q$. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.

A categorification of cluster algebras of type B and C through symmetric quivers

Abstract

We express cluster variables of type and in terms of cluster variables of type . Then we associate a cluster tilted bound symmetric quiver of type to any seed of a cluster algebra of type and . Under this correspondence, cluster variables of type (resp. ) correspond to orthogonal (resp. symplectic) indecomposable representations of . We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.
Paper Structure (14 sections, 15 theorems, 40 equations, 16 figures)

This paper contains 14 sections, 15 theorems, 40 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Cluster algebras of geometric type
  4. Combinatorial description of the cluster complex of type $A_n$
  5. Combinatorial description of the cluster complex of type $B_n$/$C_n$
  6. Cluster algebras of type $B$ and $C$ with principal coefficients
  7. Cluster expansion formula for cluster algebras of type $B$ and $C$
  8. Type B
  9. Type C
  10. The categorification
  11. Symmetric quivers and their representations
  12. $\rho$-orbits as orthogonal and symplectic representations
  13. Categorical interpretation of Theorem \ref{['theorem1']} in the acyclic case
  14. Proof of Theorem \ref{['theorem 2']}

Key Result

Theorem 1

Let $\mathcal{A}^B(T)$ be the cluster algebra of type $B_n$ with principal coefficients in a $\theta$-invariant triangulation $T=\{ \tau_1, \dots, \tau_{2n-1} \}$ of $\mathbf{P}_{2n+2}$. Then the $F$-polynomial $F_{ab}$ of $x_{ab}$ is given in the following way:

Figures (16)

  • Figure 1: A triangulated octagon with the elementary lamination associated to each diagonal of the triangulation (in blue).
  • Figure 2: An exchange relation in a triangulated octagon.
  • Figure 3: Exchanges in types $B_n$ and $C_n$
  • Figure 4: The matrix $B(\Bar{T})$ associated with a $\theta$-invariant triangulation of an octagon.
  • Figure 5: On the left, two $\theta$-orbits $[a,\Bar{a}]$ and $[a,b]$. On the right, their restrictions.
  • ...and 11 more figures

Theorems & Definitions (64)

  • Theorem : \ref{['theorem1']}
  • Theorem : \ref{['cat_interpr']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Example 2.4
  • Proposition 2.5
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • ...and 54 more