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Super Caldero--Chapoton map for type $A$

İlke Çanakçı, Francesca Fedele, Ana Garcia Elsener, Khrystyna Serhiyenko

Abstract

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichm{ü}ller theory to define a super version of the cluster algebra of type $A$ and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type $A$.

Super Caldero--Chapoton map for type $A$

Abstract

One can explicitly compute the generators of a surface cluster algebra either combinatorially, through dimer covers of snake graphs, or homologically, through the CC-map applied to indecomposable modules over the appropriate algebra. Recent work by Musiker, Ovenhouse and Zhang used Penner and Zeitlin's decorated super Teichm{ü}ller theory to define a super version of the cluster algebra of type and gave a combinatorial formula to compute the even generators. We extend this theory by giving a homological way of explicitly computing these generators by defining a super CC-map for type .
Paper Structure (22 sections, 24 theorems, 55 equations, 23 figures)

This paper contains 22 sections, 24 theorems, 55 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Cluster algebras from surfaces
  3. Cluster algebras from marked surfaces
  4. Snake graph formula
  5. Super lambda lengths
  6. The default orientation and the positive order
  7. Snake graphs and double dimer covers
  8. Lattice structure in double dimer covers
  9. Representation theoretic interpretation of the snake graph formula
  10. The CC-map
  11. Cluster category of type $A$
  12. The Frobenius category $\mathcal{C}_{\mathcal{F}}$
  13. The module category $\mathrm{mod} \, A_T$
  14. Tensoring with the algebra of dual numbers
  15. Associating $\mu$-invariants to $N$.
  16. ...and 7 more sections

Key Result

Theorem 2.9

MSW11 Suppose $(S,M)$ is a marked surface and $T$ is a triangulation on $(S,M)$. Let $\gamma$ be an arc that is not in $T$, $x_\gamma$ be the corresponding cluster variable in the cluster algebra $\mathcal{A}(S,M)$ and $\mathcal{G}_\gamma$ be its snake graph with respect to $T$. Then $x_\gamma=x_{\m

Figures (23)

  • Figure 1: Ptolemy transformation: $ef=ac+bd$.
  • Figure 2: A triangulation $T$ of the octagon and its associated quiver $Q_T$, highlighted in red in the figure. The cluster algebra $\mathcal{A}(S,M)$ has initial seed $(\mathbf{x}_T, Q_T)=(\{ x_1, \ldots, x_5 \}, 1 \leftarrow 2\leftarrow 3 \to 4 \leftarrow 5 )$.
  • Figure 3: On the left a diagonal $\gamma$ on a triangualted octagon, in the middle is its snake graph $\mathcal{G}_\gamma$ and on the right is one of its dimer covers $P$ highlighted in thick blue.
  • Figure 4: The lattice $\mathcal{L(\mathcal{G})}$ of the snake graph $\mathcal{G}_{\gamma}$ for the previous example. In the figure, $\mathrm{wt}(P)$ is indicated for each dimer cover $P$.
  • Figure 5: The equivalence on orientations determining the spin structures on $S$.
  • ...and 18 more figures

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Example 2.10
  • ...and 80 more