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Mutation of theta functions

Nathan Reading, Salvatore Stella

Abstract

We give an account of mutation of theta functions in cluster scattering diagrams, starting with a notion of mutation that is related to, but different from, the notion of mutation defined by Gross, Hacking, Keel, and Kontsevich. This different approach to mutation leads to several applications. Three of the applications simplify the process of computing structure constants for multiplication of theta functions, and these are used in another paper on cluster scattering diagrams of affine type. Notable in these three applications is the appearance of mutation symmetries and dominance regions. The other two applications have to do with pointed reduced bases, a variation on the pointed bases of Fan Qin. We give a characterization of pointed reduced bases analogous to Qin's characterization of pointed bases. All of these applications take place in a version of Gross, Hacking, Keel, and Kontsevich's canonical algebra that can be constructed for an arbitrary exchange matrix.

Mutation of theta functions

Abstract

We give an account of mutation of theta functions in cluster scattering diagrams, starting with a notion of mutation that is related to, but different from, the notion of mutation defined by Gross, Hacking, Keel, and Kontsevich. This different approach to mutation leads to several applications. Three of the applications simplify the process of computing structure constants for multiplication of theta functions, and these are used in another paper on cluster scattering diagrams of affine type. Notable in these three applications is the appearance of mutation symmetries and dominance regions. The other two applications have to do with pointed reduced bases, a variation on the pointed bases of Fan Qin. We give a characterization of pointed reduced bases analogous to Qin's characterization of pointed bases. All of these applications take place in a version of Gross, Hacking, Keel, and Kontsevich's canonical algebra that can be constructed for an arbitrary exchange matrix.
Paper Structure (19 sections, 37 theorems, 42 equations, 2 figures)

This paper contains 19 sections, 37 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scattering diagrams and theta functions
  3. Context for scattering diagrams
  4. Scattering diagrams
  5. Theta functions
  6. Structure constants
  7. Matrix mutation, mutation maps, and the mutation fan
  8. The choice of coefficients
  9. The small canonical algebra
  10. Definition of the small canonical algebra
  11. Comparison with the Gross-Hacking-Keel-Kontsevich approach
  12. Bases and reduced bases for the small canonical algebra
  13. Mutation of theta functions
  14. Applications of mutation of theta functions
  15. Mutation symmetry
  16. ...and 4 more sections

Key Result

Theorem 1.1

Theta functions relative to mutation-equivalent exchange matrices with signed-nondegenerating coefficients can be obtained from one another by multiplication with an appropriate Laurent monomial in frozen variables.

Figures (2)

  • Figure 1: Broken lines for $\vartheta_{[-2,3]}=\vartheta_{Q,[-2,3]}$
  • Figure 2: $\operatorname{Scat}(\mu_1({\tilde{B}}))$ and broken lines for $\vartheta^{\mu_1({\tilde{B}})}_{\eta_1^B(Q),\eta_1^B([-2,3])}$

Theorems & Definitions (74)

  • Theorem 1.1
  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 64 more