Mutation of Theta Functions

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.