Theta functions and the mutation fan

Abstract: The setting for this work is the cluster scattering diagram defined by Gross, Hacking, Keel, and Kontsevich (GHKK).  The cluster scattering diagram is a collection of walls (codimension-1 cones plus some additional algebraic data).  Theta functions (one for each g-vector) include the cluster monomials and form a basis for the cluster algebra (or often something larger).  Explicit constructions of cluster scattering diagrams and explicit computations of theta functions are hopelessly complicated in general, but I believe that eventually there will be combinatorial models in all mutation-finite types.  I'll mention work with Salvatore Stella on combinatorial models in affine type, and work with Greg Muller and Shira Viel on the surfaces case. But I will spend most of the time discussing a result (with Stella) that I think will make it possible to complete these combinatorial constructions of theta functions.

The mutation fan encodes the piecewise-linear geometry of matrix mutation.  The result is:  If you take a product of theta functions whose g-vectors are all in one cone of the mutation fan, the product expands as a sum of theta functions whose g-vectors are all in one cone of the mutation fan.  The result seems natural and in some sense unsurprising, but it requires some work and it is quite useful.  The result requires two serious changes in point of view from the GHKK setup:  Taking a different point of view on what "mutation of scattering diagrams" means and demoting "frozen variables" to the status of coefficients.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)