Positivity in Cluster Algebras and Their Generalizations

Abstract

The theory of bf cluster algebras gives us a combinatorial framework for understanding the previously opaque nature of certain algebras. Each cluster algebra is generated by its cluster variables, which can be obtained via the recursive process of mutation. One remarkable property of cluster algebras is Laurent positivity, which means that every cluster variable can be written in a nice form; specifically, as a Laurent polynomial with positive integer coefficients in the initial cluster variables. Laurent positivity for cluster algebras unifies positivity phenomena in a variety of contexts, including Teichmuller theory, Gromov-Witten theory, string theory, and tropical geometry. Laurent positivity was conjectured by Fomin and Zelevinsky when they introduced cluster algebras in 2002, but the proof remained elusive for over a decade. There have since been two proofs: a combinatorial approach by Lee and Schiffler, and a geometric approach by Gross, Hacking, Keel, and Kontsevich using a novel connection to scattering diagrams. Scattering diagrams themselves are powerful tools, originating from mirror symmetry, where they track how certain geometric invariants (Gromov--Witten invariants and Donaldson--Thomas invariants) change under varying stability conditions. Every cluster algebra is associated with a cluster scattering diagram that encodes algebraic relations between cluster variables, making them a useful tool in cluster algebra theory. The work in this dissertation unifies these methods, aiming to deepen our understanding of positivity in both cluster algebras and scattering diagrams. In Chapter 3, which is joint work with Kyungyong Lee and Lang Mou, we prove positivity for generalized cluster algebras of all ranks, confirming a 2014 conjecture of Chekhov--Shapiro. We achieve this by giving a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank 2 generalized cluster scattering diagrams. This formula enumerates a new class of Dyck path objects, called tight gradings, implying positivity of the scattering diagrams in rank 2. In Chapter 4, which is joint work with Kyungyong Lee, we construct an explicit bijection between broken lines on scattering diagrams and compatible pairs on Dyck paths, which both play crucial roles in the proofs of cluster algebra positivity. In Chapter 5, we give a new expansion formula for quantum cluster variables using colored subpaths of Dyck paths, leveraging a connection we make to the compatible pair framework.