Higher Dimer Models for Cluster Algebras and Categorification of Non-Orientable Surfaces

Abstract

Cluster algebras were first defined by Fomin and Zelevinksy in the early 2000’s in an effort to study problems regarding dual canonical bases and total positivity. Since their axiomatization, these algebras have proven to be connected to a wide range of mathematical fields as well as in physics. Much of this thesis is dedicated to deepening our understanding of these algebras in various settings. To start, cluster algebras are not algebras that we think of in terms of generators and relations. Instead, the generators, known as cluster variables, are defined combinatorially through a recursive process. One main aspect of this thesis is trying to pin down these generators with direct, non-recursive methods. In this process, we will see that the methods we develop shed light on deep connections between the rich combinatorics and representation theory of these algebras. With that being said, the first goal of the thesis is to create expansion formulae for the generators of certain cluster algebras. Namely, we create graph theoretic combinatorics called higher dimer models to model cluster variables for cluster algebras in type Dn and in the Grassmannian Gr(3,n). We prove our models indeed give the correct formulae by connecting our combinatorics to quiver representation and SLn-representation theory respectively. In a parallel track, cluster theorists have also been interested in generalizing the notion of a cluster algebra in various ways. For instance, in some early work by Fomin, Shapiro and Thurston, a cluster structure was defined for orientable triangulated surfaces. The second goal of the thesis is to explore a generalization of cluster algebras from orientable surfaces to the non-orientable setting. Dupont and Palesi defined a cluster-like structure for non-orientable surfaces known as a quasi-cluster algebra. We aim to explore how much of the theory from orientable surfaces has a natural non-orientable analogue. Specifically, we investigate two directions: a possible categorification for these algebras and a matrix formulae for their generators. In the former, we categorify the non-orientable analogue of triangulations; that is, we create a dictionary between topological objects on the surfaces and representation theory of a certain quiver algebra. This is the first step towards the additive categorification of quasi-cluster al- gebras. In the ladder, we develop ways to non-recursively compute the generators for quasi-cluster algebras using linear algebraic methods. We prove these methods are more meaningful by showing that they give a linear algebraic formulation of the famous skein relations coming from various fields in topology.