Categorifications of Cluster Varieties

Abstract

In this thesis we study additive categorifications of algebraic varieties whose coordinate rings admit a cluster algebra structure. The presence of a cluster structure has important implications for the geometry of a variety including the existence of canonical linearly independent sets of regular functions. Additionally, methods for studying cluster algebras can be applied to obtain insights that are beyond the reach of conventional geometric tools. In order to associate a cluster algebra structure to a variety one needs to construct an initial seed of regular functions that generate the coordinate ring, and where each cluster variable generated from the initial seed is indeed a regular function. In the case of Grassmannians and their open positroid subvarieties, this initial seed is described by a strand diagram in the disk called a Postnikov diagram. Our first objective is to study the cluster structure on the homogeneous coordinate ring of the Grassmannian using the additive categorification introduced by Jensen King and Su. This abstract approach allows us to apply methods from representation theory, algebraic geometry and combinatorics. We use these tools to describe a class of (𝑘 − 1)-exact sequences present in the subcategory of Plücker modules. In the process, we generalise a construction of Jasso involving 𝑛-exact sequences formed by acyclic complexes. The second goal of this thesis is to develop a method for categorifying general cluster varieties. The mutation that defines these cluster structures is determined from the combinatorics of a frozen quiver with potential called a dimer quiver. In order to construct a categorification from those of simpler cluster varieties, we introduce a gluing operation for dimer quivers on surfaces. We use this operation to recover the boundary algebra obtained from a gluing when certain consistency conditions are met. As an application we determine the boundary algebra of a class of homogeneous strand diagrams on the annulus and we provide a method to determine the boundary algebra of the initial seed of a positroid variety through a decomposition into glued dimer quivers. Finally, we investigate a categorical interpretation of freezing cluster variables using reductions of stably 2-Calabi-Yau Frobenius extriangulated categories introduced by Faber, Marsh and Pressland. This work provides a first step towards constructing cluster categories from a gluing decomposition of cluster varieties.