Degree Growth in Graded Cluster Algebras

This thesis focuses on graded cluster algebras, looking specifically at degree growth. We begin by considering the rank 3 skew-symmetric case, building on earlier work by Booker-Price. We establish the existence of fastest growing paths, and compare the behaviour for different initial conditions. The central part of the thesis concerns the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian. We construct a distinguished mutation path with certain nice properties. In particular, we suggest a way of using this mutation path to define a partial order on cluster variables, making use of perfect matchings on the exchange quivers. We show that, at least in the finite type case, the partial order we obtain coincides with the ‘standard’ partial order which appears in work of Lenagan and Rigal on quantum graded algebras with a straightening law. We hope that the connection with Lenagan and Rigal’s work could be used to transfer the techniques they use in order to establish the homological properties of other classes of (quantum) cluster algebra. In the final part of the thesis we show that, under mild assumptions, the Segre product of two graded cluster algebras has a natural cluster structure.