Applications of Graded Methods to Cluster Variables in Arbitrary Types

TL;DR

This thesis explores the properties of gradings in various cluster algebras, classifying degrees and variables, especially in infinite types, and linking surface-based gradings to combinatorial structures.

Contribution

It provides a comprehensive classification of gradings in finite and infinite type cluster algebras, introduces new conditions for infinite degrees, and connects surface gradings with combinatorial models.

Findings

Mutation-cyclic matrices yield finitely many degrees with positive degrees.

Mutation-acyclic matrices produce infinitely many degrees.

Surface-based gradings are isomorphic to valuation function spaces.

Abstract

This thesis is concerned with studying the properties of gradings on several examples of cluster algebras, primarily of infinite type. We first consider two finite type cases: $B_n$ and $C_n$, completing a classification by Grabowski for coefficient-free finite type cluster algebras. We then consider gradings arising from $3 \times 3$ skew-symmetric matrices. We show that the mutation-cyclic matrices give rise to gradings in which all degrees are positive and have only finitely many associated cluster variables (excepting one particular case). For the mutation-acyclic matrices, we prove that all occurring degrees have infinitely many variables. We provide a sufficient condition for a graded cluster algebra generated by a quiver to have infinitely many degrees, based on the presence of a subquiver in its mutation class. We use this to show that the cluster algebras for (quantum) coordinate rings of matrices and Grassmannians contain cluster variables of all degrees in $\mathbb{N}$. Next we consider the list (given by Felikson, Shapiro & Tumarkin) of mutation-finite quivers that do not correspond to triangulations of marked surfaces. We show that $X_7$ gives rise to only two degrees, both with infinitely many variables, and that $\widetilde{E}_6$, $\widetilde{E}_7$ and $\widetilde{E}_8$ give rise to infinitely many variables in some degrees. Finally, we study gradings arising from marked surfaces (see Fomin, Shapiro & Thurston). We adapt a definition by Muller to define the space of valuation functions on such a surface and prove combinatorially that it is isomorphic to the space of gradings on the associated cluster algebra. We illustrate this theory by applying it to the annulus with $n+m$ marked points. We show that the standard grading is of mixed type. We also give an alternative grading in which all degrees have infinitely many cluster variables.