Applications of Graded Methods to Cluster Variables in Arbitrary Types (1803.02341v1)

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.