English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

This thesis is concerned with studying the properties of gradings on severalexamples of cluster algebras, primarily of infinite type.  We first consider two finite type cases: $B_n$ and $C_n$, completing aclassification 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 alldegrees are positive and have only finitely many associated cluster variables(excepting one particular case). For the mutation-acyclic matrices, we provethat all occurring degrees have infinitely many variables.  We provide a sufficient condition for a graded cluster algebra generated by aquiver to have infinitely many degrees, based on the presence of a subquiver inits mutation class. We use this to show that the cluster algebras for (quantum)coordinate rings of matrices and Grassmannians contain cluster variables of alldegrees in $\mathbb{N}$.  Next we consider the list (given by Felikson, Shapiro & Tumarkin) ofmutation-finite quivers that do not correspond to triangulations of markedsurfaces. We show that $X_7$ gives rise to only two degrees, both withinfinitely 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 valuationfunctions on such a surface and prove combinatorially that it is isomorphic tothe space of gradings on the associated cluster algebra. We illustrate thistheory by applying it to the annulus with $n+m$ marked points. We show that thestandard grading is of mixed type. We also give an alternative grading in whichall degrees have infinitely many cluster variables.

Applications of Graded Methods to Cluster Variables in Arbitrary Types