Researcher ORCID Identifier
Open Access Senior Thesis
Bachelor of Science
© 2021 Feiyang Lin
Motivated by existing results about the Kronecker cluster algebra, this thesis is concerned with two families of cluster algebras, which are two different ways of generalizing the Kronecker case: rank-two cluster algebras, and cluster algebras of type An,1. Regarding rank-two cluster algebras, our main result is a conjectural bijection that would prove the equivalence of two combinatorial formulas for cluster variables of rank-two skew-symmetric cluster algebras. We identify a technical result that implies the bijection and make partial progress towards its proof. We then shift gears to study certain power series which arise as limits of ratios of F-polynomials in cluster algebras of type An,1. With several different perspectives in mind, including that of continued fractions, path-ordered products and the surface model, we state and prove various equivalent formulas for these power series. In our study of these two families, we make use of a product formula for F-polynomials, called Gupta's formula, which is applicable to all cluster algebras of geometric type. We dedicate one of our chapters to an exposition of this formula. Though Gupta's formula has previously appeared in different notations, and in that sense is not new, we believe that our statement and proof of the formula provides a new approach to the formula which is elementary and combinatorial.
Lin, Feiyang, "On Rank-Two and Affine Cluster Algebras" (2021). HMC Senior Theses. 251.