Researcher ORCID Identifier

0000-0003-0775-7009

Graduation Year

2021

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Gregg Musiker

Reader 2

Dagan Karp

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© 2021 Feiyang Lin

Abstract

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.

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