Cluster algebras: Positivity conjecture

Abstract

A well-known fact is that cluster variables of a cluster algebra can be expressed as Laurent polynomials in the variables of any given cluster (The Laurent phenomenon). Sergey Fomin and Andrei Zelevinsky conjectured in 2002 that the coefficients of these Laurent polynomials are nonnegative integer linear combinations over the coefficient group of the cluster algebra (The Positivity conjecture). Since then special cases of this conjecture have been proven. In this thesis we will investigate this conjecture. We will introduce coefficient matrices, which we will use to give a new and slightly stronger proof of the Laurent phenomenon, and we will discuss these coefficient matrices in relation with the Positivity conjecture.