On the relations for the cluster tilted algebra resulting from a monomial tilted algebra

First constructed by Fomin and Zelevinski 13, cluster algebras have been studied from many different perspectives. One such perspective is the study of cluster tilted algebras. We focus on when C is a monomial tilted algebras and C˜ its associated cluster tilted algebra. We show the set of partial derviatives of the Keller potential form a minimal set of relations for C˜ and we show that if C is also Koszul, then there are overlap relations that can be used to determine if C˜ is Koszul. We use the tools of noncommutative Gröbner basis theory to prove these results.