Seminar

Cluster algebras were introduced and studied in a series of articles by Fomin and Zelevinsky in [FZ02,FZ03,FZ07] and by Berenstein–Fomin–Zelevinsky in [BFZ05]. They admit connections to several branches of mathematics such as representation theory, geometry, and combinatorics. These algebras are defined by generators obtained recursively form an initial data (a quiver or a matrix). During this talk we will define cluster algebras and show examples. Provided the cluster algebra is acyclic (hence it is a Krull domain), we will show how to compute the associated class group just by looking at the initial data. As a result, we determine when a cluster algebra is a unique factorization domain. Eventually, such proofs rest entirely on tableau combinatorics.