Cluster algebras and representation theory

Cluster algebras are combinatorially defined commutative algebras defined by S. Fomin and A. Zelevinsky in 2001 in their study of the dual canonical basis of a quantum group associated to a simple Lie algebra. A cluster algebra is defined as a subring of a field of rational functions, generated by overlapping sets of generators, or clusters, which are generated from an initial cluster via a mutation process. Motivation for cluster algebras also came from total positivity criteria for matrices and the theory of integer recurrences. Cluster algebras play a role in various fields, including combinatorics, reductive algebraic groups, the geometry of surfaces, integrable systems and the representation theory of finite dimensional algebras.

There is a beautiful relationship between the combinatorics of cluster algebras, Lie theory, and the corresponding representation theory. For example, the number of clusters in a cluster algebra associated to the special linear group SL(C) is given by a Catalan number, and the number of tilting modules over a corresponding finite dimensional algebra (a path algebra) is also a Catalan number. Cluster representation theory aims to develop explanations and generalize such coincidences, while developing new representation theoretic ideas and techniques. The aim of the PhD project is to generalize the cluster combinatorics described above and to find and develop corresponding representation-theoretic models.

Essential prerequisites include undergraduate courses in abstract algebra (such as group theory, ring theory or representation theory), while courses in combinatorics would be useful.