Seminar Cluster Algebras


A square matrix with real entries is called totally positive, if all its minors have positive values. To test whether a given matrix is totally positive is an old question. Since the beginning of 20th century, many mathematicians coming from different domains made their contributions to totally positive matrices: Pólya, Schoenberg, Fekete, Gantmacher, Krein, Karlin, Whitney, Gasca, Peña, Pinkus, to name but a few. In the beginning of 1990s, Lusztig brought the powerful (and meanwhile complicated) tools from representation theory of quantum groups into the study of these matrices, which becomes an active research direction in the last 30 years.

Motivated by the work of Lusztig, Fomin and Zelevinsky introduced cluster algebra in a seminal work in 2002 in order to study the total positivity in the work of Lusztig from a combinatorial point of view. It turned out later that cluster algebra is a structure which exists widely in Lie theory (Grassmann variety, flag variety), representation theory (Lie algebra and quiver representation), symplectic geometry (Poisson structure, action-angel coordinates), complex analysis (Teichmüller spaces), mathematical physics (scattering amplituhedron, thermodynamic system), combinatorics and so on.

The goal of this seminar is to understand the motivation, basic properties and examples of cluster algebras.