Cluster algebras were introduced by S. Fomin and A. Zelevinsky as a framework for studying total positivity and canonical bases in semisimple groups. They have since appeared in a wide range of mathematical areas, including Teichmüller theory, Poisson geometry, quiver representations, Lie theory, algebraic geometry, algebraic combinatorics, and even in partial differential equations (in the equations describing shallow water waves). This project will bring new Coxeter-theoretic tools to bear on the study of cluster algebras, in order to greatly expand the class of well-understood cluster algebras, and to prove new results even in finite type. Much of the research will use the combinatorics and geometry of sortable elements and Cambrian fans, developed by the investigator in collaboration with D. Speyer.
This project brings together two streams of mathematical research that both have deep connections to a broad array of mathematical fields. Coxeter groups are an algebraic abstraction based on reflective symmetry, and they have played a role in some of the important mathematical developments of the past century. Cluster algebras are a more recent discovery, but have already shown surprising applications in unexpected areas. The application of Coxeter-theoretic tools is one of several promising approaches to the structural study of cluster algebras. The interaction also goes in the other direction, as cluster algebras bring to light new ways of understanding Coxeter groups.
