The representation theory of an algebra gives rise to various interesting geometrical objects, such as the g-vector fan and Newton polytopes of representations. Classical objects such as the associahedron can be realized in this way, and these constructions have interesting applications in the categorification of cluster algebras.
In this talk, I will associate to any representation-finite algebra another geometrical object, an affine variety which is closely related to the polytopes mentioned above. We will see how this variety reflects the tau-tilting theory of the algebra, and how F-polynomials of representations give a parametrization of it.
This is a report on ongoing work with Nima Arkani-Hamed, Hadleigh Frost, Giulio Salvatori and Hugh Thomas.