PIMS-USask Geometry, Algebra and Physics Seminar: David Hernandez

The structure of the eigenvalues of a quantum system, that is of its spectrum, is crucial to its study. The spectrum of the "ice model" (six vertex model) was computed in the seminal work of Baxter. It has a remarkable form involving polynomials and the famous Baxter relations. Later, it was conjectured that there is an analog form for the spectrum of more general quantum integrable systems.

We will discuss how, using the modern mathematical point of view of representation theory, these (Baxter) polynomials occur in a natural way. Besides, Baxter relations can be categorified using relevant categories of representations of quantum groups. This leads to a proof of the general conjecture (joint results with Jimbo and with Frenkel). We will also discuss other recent applications for the representation theory of truncated quantum groups (quantized Coulomb branches), in the framework of symplectic duality (3d mirror symmetry).