The shuffle conjecture was a big open problem in algebraic combinatorics which gave a combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of certain symmetric functions indexed by Dyck paths. This conjecture was finally solved after 14 years by Carlsson and Mellit by the introduction of a new interesting algebra denoted Aq,t. This algebra arises as an extension of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. In later work by Carlsson, Mellit, and Gorsky this algebra and its representation was realized using parabolic flag Hilbert schemes and was also shown to contain the generators of the elliptic Hall algebra. I will discuss a new topological formulation of Aq,t and its representation over a thickened annulus and a categorification thereof over the derived trace of the Soergel category. This is joint work with Matt Hogancamp.
This video is part of the University of Georgia‘s Algebra seminar.
