A Skein theoretic Carlsson-Mellit algebra

Abstract: The Carlsson-Mellit algebra arose for the first time in the proof of the shuffle conjecture, which gives an explicit combinatorial formula for the Frobenius character of the space of diagonal harmonics in terms of parking functions. Its polynomial representation, given by certain complicated plethystic operators over extensions of the ring of symmetric functions, plays a particularly important role as it encodes much of the underlying combinatorial theory. By various results of Gorsky, Mellit and Carlsson it was shown that this algebra can be used to construct generators of the Elliptic Hall algebra in addition to having deep connections to the homology of torus knots. Thus, a natural starting point in the search to categorify these structures is the categorification of the Carlsson-Mellit algebra and its polynomial representation.

In this talk I will explain joint work with Matt Hogancamp where we constructed a purely skein theoretic formulation of this algebra and realized its generators as certain braid diagrams on a thickened annulus. Consequently, we used this framework to categorify the polynomial representation of the Carlsson-Mellit algebra as a family of functors over the derived trace of the Soergel category.

mathematical physicsalgebraic geometrydifferential geometrygeometric topologyoperator algebrasrepresentation theorysymplectic geometry

Audience: researchers in the topic

Geometry, Physics, and Representation Theory Seminar

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