University of Minnesota Combinatorics Seminar: 2021-2022

Date: 10/15/2021

Speaker: Nicolle Gonzalez

A diagrammatic Carlsson-Mellit algebra
The shuffle conjecture was an open problem 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 via the introduction of a new algebra denoted $A_{q,t}$. This algebra, a cousin of the DAHA, arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. Carlsson, Gorsky, and Mellit showed this algebra and its representation could be realized using parabolic flag Hilbert schemes and in addition to containing the generators of the elliptic Hall algebra. Despite the various formulations of this algebra, computations within it are complicated and non-intuitive. In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of $A_{q,t}$ and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of $A_{q,t}$ and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations as it allows us to realize symmetric functions and their related operators as topological diagrams that can be wiggled and thus maneuvered and understood much more easily than their algebraic counterparts.

A diagrammatic Carlsson-Mellit algebra

The shuffle conjecture was an open problem 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 via the introduction of a new algebra denoted $A_{q,t}$. This algebra, a cousin of the DAHA, arises as an extension of two copies of the affine Hecke algebra by certain raising and lowering operators and acts on the space of symmetric functions via certain complicated plethystic operators. Carlsson, Gorsky, and Mellit showed this algebra and its representation could be realized using parabolic flag Hilbert schemes and in addition to containing the generators of the elliptic Hall algebra. Despite the various formulations of this algebra, computations within it are complicated and non-intuitive.

In this talk I will discuss joint work with Matt Hogancamp where we construct a new topological formulation of $A_{q,t}$ and its representation as certain braid diagrams on an annulus. In this setting many of the complicated algebraic relations of $A_{q,t}$ and applications to symmetric functions are trivial consequences of the skein relation imposed on the pictures. In particular, many difficult computations become simple diagrammatic manipulations as it allows us to realize symmetric functions and their related operators as topological diagrams that can be wiggled and thus maneuvered and understood much more easily than their algebraic counterparts.