Nicolle Gonzalez: A topological approach to the shuffle conjecture
The shuffle conjecture gives a formula for the Frobenius character of the space of diagonal harmonics as a sum over word parking functions. After fifteen years the conjecture was finally proved by Carlsson and Mellit using new algebraic techniques and introducing the so-called "Dyck path" algebra which arises as an extension of two copies of the affine Hecke algebra. Their proof relies on the fact that a representation of this Dyck path algebra can be used to generate all the operators and symmetric functions that are connected to the shuffle conjecture. However, this representation is via plethystic operations and thus generally complicated to compute with.
I will discuss an alternate formulation of this algebra and its representation obtained topologically in which the generators are closures of braids on an annulus. This new skein-theoretic formulation allows us to reinterpret all the symmetric functions and their operators as diagrams instead of complicated plethystic expressions. I will describe how one can obtain familiar objects like the $\nabla$ operator, Schur and Hall-Littlewood polynomials, and others diagrammatically and how this new framework (in which plethysm does not appear at all) might be used to resolve positivity questions about $\nabla$ and related results. This is joint work with Matt Hogancamp.