Planar algebras for the Young graph and the Khovanov Heisenberg category

TL;DR

This paper constructs planar algebras related to the Young graph, linking diagrammatic categorification with asymptotic representation theory, especially through harmonic functions and the Khovanov Heisenberg category.

Contribution

It introduces a new class of planar algebras associated with Young diagrams, connecting them to the Khovanov Heisenberg category and harmonic functions.

Findings

Planar algebra structure defined via state sums over Young diagrams.

Relations of the Khovanov Heisenberg category are recovered within the planar algebra.

Identification of elements with functions like moments, cumulants, and characters.

Abstract

This paper studies planar algebras of Jones' style associated with the Young graph. We first see that, given a positive real valued function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the function is harmonic and related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagramatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagramatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups.