Planar algebras for the Young graph and the Khovanov Heisenberg category

TL;DR

The work constructs Jones-style planar algebras from the Young graph by filling planar tangles with Young diagrams and weights given by a harmonic function, with the Plancherel case yielding a crossing element. It shows that the defining relations of the Khovanov Heisenberg category arise as planar-algebra relations in this framework, and that normalized characters and Boolean cumulants become explicit planar-algebra elements, thereby connecting diagrammatic categorification with asymptotic representation theory. The approach provides a self-contained, diagrammatic route to the Rattan--Śniady conjecture and illuminates how combinatorics of Young diagrams can be encoded in planar algebra language, while suggesting broad generalizations to other branching graphs, deformations, and cumulant theories. Overall, the paper bridges diagrammatic categorification and asymptotic representation theory through a concrete, constructive planar-algebra model tied to harmonic analysis on the Young graph.

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.

Figures (8)

• Figure 1: A morphism from $Q_{-}\otimes Q_{+}$ to $Q_{-}\otimes Q_{+}\otimes Q_{+}\otimes Q_{-}$.
• Figure 2: Young graph. The vertical direction (downwards) indicates the weights of Young diagrams.
• Figure 3: Profile (red) of the Young diagram $\lambda = (7,4,2,1)$. In this example, the local minima and maxima are $(\mathsf{x}_{1},\mathsf{x}_{2},\mathsf{x}_{3},\mathsf{x}_{4},\mathsf{x}_{5})=(-4,-2,0,3,7)$ and $(\mathsf{y}_{1},\mathsf{y}_{2},\mathsf{y}_{3},\mathsf{y}_{4})= (-3,-1,2,6)$.
• Figure 4: Planar tangle.
• Figure 5: Composition of planar tangles.

Theorems & Definitions (44)

• Theorem 1.1: Koshida2023, conjectured in RS2008
• Remark 1.2
• Example 2.1
• Remark 2.2
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4
• Remark 3.5
• Proposition 3.6