A graphical calculus for the Jack inner product on symmetric functions

TL;DR

This paper introduces a graphical calculus based on Frobenius superalgebras that models the Jack inner product on symmetric functions, providing a visual and algebraic framework for understanding this mathematical structure.

Contribution

It develops a novel graphical calculus for Frobenius superalgebras and identifies a diagrammatic space with the Jack inner product space of symmetric functions.

Findings

Graphical calculus models the Jack inner product.

Algebras of closed diagrams correspond to symmetric functions.

Inner product space realized through annular diagram actions.

Abstract

Starting from a graded Frobenius superalgebra $B$, we consider a graphical calculus of $B$-decorated string diagrams. From this calculus we produce algebras consisting of closed planar diagrams and of closed annular diagrams. The action of annular diagrams on planar diagrams can be used to make clockwise (or counterclockwise) annular diagrams into an inner product space. Our main theorem identifies this space with the space of symmetric functions equipped with the Jack inner product at Jack parameter $\operatorname{dim} B_\mathrm{even} - \operatorname{dim} B_\mathrm{odd}$. In this way, we obtain a graphical realization of that inner product space.