On vertex operator realizations of Jack functions

TL;DR

This paper develops a vertex operator algebra framework to express Jack symmetric functions, including rectangular and marked rectangular shapes, using generalized homogeneous symmetric functions.

Contribution

It introduces a novel vertex operator approach to realize Jack functions of specific shapes, expanding the algebraic tools available for symmetric function theory.

Findings

Derived a general formula for vertex operator products in terms of symmetric functions

Realized Jack functions of rectangular shapes via vertex operators

Extended realization to marked rectangular shapes

Abstract

On the vertex operator algebra associated with rank one lattice we derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. As an application we realize Jack symmetric functions of rectangular shapes as well as marked rectangular shapes.