Jack vertex operators and realization of Jack functions

TL;DR

This paper introduces an iterative method to realize Jack functions from rectangular shapes, proves positivity conjectures, and establishes a basis of symmetric functions using Jack vertex operators, with applications to Dyson integrals.

Contribution

It presents a novel iterative approach to construct Jack functions, proves cases of Stanley's positivity conjecture, and introduces a basis of symmetric functions via Jack vertex operators.

Findings

Realization of Jack functions from rectangular shapes.

Proof of some cases of Stanley's positivity conjecture.

New evaluation formulas for Dyson integrals and Vandermonde powers.

Abstract

We give an iterative method to realize general Jack functions from Jack functions of rectangular shapes. We first show some cases of Stanley's conjecture on positivity of the Littlewood-Richardson coefficients, and then use this method to give a new realization of Jack functions. We also show in general that vectors of products of Jack vertex operators form a basis of symmetric functions. In particular this gives a new proof of linear independence for the rectangular and marked rectangular Jack vertex operators. Thirdly a generalized Frobenius formula for Jack functions was given and was used to give new evaluation of Dyson integrals and even powers of Vandermonde determinant.