Applications of Laplace-Beltrami operator for Jack polynomials

TL;DR

This paper introduces a new method to analyze the Laplace-Beltrami operator on symmetric functions, leading to explicit formulas and combinatorial descriptions of Jack symmetric functions and their coefficients.

Contribution

It provides a novel explicit computational approach for Jack symmetric functions, including formulas for Littlewood-Richardson coefficients and action of Virasoro operators.

Findings

Derived a combinatorial formula for Jack functions

Established a determinantal formula for Jack functions

Obtained explicit action formulas for Virasoro operators

Abstract

We use a new method to study the Laplace-Beltrami type operator on the Fock space of symmetric functions, and as an example of our explicit computation we show that the Jack symmetric functions are the only family of eigenvectors of the differential operator. As applications of this explicit method we find a combinatorial formula for Jack symmetric functions and the Littlewood-Richardson coefficients in the Jack case. As further applications, we obtain a new determinantal formula for Jack symmetric functions. We also obtained a generalized raising operator formula for Jack symmetric functions, and a formula for the explicit action of Virasoro operators. Special cases of our formulas imply Mimachi-Yamada's result on Jack symmetric functions of rectangular shapes, as well as the explicit formula for Jack functions of two rows or two columns.