Jack polynomials, $\hbar$-dependent KP hierarchy and affine Yangian of ${\mathfrak{gl}}(1)$

TL;DR

This paper explores the connections between Jack polynomials, the $ abla$-dependent KP hierarchy, and the affine Yangian of ${rak{gl}}(1)$, revealing parameter relations and their roles in tau functions.

Contribution

It establishes explicit parameter relations linking Jack polynomials, $ abla$-KP hierarchy, and affine Yangian of ${rak{gl}}(1)$, and shows how Jack polynomials describe tau functions.

Findings

Parameter relations: $eta= abla^2$, $h_1= abla$, $h_2=- abla^{-1}$.

Vertex operators in Jack polynomials match those in $ abla$-KP hierarchy.

Jack polynomials serve as tau functions for the $ abla$-KP hierarchy.

Abstract

In this paper, we discuss the relations between the Jack polynomials, $\hbar$-dependent KP hierarchy and affine Yangian of ${\mathfrak{gl}}(1)$. We find that $\alpha=\hbar^2$ and $h_1=\hbar, \ h_2=-\hbar^{-1}$, where $\alpha$ is the parameter in Jack polynomials, and $h_1,\ h_2$ are the parameters in affine Yangian of ${\mathfrak{gl}}(1)$. Then the vertex operators which are in Jack polynomials are the same with that in $\hbar$-KP hierarchy, and the Jack polynomials can be used to describe the tau functions of the $\hbar$-KP hierarchy.