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

Abstract

In this paper, we discuss the relations between the Jack polynomials, $\hslash$-dependent KP hierarchy and affine Yangian of $\mathfrak{gl}(1)$. We find that $\alpha ={\hslash }^2$ and $h_1=\hslash ,h_2=-{\hslash }^{-1}$, 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 $\hslash$-KP hierarchy, and the Jack polynomials can be used to describe the tau functions of the $\hslash$-KP hierarchy.