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

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