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

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.