Symmetric deformed 2D/3D Hurwitz-Kontsevich model and affine Yangian of gl(1)

Since the ($\beta$-deformed) Hurwitz Kontsevich model corresponds to the special case of affine Yangian of $\mathfrak{g}\mathfrak{l}(1)$. In this paper, we construct two general cases of the $\beta$-deformed Hurwitz Kontsevich model. We find that the $W$-operators of these two models can be represented by the generators $e_k,f_k,{\psi }_k$ of the affine Yangian of $\mathfrak{g}\mathfrak{l}(1)$, and the eigenstates (the symmetric functions $Y_{\lambda }$ and 3-Jack polynomials) can be obtained from the 3D Young diagram representation of affine Yangian of $\mathfrak{g}\mathfrak{l}(1)$. Then we can see that the $W$-operators and eigenstates are symmetric about the permutations of coordinate axes.