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 gl(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(kappa), f(kappa),psi(kappa) of the affine Yangian of gl(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 gl(1). Then we can see that the W-operators and eigenstates are symmetric about the permutations of coordinate axes.