Symmetric deformed 2D/3D Hurwitz–Kontsevich model and affine Yangian of $${\mathfrak {gl}}(1)$$

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