Symmetric Functions and 3D Fermion Representation of $\pmb {W_{1+\infty }}$ Algebra

Abstract

In this paper, we consider the actions of affine Yangian and $W_{1+\infty }$ algebra on three cases of symmetric functions. The first one is Schur functions of 2D Young diagrams. It is known that affine Yangian and $W_{1+\infty }$ algebra can be represented by 1 Boson field with center 1 in this case. The second case is the symmetric functions $Y_\lambda ({\mathbf{p}})$ of 2D Young diagrams which we defined. They become Jack polynomials when $h_1=h, h_2=-h^{-1}$. In this case affine Yangian and $W_{1+\infty }$ algebra can be represented by 1 Boson field with center $-h_\epsilon /\sigma _3$. The third case is 3-Jack polynomials of 3D Young diagrams who have at most N layers in z-axis direction. We show that in this case affine Yangian and $W_{1+\infty }$ algebra can be represented by N Boson field with center $-h_\epsilon /\sigma _3$. At each case, we define the Fermions $\Gamma _m$ and $\Gamma _m^*$ and use them to represent the $W_{1+\infty }$ algebra.