Symmetric Functions and 3D Fermion Representation of W1+infinity Algebra

In this paper, we consider the actions of affine Yangian and W1+infinity algebra on three cases of symmetric functions. The first one is Schur functions of 2D Young diagrams. It is known that affine Yangian and W1+infinity algebra can be represented by 1 Boson field with center 1 in this case. The second case is the symmetric functions Y-lambda(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 W1+infinity 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 W1+infinity 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 W1+infinity algebra.