KP hierarchy, affine Yangian and $W_{1+\infty}$ algebra

Abstract

3D (dimensional) Young diagrams are a generalization of 2D Young diagrams. We want to construct the structures on 3D Young diagrams paralleled to that on 2D Young diagrams. We have already obtained the 3D Bosons, 3D Fermions and 3-Jack polynomials, which are the symmetric functions on 3D Young diagrams. In this paper, we realize the KP hierarchy by the fields $V_j(z)$ in the $W_{1+\infty}$ algebra, where $V_j(z)$ are constructed by $J_j(z)$, which are the Bosonic fields associated the 2D Young diagrams on the slices $z=j$ of 3D Young diagrams. We also show the correspondence between the Bosonic Fock spaces and Fermionic Fock space for the cases: Schur functions, Jack polynomials and 3-Jack polynomials.