3D bosons, 3-Jack polynomials and affine Yangian of mathfrak{gl}(1)

Abstract

3D (3 dimensional) Young diagrams are a generalization of 2D Young diagrams. In this paper, We consider 3D Bosons and 3-Jack polynomials. We associate three parameters h1, h2, h3 to y, x, z-axis respectively. 3-Jack polynomials are polynomials of Pn,j, n ≥ j with coefficients in ℂ(h1, h2, h3), which are the generalization of Schur functions and Jack polynomials to 3D case. Similar to Schur functions, 3-Jack polynomials can also be determined by the vertex operators and the Pieri formulas.