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

TL;DR

This paper introduces 3-Jack polynomials, a 3D generalization of Schur and Jack polynomials, associated with 3D Young diagrams and parameters, and explores their algebraic properties and connections to 3D Bosons.

Contribution

It develops the theory of 3-Jack polynomials, extending classical symmetric functions to three dimensions, and relates them to 3D Bosons and affine Yangian of gl(1).

Findings

3-Jack polynomials generalize Schur and Jack functions to 3D.

They can be characterized by vertex operators and Pieri formulas.

The paper establishes algebraic structures linking 3D Young diagrams and affine Yangian.

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 $h_1,h_2,h_3$ to $y,x,z$-axis respectively. 3-Jack polynomials are polynomials of $P_{n,j}, n\geq j$ with coefficients in $\mathbb C(h_1,h_2,h_3)$, 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.