3D Bosons and $W_{1+\infty}$ algebra

TL;DR

This paper constructs 3D Bosons from 2D Young diagrams using the Yang-Baxter equation, and demonstrates their representation of the $W_{1+ abla}$ algebra along with Littlewood-Richardson rules for 3-Jack polynomials.

Contribution

It introduces a novel 3D Boson framework derived from 2D diagrams and establishes their algebraic representation and combinatorial rules.

Findings

Construction of 3D Bosons from 2D Young diagrams.

Representation of $W_{1+ abla}$ algebra using 3D Bosons.

Derivation of Littlewood-Richardson rule for 3-Jack polynomials.

Abstract

In this paper, we consider 3D Young diagrams with at most $N$ layers in $z$-axis direction, which can be constructed by $N$ 2D Young diagrams on slice $z=j$, $j=1,2,\cdots, N$ from the Yang-Baxter equation. Use 2D Bosons $\{a_{j,m},\ m\in\Z\}$ associated to 2D Young diagrams on the slice $z=j$, we constructed 3D Bosons. Then we show the 3D Boson representation of $W_{1+\infty}$ algebra, and the Littlewood-Richardson rule for 3-Jack polynomials from the actions of 3D Bosons on 3D Young diagrams.