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

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 $\{a_{j,m},m\in \mathbb{Z}\}$ associated to 2D Young diagrams on the slice $z=j$, we constructed 3D Bosons. Then we show the 3D Boson representation of $W_{1+\mathrm{\infty }}$ algebra, and the Littlewood-Richardson rule for 3-Jack polynomials from the actions of 3D Bosons on 3D Young diagrams.