Every slice of a 3D Young diagram on the plane z = n in the coordinate system O - xyz is a 2D Young diagram. In this paper, we show how Jack polynomials of 2D Young diagrams constitute a Jack polynomial of 3D Young diagram, which is called 3-Jack polynomial. We give the specific expressions of 3-Jack polynomials of 3D Young diagrams of total box number less than 4, and we find a method to obtain 3-Jack polynomials for every 3D Young diagram. 3-Jack polynomials become Jack polynomials of 2D Young diagrams when 3D Young diagrams have only one layer in the z-axis direction.

中文翻译：

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3-JACK 多项式和 Yang--Baxter 方程

坐标系O - xyz中平面z = n上的 3D Young 图的每个切片都是 2D Young 图。在本文中，我们展示了 2D Young 图的 Jack 多项式如何构成 3D Young 图的 Jack 多项式，称为 3-Jack 多项式。我们给出了总箱数小于4的3D Young图的3-Jack多项式的具体表达式，并找到了一种获取每个3D Young图的3-Jack多项式的方法。当3D Young图在z轴方向只有一层时，3-Jack多项式成为2D Young图的Jack多项式。