Affine Yangian and 3-Schur functions

Abstract

3D (3 dimensional) Young diagram is a generalization of 2D Young diagram. In this paper, from the orthogonality of 3D Young diagrams and the properties in affine Yangian and its MacMahon representation, we obtain the Schur functions corresponding to 3D Young diagrams, which are called 3-Schur functions. 3-Schur functions are a generalization of Schur functions in the sense that when h1=1,h2=−1,h3=0, the 3-Schur functions of 3D Young diagrams become Schur functions of 2D Young diagrams, which is a special case of h1=h,h2=−1h,h3=1h−h. When h1=h,h2=−1h,h3=1h−h, the 3-Schur functions turn into the Jack symmetric polynomials of 2D Young diagrams by multiplying a coefficient. We will see that 3-Schur functions are symmetric about three coordinate axes.