Affine Yangian and Schur functions on plane partitions of 4

Abstract

In this paper, we first calculate the orthogonal basis of the vector space spanned by e_ie_je_ke₀|0⟩, where e_j are the generators of the affine Yangian of $\mathfrak{g}\mathfrak{l}(1)$ . The elements of this orthogonal basis correspond to three dimensional bosons. Then, we calculate the Schur functions of plane partitions of 4, we find that the plane partitions become Young diagrams, and the Schur functions on plane partitions become Schur functions on Young diagrams when h₁ = 1, h₂ = −1, and h₃ = 0.