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 eiejeke0|0⟩, where ej are the generators of the affine Yangian of gl(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.

3-Schurs from explicit representation of Yangian $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$. Levels 1–5

“…Our results for 3-Schur polynomials reproduce the previously known 3-Schur functions of [56,57] at the levels 1-3, but deviates from those from [58] at level 4 -the first level where the truly three-dimensional plane partitions appear, and the problem becomes truly JHEP11(2023)165 difficult. Presumably, this difference can be eliminated by the change of basis of timevariables.…”

3D bosons, 3-Jack polynomials and affine Yangian of $$ \mathfrak{gl}(1) $$

“…× (1 + h 2 h 3 )(2 + h 2 h 3 )(3 + h 2 h 3 )P 4 1 + 6h This expression is slightly different from that in [18] since here we choose…”

“…For example, P n,1 = p n,1 and P 2,2 = p 2,2 . The central charges c 2 and c 3 match P 2,2 , P 2,2 and P 3,3 , P 3,3 in [11] respectively, and c 4 matches P 4,4 , P 4,4 in [18]. Here we choose P 4,4 = − → E 13 |0 , where − → E 13 |0 is defined in [18]:…”