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

Abstract

We suggest an ansatz for representation of affine Yangian $Y(\hat{ \mathfrak{gl}}_1)$ by differential operators in the triangular set of time-variables ${\bf P}_{a,i}$ with $1\leqslant i\leqslant a$, which saturates the MacMahon formula for the number of $3d$ Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting "cut-and-join" generators $\psi_n$ of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods.