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

We suggest an ansatz for representation of affine Yangian $Y({\widehat{\mathfrak{g}\mathfrak{l}}}_1)$ by differential operators in the triangular set of time-variables ${\mathbf{P}}_{a,i}$ with $1⩽i⩽a$, 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.