3-Schurs from explicit representation of Yangian $$ \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) $$. Levels 1–5
А. Морозов, Nikita Tselousov
Abstract
A bstract We suggest an ansatz for representation of affine Yangian $$ Y\left({\hat{\mathfrak{gl}}}_1\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>gl</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> </mml:math> by differential operators in the triangular set of time-variables P a , i with 1 ⩽ i ⩽ a , which saturates the MacMahon formula for the number of 3 d 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 ψ 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.