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

Abstract

We suggest an ansatz for representation of affine Yangian $Y({\widehat{\mathfrak{gl}}}_1)$ 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.

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Cited by in corpus (6)

• Simple Representations of BPS Algebras: the case of $Y({\widehat{\mathfrak{gl}}}_2)$
• Wall-Crossing Effects on Quiver BPS Algebras
• Algorithms for representations of quiver Yangian algebras
• Super-Hamiltonians for super-Macdonald polynomials
• Weyl Mutations in Quiver Yangians
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations