Closed normal-ordered formulas for higher affine-Yangian modes

Determine closed normal-ordered expressions, in the paper’s Jack Fock-space conventions, for the affine-Yangian raising and lowering modes beyond the explicitly computed modes e₂ and f₂.

Background

The paper identifies the affine Yangian of 𝔤𝔩₁ as the correct deformation of the classical 𝒲₁+∞ construction and gives recursive relations generating all raising and lowering modes from the first modes and the Cartan mode ψ₃. It explicitly computes the first modes and the next modes e₂ and f₂.

However, the recursive construction does not provide closed normally ordered formulas in the chosen conventions at arbitrary order. The conclusion therefore records the derivation of such formulas for higher modes as an unresolved problem in general.

References

Determining their closed normal-ordered expressions in our conventions remains a separate problem in general; \cref{prop:first-Yangian-modes,prop:second-Yangian-modes} carry it out explicitly through $e_2$ and $f_2$.

— Jack Content Operators and the Deformed ${\mathcal W}_{1+\infty}$ Algebra  (2609.10284 - Thibon, 9 Sep 2026) in Conclusion, Section “Conclusion”