Quantum divided difference operators for indices at least three

Determine an explicit formula for operators ∂_k^q, for every k ≥ 3, satisfying ∂_k^q(S_w^q) = S_{ws_k}^q when w(k) > w(k+1) and ∂_k^q(S_w^q) = 0 when w(k) < w(k+1), provided such operators exist for the q-quantum Schubert polynomials S_w^q.

Background

The paper constructs an action of the symmetric group on the polynomial ring with quantum parameters q_{ij} and defines divided difference operators ∂k on this ring. For q-quantum Schubert polynomials, the paper proves that these operators generally produce an additional correction term η_k(S_wq), rather than directly yielding S{ws_k}q.

For k = 1 and k = 2, the authors define modified operators ∂_kq that remove the correction term and satisfy the desired quantum divided-difference relation. They state that extending this construction to k ≥ 3 is more complicated and leave the existence and explicit form of suitable operators unresolved.

References

If there exists \partial_kq satisfying eq:Intro_quantum_divided_difference for $k \geq 3$, then it would be interesting to find its formula.

— Symmetric group actions on the coordinate ring of the lower unipotent group and the polynomial ring with quantum parameters  (2609.28005 - Horiguchi, 23 Sep 2026) in Section 1, Introduction, subsection “Divided difference operators”