Algebraic relations among q-deformed vertex operators

Determine the algebraic relations satisfied by the q-deformed vertex operators 𝒟_1(z),…,𝒟_{r−1}(z) acting on the ring of r multi-symmetric functions.

Background

The paper introduces q-analogues 𝒟_i of the plethystic operators D_i and packages them into vertex operators 𝒟_i(z). These operators are defined by plethystically shifting adjacent alphabets and multiplying by a q-deformed plethystic exponential. The paper notes that their algebraic relations are not apparent and leaves their determination unresolved; understanding such relations could provide a broader operator-theoretic structure for multi-symmetric Schur functions.

References

It is an interesting, and seemingly nontrivial, question to determine the algebraic relations the vertex operators $\mathcal{D}1(z),\ldots,\mathcal{D}{r-1}(z)$ satisfy.

— Multi-Symmetric Schur Functions  (2502.08738 - Weising, 12 Feb 2025) in Final Remark, Section 6.2, “Weyl Symmetrizers and Bernstein Operators”