Integral normal forms for stable Jack multiplication operators

Establish that, for every reduced cycle type μ, the stable multiplication operator Δ_μ(α), after collecting identical normally ordered blocks, has coefficients in ℤ[α].

Background

The paper constructs explicit normally ordered formulas for the first two stable multiplication operators, Δ₂(α) and Δ₃(α), and proves that their collected coefficients lie in ℤ[α]. The Sergeev–Veselov normal-ordering recursion introduces negative powers of α and an auxiliary formal dimension p₀, but these cancel in the first two cases.

The conjecture asks whether the same integral normal-form phenomenon holds uniformly for every stable multiplication operator Δ_μ(α) associated with a reduced cycle type. It is intended to provide an operator-level explanation of polynomiality, rather than a new abstract polynomiality theorem for Jack structure constants. The paper notes that an additional integral control of the triangular basis change would be needed to derive polynomiality of the stable class coefficients from this conjecture.