Graham positivity for the factorial Pieri coefficients

Establish Graham positivity for the Pieri-rule coefficients of the double Schur functions, namely prove that these coefficients belong to the polynomial ring generated by the differences \(\alpha_{i-1}-\alpha_i\) with nonnegative integer coefficients.

Background

The paper derives contour-integral and symmetric-function formulas for the coefficients in the Pieri rule for double Schur functions. These formulas give compressed sums of monomials and, after common factors are removed, exhibit no monomial cancellations.

The authors explicitly note that their formulas do not immediately establish Graham positivity, a positivity property asserting membership in Z⩾0[αi−1−αi∣i∈Z]\mathbb{Z}_{\geqslant 0}[\alpha_{i-1}-\alpha_i\mid i\in\mathbb{Z}]. They obtain only a partial indication of this phenomenon by specializing all parameters to a common value.

References

Let us note that our formula eq:Aklambda_sf does not immediately imply Graham positivity, which means the coefficients belong to $Z_{\geqslant 0}[\alpha_{i-1} - \alpha_i \mid i \in Z]$.

— On the Boson-Fermion Correspondence for Factorial Schur Functions  (2502.02841 - Bump et al., 5 Feb 2025) in Section "Skew-Pieri rule", immediately following Proposition \ref{prop:fun_pieri}