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Schur expansion positivity and real-rootedness in the integral Jack-to-Schur expansion (Alexandersson–Haglund–Wang)

Investigate and prove that for partitions λ,μ of size d, the coefficients a_k(λ,μ) and b_{d−k}(λ,μ) in the Schur expansion of the integral Jack polynomial J_λ(x;τ) satisfy a_k(λ,μ)∈ℤ_{≥0} and b_{d−k}(λ,μ)∈ℤ_{≥0}, and that the polynomials ∑_{k=0}^d a_k(λ,μ) z^k and ∑_{k=0}^d b_{d−k}(λ,μ) z^k are real-rooted.

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Background

This conjecture, originally due to Alexandersson–Haglund–Wang, is recalled here for completeness. It concerns integrality, positivity, and real-rootedness properties of the Schur expansion coefficients of integral Jack polynomials, with coefficients expressed via binomial basis transforms.

The authors include this conjecture to highlight related positivity phenomena in Jack theory that complement the semiring and evaluation-positivity programs studied in this paper.

References

Conjecture [AHW] Let λ,μ be partitions of d, and define the expansion coefficients a_k(λ,μ) and b_k(λ,μ) as follows: v_{λμ}(τ) = ∑{k=0}{d−1} a_k(λ,μ) (τ+k choose d) = ∑{k=1}{d} b_{d−k}(λ,μ) (τ choose k) k!. Then a_{k}(λ,μ) and b_{d−k}(λ,μ) are in ℤ{≥0}. Furthermore, the polynomials ∑{k=0}d a_k(λ,μ) zk and ∑{k=0}d b{d−k}(λ,μ) zk have only real zeros.

— Majorization via positivity of Jack and Macdonald polynomial differences (2509.19649 - Chen et al., 24 Sep 2025) in Conjecture (AHW), Section 4.3