Stanley’s positivity conjecture for Jack Littlewood–Richardson coefficients

Prove that the normalized Jack Littlewood–Richardson coefficients c^\theta_{\mu,\nu} are nonnegative for every \theta>0; in its stronger form, establish that the corresponding structure constants in the J-normalization are polynomials in \theta^{-1} with nonnegative integer coefficients.

Background

The paper defines normalized Jack Littlewood–Richardson coefficients through the product expansion of normalized Jack polynomials. These coefficients define a signed convolution of probability measures on partitions; if they were nonnegative for all positive Jack parameters, that convolution would automatically be a probability measure and would yield the associated quantized \gamma-convolution probabilistically.

The conjecture is known in the representation-theoretic cases \theta=1/2,1,2, but the general positivity assertion remains unresolved. The paper notes that Stanley’s original conjecture is stronger than the nonnegativity statement used for the convolution construction.

References

One of the celebrated conjectures of Stanley states that the Jack Littlewood--Richardson coefficients are (up to a proper normalization) polynomials in $\theta$ with nonnegative integer coefficients. This conjecture is much stronger than the assumption that $N{(1)}\boxplus\theta_N{(2)}$ are probability measures. Even though Stanley's conjecture is still wide open, we will actually show the existence of the quantized $\gamma$-convolution for a large class of measures.

— Discrete $N$-particle systems at high temperature through Jack generating functions  (2502.13098 - Cuenca et al., 18 Feb 2025) in Remark \ref{rem:stanley}, Section 5.1 (Application 1)