Stanley famously conjectured ("Some combinatorial properties of Jack symmetric functions" Adv. in Math. (77) 1989, doi:10.1016/0001-8708(89)90015-7, MR1014073, Zbl 0743.05072) that the Jack Littlewood-Richardson coefficient $\langle J_\mu J_\nu, J_\lambda\rangle$ is a non-negative integer polynomial in the Jack deformation parameter $\alpha$. He also proposed a 'stronger' form of this conjecture when the corresponding Schur LR coefficient satisfies $c_{\mu\nu}^{\lambda} = 1$, namely that in this case there should be a formula of the type
$$
\langle J_\mu J_\nu, J_\lambda\rangle = \left(\prod_{b\in \mu} A_\mu(b)\right)\left(\prod_{b\in \nu} B_\nu(b)\right)\left(\prod_{b\in \lambda} C_\lambda(b)\right)
$$
where $A,B,C$ are a choice of upper or lower hook length for every box $b$ in each of the three partitions. Such a formula is manifestly a non-negative integral polynomial in $\alpha$.
My question is: are there any (even loose) conjectural forms for the structure of the Jack LR coefficients in any case where $c_{\mu\nu}^{\lambda} > 1$?
Best Answer
I have a new preprint out today regarding this question: The Stanley Conjecture Revisited.
In this paper, I conjectured that all Jack LR coefficients obey a 'windowing' property, that allows to reduce the computation of these coefficients to a smaller sub-problem.
Assuming this conjecture, many explicit formulas are presented for the Jack LR coefficient $c_{21,21}^{321}(\alpha)$, which is the simplest case that has Schur LR $c=2$.