Littlewood-Richardson coefficients for Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$.

We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle p_\mu,p_\nu\rangle_\alpha := \delta_{\mu,\nu} \alpha^{\ell(\mu)} z_\mu$, where $z_\mu = \prod_i i^{m_i(\mu)} m_i(\mu)!$ is the standard numerical factor and $p_\mu$ denotes the power-sum symmetric function.

For any partition $\lambda$, let us denote by $J_\lambda$ the Jack symmetric function. These functions are uniquely determined (up to a normalization constant - here I use the same normalization as used in Macdonald's book) by their triangular expansion with respect to the monomial symmetric functions and by being orthogonal with respect to $\langle \cdot,\cdot\rangle_\alpha$.

A famous conjecture of Stanley states that $f_{\mu,\nu}^\lambda(\alpha) := \langle J_\mu J_\nu,J_\lambda \rangle_\alpha$ is a polynomial in $\alpha$ with non-negative integer coefficients. As far as I know, it is still open.

Some background is given in Luc Lapointe, Luc Vinet, A Rodrigues formula for the Jack polynomials and the Macdonald-Stanley conjecture.

Here is my question: is it known that for any positive real value of $\alpha$ the corresponding Littlewood-Richardson coefficients are non-negative, i.e. $f_{\mu,\nu}^\lambda(\alpha) \geq 0$?

• I have not heard of such a result; it seems to be almost as hard as Stanley's conjecture. Jun 4 '14 at 19:11
• I have not heard too, that is, why I am asking. From one hand side it seems to be as hard as Stanley's conjecture. However, one can imagine, that this kind of positivity can be obtained using methods that will not help in solving Stanley's conjecture. For example, if we know that at some point $\alpha$ LR coefficients are stricty positive, then we know that also in some neighborhood. Then, possibly, one can study behaviour of these coefficients in this neighborhood and try to extend the interval of positivity. This is very naive and totally speculative, since I don't know how to do it. Jun 4 '14 at 19:49
• I guess that it might be possible to prove for $\alpha=2$, which correspond to Zonal spherical functions, since there is some more representation theory available there. Jul 26 '15 at 0:08

The conjecture above extends to the shifted Jack polynomials./ We can prove that the analogous coefficients are polynomials (up to a power of $$\alpha$$), and for any fixed $$\alpha\geq 0$$, the coefficient $$c^{\lambda}_{\mu\nu}(\alpha)$$ is non-negative, whenever $$|\lambda|-|\mu| \leq 1$$. One can prove this by using a recursion for the $$c^{\lambda}_{\mu\nu}(\alpha)$$, similar to what Molev-Sagan did for factorial Schur polynomials.