Littlewood-Richardson coefficients for Jack symmetric functions

Question

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. $f_{\mu,\nu}^\lambda(\alpha) \geq 0$?

Answer 1

I have some small partial results on this together with V. Feray (at the moment unpublished, but we plan to write about this on the Open problems in Algebraic Combinatorics blog), and put the notes on arxiv, hopefully before the end of the year (now available!).

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.