Jack symmetric functions and their inner products

Question

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials".

Let $\Lambda$ be the ring of symmetric functions over $\mathbb{C}$ (in countably infinitely many variables $x_i$). For a partition $\mu=(\mu_1,\mu_2,\dots)$ let $p_\mu$ denote the power sums, i.e., $p_\mu=p_{\mu_1}p_{\mu_2}\cdots$ and $p_{\mu_i}=\sum_k x_k^{\mu_i}$.

Let $\langle\ ,\ \rangle^{\alpha}$ be the inner product given by $\langle p_\mu,p_\nu\rangle=\delta_{\mu,\nu}\alpha^{\ell(\mu)}z_\mu$, where $\alpha\in\mathbb{C}$, $\ell(\mu)$ is the length of the partition $\mu$, $z_\mu=\prod_{i\in\mu}i^{m_i(\mu)}\cdot m_i(\mu)!$ and $m_i(\mu)$ is the multiplicity of $i$ in $\mu$.

The Jack symmetric functions $P_\mu^{(\alpha)}$ form a basis of $\Lambda$ indexed by partitions and are characterised by 2 properties:

1. $P_\mu^{(\alpha)}=m_\mu + \sum_\nu u_{\mu,\nu}(\alpha)m_\nu$, where the $m_\nu$ are symmetric monomials, the $u_\nu(\alpha)\in\mathbb{C}$ and the summation index runs over all partitions $\nu <\mu$ in the dominance ordering.
2. The Jack symmetric functions are mutually orthogonal $\langle P_\mu^{(\alpha)} ,P_\nu^{(\alpha)} \rangle^{\alpha}=0$ if $\mu\neq\nu$.

In fact explicit formulae for $\langle P_\mu^{(\alpha)}, P_\mu^{(\alpha)}\rangle^{\alpha}$ are know, but rather complicated, so I won't list them here.

Question: What happens when I vary the parameter $\alpha$ of the inner product, but not for the Jack polynomials? Does anyone know how to evaluate $\langle P_\mu^{(\alpha)} ,P_\nu^{(\alpha)} \rangle^{\beta}$, $\beta\in\mathbb{C}$? I am most interested in the case $\beta=-2\alpha$.

Answer 1

You ask what happens when you vary the parameter $\alpha$ of the inner product, but not for the Jack polynomials. I cannot say much in the general case, but one instance of interest for me was when the Jack polynomials under consideration are the Schur polynomials ($\alpha=1$). Then the inner product is preserved under duality of partitions $\lambda \mapsto \lambda^*$: $$ \langle J^{(1)}_\lambda,J^{(1)}_\mu \rangle^\beta = \langle J^{(1)}_{\lambda^*},J^{(1)}_{\mu^*} \rangle^\beta. $$

See Inverse Participation Ratios in the XX spin chain for more details.