Normalization of Jack polynomial integral-scalar product?

Question

In eq. (10.35) of his book "Symmetric functions and Hall polynomials" I.G.Macdonald gives the following scalar product, under which Jack polynomials with different partitions $\mu\neq\lambda$ are orthogonal

$$\langle J^\alpha_\lambda(z_1,z_2),J^\alpha_\mu(z_1,z_2)\rangle'_2=\frac{1}{2}\int_T J^\alpha_\lambda(z_1,z_2)\overline{J^\alpha_\mu(z_1,z_2)}\prod_{i\neq j}\left(1-\frac{z_i}{z_j}\right)^{1/\alpha}dz^2$$

where the integration contour is $T=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|=1,|z_2|=1\}$. Therefore, the integral equals $c_{\lambda,\alpha}\delta_{\mu,\lambda}$. However, Macdonald does not give the normalization $c_{\lambda,\alpha}$ for the scalar product. Is the normalization known?

Answer 1

Yes, my friend. Take $J_\lambda^{(\alpha)}$ in the J-normalization. Let $n$ be the number of variables (which for you is $2$). Let $\lambda'$ denote the conjugate partition to $\lambda$. Define $$ C_\lambda^{(\alpha)}=\prod_{(i,j) \in \lambda}(\alpha(\lambda_i-j)+\lambda_j'-i+1)(\alpha(\lambda_i-j)+\lambda_j'-i+\alpha) $$ and $$ \mathcal{N}_\lambda^{\alpha}(n)=\prod_{(i,j) \in \lambda} \frac{n+(j-1)\alpha-(i-1)}{n+j\alpha-i}. $$ In this notation, $$ \int \cdots \int \frac{d\theta_1}{2 \pi} \cdots \frac{d\theta_n}{2 \pi} |J_\lambda^{(\alpha)}|^2 \prod_{1 \leq j < k \leq n} |e^{i \theta_k}-e^{i \theta_j}|^{2/\alpha}= \mathcal{N}_\lambda^{\alpha}(n) C_\lambda^{(\alpha)} \frac{\Gamma(1+n/\alpha)}{\Gamma(1+1/\alpha)^n}. $$ Moments of traces of circular beta-ensembles, Tiefeng Jiang and Sho Matsumoto