Jack function

Remarks

Example: in degree $2$, $P_{(2)}^{(\alpha)} = m_{(2)} + \frac{2}{1+\alpha} m_{(1,1)}$ and $P_{(1,1)}^{(\alpha)} = m_{(1,1)}$. Check: $m_{(2)} = p_2$ and $m_{(1,1)} = \frac12 (p_1^2 - p_2)$, so $P_{(2)}^{(\alpha)} = \frac{\alpha}{1+\alpha} p_2 + \frac{1}{1+\alpha} p_1^2$. With $\langle p_2, p_2 \rangle_\alpha = 2\alpha$ and $\langle p_1^2, p_1^2 \rangle_\alpha = 2\alpha^2$, one gets $\langle P_{(2)}^{(\alpha)}, m_{(1,1)} \rangle_\alpha = \frac12 \left( -\frac{2\alpha^2}{1+\alpha} + \frac{2\alpha^2}{1+\alpha} \right) = 0$. The normalization gives $J_{(2)}^{(\alpha)} = (1+\alpha) m_{(2)} + 2 m_{(1,1)}$ and $J_{(1,1)}^{(\alpha)} = 2 m_{(1,1)}$.

Non-example: the Schur function $s_{(2)} = m_{(2)} + m_{(1,1)}$ satisfies clause 1 but fails clause 2 for $\alpha \ne 1$, since $\langle s_{(2)}, m_{(1,1)} \rangle_\alpha = \frac{\alpha^2 - \alpha}{2}$.

At $\alpha = 1$ the form is the Hall inner product, $P_\lambda^{(1)} = s_\lambda$, and $J_\lambda^{(1)}$ is $s_\lambda$ times the product of the hook lengths of $\lambda$. At $\alpha = 2$, $J_\lambda^{(2)}$ are the zonal polynomials. Existence of the family is a theorem; it is not automatic, because dominance is only a partial order. Restricting to $n$ variables gives the Jack polynomials $J_\lambda^{(\alpha)}(x_1, \dots, x_n)$, which vanish when $\ell(\lambda) > n$.