I am studying the generalisation of the Selberg's integral by using Jack polynomial as outlined by Forrester and Warnaar (arXiv: 0710.3981). They write down the symmetric Jack polynomial as \begin{equation} P_\lambda^{(\alpha)} = m_\lambda + \sum_{\mu < \lambda}a_{\lambda \mu} m_\mu \end{equation} where $m_\lambda$ is the monomial symmetric function for the partition $\lambda$. I basically want to play around with the equation number (2.46) in the reference mentioned above. Could someone give me examples of simple Jack polynomials? Also, I don't understand the significance of the parameter $\alpha$ in $P_\lambda^{(\alpha)}$.
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1$\begingroup$ Jack polynomials are a family of symmetric polynomials with one parameter, here called $\alpha$. At $\alpha=1$ you get Schur, and for $\alpha=2$ zonal symmetric polynomials. (Jack polynomials are in turn generalised Macdonald polynomials, from which they are obtained by setting $t = q^\alpha$ and letting $q \to 1$.) $\endgroup$ – Jules Lamers Aug 4 at 3:46
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$\begingroup$ PS. Lest it might confuse someone let me correct my previous comment: in Macdonald's notation it's $q=t^{\alpha}$, rather than the reverse. (This is related to the coupling of the quantum Calogero--Sutherland model $g\,(g-1)$ as $g=1/\alpha$.) $\endgroup$ – Jules Lamers Nov 16 at 4:27
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Try this set of notes on the Jack symmetric functions / polynomials: https://tcjpn.wordpress.com/2016/11/27/a-note-on-the-jack-symmetric-functions-polynomials/
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$\begingroup$ So basically the coefficients $a_{\lambda\mu}$ get fixed in terms of the parameter $\alpha$, right? $\endgroup$ – morgoth Aug 4 at 13:17
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