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I implemented the Jack polynomials with a symbolic Jack parameter $\alpha$ in their coefficients ($\alpha=1$ for Schur polynomials, $\alpha=2$ for zonal polynomials). From my implementation (and also from the definition I think), the coefficients are fractions of polynomials in $\alpha$. But each time I compute a $J$-Jack polynomial, I observe that they actually are polynomials in $\alpha$ (I didn't check yet for the $P$-Jack polynomials and the $Q$-Jack polynomials). Is it an established fact, and are there some references about it?

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  • $\begingroup$ Coefficients in what basis? $\endgroup$ Apr 2 at 12:59
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    $\begingroup$ the combinatorial formula in Wikipedia is a polynomial in $\alpha$: en.wikipedia.org/wiki/Jack_function $\endgroup$ Apr 2 at 13:03
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    $\begingroup$ But being a symmetric function, you can group together all the terms for the different permutations of a monomial into the monomial symmetric function, no? (Anyways sorry this discussion is probably irrelevant seeing as you got the answer you were looking for...) $\endgroup$ Apr 2 at 13:26
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    $\begingroup$ @SamHopkins Thank you for this remark! This can considerably shorten the expressions of the Jack polynomials. I never had this idea. $\endgroup$ Apr 2 at 14:56
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    $\begingroup$ @CarloBeenakker, re, since that seems to answer the question, maybe you can post it as an answer? $\endgroup$
    – LSpice
    Apr 2 at 15:04

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Following up on the suggestion of LSpice, to remove this from the "unanswered queue":

the combinatorial formula in Wikipedia, due to Knop and Sahi, is a polynomial in $\alpha$.

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  • $\begingroup$ And to settle a subquestion: by definition, the '$P$-normalisation' is monic, $P_\lambda = m_\lambda + \text{lower}$, so the lower terms have coefficients that are rational functions of $\alpha$. It differs from the '$J$-normalisation by a factor that depends on $\alpha$ and remove the denominator. $\endgroup$ Apr 7 at 14:04

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