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The Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ appear in the expansion of a product of Schur functions into Schur functions, $s_{\mu}(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x)$. There is a combinatorial rule for computing these coefficients.

Zonal polynomials are somewhat similar to Schur functions (both are particular cases of Jack polynomials and are related to representation theory). Is there a combinatorial rule for computing the analogues of the LR coefficients for zonal polynomials?

There is already a similar question (Littlewood-Richardson coefficients for Jack symmetric functions) for Jack polynomials, but I am interested in a specific and certainly simpler case.

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It seems that this paper, Thm. 4.1, gives an expression for such coefficients, in terms of Zonal characters. This seem to be the closest so far.

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    $\begingroup$ Sadly, right after that theorem, the authors state that "To this date, there is no combinatorial rule, like the Littlewood-Richardson rule, for computing the coefficients $b^\lambda_{\mu\nu}$." But this paper is 27 years old, so one might hope for some development since then... maybe particular cases of $b^\lambda_{\mu\nu}$ are known? $\endgroup$
    – Marcel
    Commented Feb 28, 2019 at 18:25
  • $\begingroup$ Name of this paper: Bergeron and Garsia - Zonal polynomials and domino tableaux. $\endgroup$
    – LSpice
    Commented Sep 12, 2021 at 1:15

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