# Littlewood-Richardson coefficients for zonal polynomials

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.

• 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? Feb 28 '19 at 18:25