Skew Schur polynomials are defined as $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$, where the Littlewood–Richardson coefficients $c^\lambda_{\mu\nu}$ satisfy $s_\mu(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x)$.

However, it is possible to compute the skew Schur polynomials without knowing the Littlewood–Richardson coefficients, because there is a formula expressing them as a determinant, the Jacobi–Trudi formula.

LIkewise, skew zonal polynomials are defined as $Z_{\lambda/\mu}=\sum_\nu b^\lambda_{\mu\nu}Z_\nu$, where the coefficients $b^\lambda_{\mu\nu}$ satisfy $Z_\mu(x)Z_\nu(x)=\sum_\lambda b^\lambda_{\mu\nu}Z_\lambda(x)$.

Is it possible to compute the skew zonal polynomials without knowing the $b$ coefficients, using some analogue of the Jacobi–Trudi formula? I am actually only interested in the simplest quantity $Z_{\lambda/\mu}(1^N)$.

(Is there a formula for skew Macdonald functions similar to Jacobi-Trudi identity? is a similar question about Macdonald polynomials. Zonal polynomials are a very particular case for which a lot more is known.)


I haven't found anything about Jacobi–Trudi for skew zonal polynomials.

For usual zonal polynomials, Kerov has shown (Generalized Hall–Littlewood symmetric functions and orthogonal polynomials) that, when $\lambda=(r,1^s)$ is a hook, then $Z_{\lambda'}$ can be written as a determinant, $$ Z_{\lambda'}\propto\det\left(\frac{\lambda_i+2s-i-j+2}{\lambda_i+2s-2i+2}e_{\lambda_i-i+j}\right). $$

On the other hand, Matsumoto has shown (Two parameters circular ensembles and Jacobi–Trudi type formulas for Jack functions of rectangular shapes) that, when $\lambda=(r^\ell)$ is of rectangular shape, then $Z_{\lambda'}$ can be written as a pfaffian, $$ Z_{\lambda'}\propto\operatorname{Pf}\left((j-i)e_{r+2\ell-i-j+1}\right). $$

These results are particular cases of a theorem of Lassalle and Schlosser (Inversion of the Pieri formula for Macdonald polynomials) which gives a recurrence relation for generic zonal polynomials. This seems to be the state of the art in terms of Jacobi–Trudi-like formulas.

It is not clear to me under what circumstances these recurrence relations can be cast in the form of a determinant or a pfaffian. It is also not clear whether something similar holds for skew zonal polynomials.


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