# Jack polynomials as determinants

Jack symmetric polynomials are known to be generalizations of Schur functions $\chi_\lambda$, for which powerful Weyl determinant formulas are known. Are there any generalizations of two determinant formulas for general Jack symmetric $P^\alpha_\lambda(x)$ functions?

The first determinant (Jacobi-Trudi) formula represents the character of the irrep GL(N) given by the partition $\lambda$ $$\chi_\lambda(x)=\det_{i,j} s_{\lambda_i-i+j}$$ where $s_k$ are elementary Schur function and the second one gives the same function as determinant $$\chi_\lambda(x)=\frac{\det_{i,j} x_i^{\lambda_j+N-j}}{\det_{i,j} x_i^{N-j}}$$ Jack symmetric polynomials are natural generalizations of Schur polynomials, and probably, to operate with them it would be useful to have as simple as possible analogs of Weyl formulas.

• At $\alpha=1$, can Jacobi-Trudi be explained as a result that follow from consideration of differential operators and eigenfunctions? Nov 14, 2019 at 4:32
Lassalle and Schlosser have obtained in Inversion of the Pieri formula for Macdonald polynomials some recurrence relations for MacDonald polynomials $$P_\lambda(x;q,t)$$ which can be restricted to Jack polynomials making $$q=t^\alpha$$ and letting $$t\to 1$$.
These recurrence relations lead to expressions for Jack polynomials as linear combination of products of elementary Schur functions, as you want. But it is not clear to me under what circumstances these relations can be written in the form of a determinant. This is true when $$\lambda=(r,1^s)$$ is a hook, in which case $$P_{\lambda}^\alpha\propto\det\left(\frac{\alpha\lambda_i+s-\alpha i+j(\alpha-1)+1}{\alpha\lambda_i+s-i+1}P^\alpha_{(\lambda_i-i+j)}\right).$$