# 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? – Student Nov 14 '19 at 4:32