A unified view of determinantal expansions for Jack polynomials (Q1591743)

The Jack polynomials \(J_{\rho}=J_{\rho}(x_1,\ldots,x_n;\alpha)\) form a basis for the space of symmetric polynomials in \(n\) variables; the specialization \(\alpha=1\) in \(J_{\rho}(x;\alpha)\) gives the Schur function \(s_{\rho}(x)\). Recently \textit{L. Lapointe, A. Lascoux} and \textit{J. Morse} [Electron. J. Comb. 7, No. 1, Research paper N1, 7 p. (2000; Zbl 0934.05123)] have expressed Jack polynomials as determinants in monomial symmetric functions \(m_\lambda\). In the paper under review the author expresses them as determinants in elementary symmetric functions \(e_\lambda\) and shows the symmetry between these two expansions. The approach in the paper is different from that of Lapointe, Lascoux and Morse; the author uses quasi Laplace-Beltrami operators arising from differential geometry and statistics instead of the Calogero-Sutherland operator in physics. Some examples are given and comments on the sparseness of the determinants under consideration.