Laplace-Beltrami operator for Jack polynomials
نویسندگان
چکیده
We introduce a Laplace-Beltrami type operator on the Fock space of symmetric functions and show that the Jack symmetric functions are the only family of eigenvectors of the differential operator, thus giving a new characterization of Jack polynomials. This was achieved by explicit computation of its action on generalized homogeneous symmetric functions. Using this new method we give a combinatorial formula for Jack symmetric functions as well as a combinatorial formula for Littlewood-Richardson coefficients in the Jack case. As further applications, we obtain a new determinant formula for Jack symmetric functions, iterative formulas in terms of generalized homogeneous functions or monomial functions, and an explicit action of Virasoro operators on Jack symmetric functions. Special cases of our formulas imply Mimachi-Yamada’s result on Jack symmetric functions of rectangular shapes, as well as the explicit formula for Jack functions of two rows or two columns.
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