nLab Jack polynomial

Jack polynomials (or Jack symmetric functions) form a class of symmetric functions $J_\lambda^\alpha$ labelled by a partition $\lambda$ and a parameter $\alpha$; by a result of I. G. Macdonald, they form a family of orthogonal polynomials. Jack polynomials can be defined as eigenfunctions of certain Laplace/Beltrami type operator coming in the theory of Calogero integrable systems and in random matrix theory. If $\alpha = 1$ the Jack polynomials become Schur polynomials.

• Henry Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sec. A: Math. Phys. Sci. 69, 1-18, 1969-70, MR289462; reprinted in: Contemp. Math. 417, Jack, Hall-Littlewood and Macdonald polynomials, 57–74, Amer. Math. Soc. 2006
• H. Jack, A class of polynomials in search of a definition, or the symmetric group parametrized, in: Jack, Hall-Littlewood and Macdonald polynomials, 75–106, AMS 2006
• Wolframwolrd: Jack polynomial
• I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Math. Monographs, 2nd enlarged ed. 1995
• Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115, MR90g:05020 doi