Jack polynomial

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

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