JournalscmhVol. 79, No. 1pp. 1–24

Pieri-type formulas for the non-symmetric Jack polynomials

  • P. J. Forrester

    University of Melbourne, Parkville, Australia
  • D. S. McAnally

    University of Melbourne, Parkville, Australia
Pieri-type formulas for the non-symmetric Jack polynomials cover

Abstract

In the theory of symmetric Jack polynomials the coefficients in the expansion of the ppth elementary symmetric function ep(z)e_p(z) times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials Eη(z)E_\eta(z) are explored. Necessary conditions for non-zero coefficients in the expansion of ep(z)Eη(z)e_p(z) E_\eta(z) as a series in non-symmetric Jack polynomials are given. A known expansion formula for ziEη(z)z_i E_\eta(z) is rederived by an induction procedure, and this expansion is used to deduce the corresponding result for the expansion of j=1,jiNzjEη(z)\prod_{j=1, \, j\ne i}^N z_j \, E_\eta(z), and consequently the expansion of eN1(z)Eη(z)e_{N-1}(z) E_\eta(z). In the general pp case the coefficients for special terms in the expansion are presented.

Cite this article

P. J. Forrester, D. S. McAnally, Pieri-type formulas for the non-symmetric Jack polynomials. Comment. Math. Helv. 79 (2004), no. 1, pp. 1–24

DOI 10.1007/S00014-003-0789-2