Pieri-type formulas for the non-symmetric Jack polynomials |
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Authors: | P J Forrester D S McAnally |
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Institution: | (1) Department of Mathematics and Statistics, University of Melbourne, 3052 Parkville, Victoria, Australia;(2) Department of Mathematics and Statistics, University of Melbourne, 3052 Parkville, Victoria, Australia |
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Abstract: | In the theory of symmetric Jack polynomials the coefficients in the
expansion of the $p$th elementary symmetric function $e_p(z)$ 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 $E_\eta(z)$ are explored. Necessary conditions for non-zero
coefficients
in the expansion of $e_p(z) E_\eta(z)$ as a series in non-symmetric
Jack polynomials are given.
A known expansion formula for $z_i E_\eta(z)$
is rederived by an induction procedure, and this expansion is used to
deduce the corresponding result for the expansion of
$\prod_{j=1, \, j\ne i}^N z_j \, E_\eta(z)$, and consequently
the expansion of $e_{N-1}(z) E_\eta(z)$. In the general $p$ case
the coefficients for special terms in the expansion are presented. |
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Keywords: | 33D80 |
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