ar X iv : 0 80 7 . 24 73 v 1 [ m at h . C O ] 1 5 Ju l 2 00 8 DIFFERENTIAL OPERATORS , SHIFTED PARTS , AND HOOK LENGTHS
نویسنده
چکیده
Pλ(y; θ), (1) where δ := (n− 1, n− 2, . . . , 1, 0) and λ are partitions, aδ = ∏ 1≤i<j≤n(yi − yj) is the Vandermonde determinant and u is a free parameter. Under a general result, S. Sahi proves [8, Theorem 5.2] the existence of a unique polynomial P ∗ μ(y; θ), now known as shifted Jack polynomials, satisfying a certain vanishing condition. In the special case θ = 1, Okounkov and Olshanski [6,7] relate P ∗ μ(y; 1) to the Schur functions sλ and call them shifted Schur polynomials. One motivation for this paper is the a result due to Andrews, Goulden and Jackson [1, Thm 2.1] stating that Theorem 1.1. For n,m ∈ P and summing over all partitions l(λ) ≤ n, we have
منابع مشابه
Differential Operators, Shifted Parts, and Hook Lengths
We discuss Sekiguchi-type differential operators, their eigenvalues, and a generalization of Andrews-Goulden-Jackson formula. These will be applied to extract explicit formulae involving shifted partitions and hook lengths. 1. Differential operators. The standard Jack symmetric polynomials Pλ(y1, . . . , yn;α) (see Macdoland, Stanley [5, 10]) as well as their shifted counter-parts (replace θ = ...
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— Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric functions can be used instead. The purpose of this note is to prove it.
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