Differential Operators, Shifted Parts, and Hook Lengths
نویسنده
چکیده
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 = 1/α; see Okounkov-Olshanski [7] and references therein) have been studied. The former appear as eigenfunctions of the Sekiguchi differential operators D(u; θ) = aδ(y) det ( y i ( yi ∂ ∂yi + (n− j)θ + u )}n
منابع مشابه
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
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...
متن کاملAuthor manuscript, published in "The Ramanujan Journal (2009) 9 pages" Hook lengths and shifted parts of partitions
— 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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تاریخ انتشار 2008