Chern - A Great Geometer of the Twentieth Century by Shing-Tung Yau (Chief Editor)

By Shing-Tung Yau (Chief Editor)

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Remark 1. In several places we already have used the expressions "leading term" or "leading coefficient". Here and in the sequel "leading" refers to the highest partition when a homogeneous symmetric polynomial is expressed in terms of bases { Up }, { Mp }, or { yp }. /p(/jfe). 4 we know that \bp = λjfy/ iy p {lk) is a constant independent of k. Therefore our goal is to obtain \bp. Now (2) 1 6 M l / p ( I j b ) = λ ibp = Z p (/ i t ). Therefore \bp is the leading coefficient of Zp. 11). We use the following recursive relation.

Then the construction could have been carried out in exactly the same way provided that cp, p £ Pn in Theorem 2 are all distinct for V. Furthermore if we examine the proof of Theorem 1 closely we find that we could take A = ΣzVΣϊ in (9) and (10). Once Theorem 1 is proved the identity involving the Wishart distribution can be derived as a special case. Although the Wishart distribution seems to be a natural candidate to take for our construction, we could have used any orthogonally invariant distribution from a purely logical point of view.

Let yp = ΣqePn (26) implies α Λ T h e n aTv = \μpa ; α where o = (θ(n)> > (i*)) Now by the uniqueness part of Lemma 4 a! u w coincides with the p-tΛ row of B up to a multiplicative constant. Therefore yp is a zonal polynomial. 2 25 Integral identities Corollary 3. (28) εwyp(A\v) = κpyp(AΣ), where A is symmetric and W is distributed according to ΊV(Σ, v). Proof. 2 and will be used to show that a particular symmetric polynomial is a zonal polynomial. See Sec. 3, Sec. 4, and Sec. 7. Remark β. In (21) zonal polynomials are defined as linear combinations of i/g's.

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