Zonal polynomials by Akimichi Takemura

By Akimichi Takemura

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Now = Mq* t(A), where 1 < j \ , . . ,jt-t ^ t are indices not included in ( ι ' i , . . 4 the leading term in / is appMp* . This verifies (ii). 1 | W| β can be absorbed into the Wishart density and we have ίvy{|W| β l/p(A- 1 lΓ- 1 )}=c I / I/p(A" 1 ) for some cυ. Therefore £Wf(AW) = \A\*cuyp(A-1) = f(A). 2 we have (A) for some c. 3 55 More on integral identities The second distribution is a "multivariate Fw distribution. There are many ways to generalize the univariate F distribution to the multivariate case.

For future references we record (20) as a corollary. Corollary 1. IfAiskXk and t(p) > k then yp{A) = 0. 1. Theorem 2. Let integers n, k and a partition p £ Pn be given with t(p) < k. Suppose that f satisfies the following conditions: (0 / e vΛfk. e. for some real numbers apq with app 7^ 0. (iii) For some constants cVy for all k X k symmetric A and for all sufficiently large degrees of freedom v. Then f = app \yp and cu = λ^p. Proof. (/*) 49 for all k X k symmetric matrix A. Therefore by Theorem 1 we have apq(cμ — λ ^ ) = 0 for all q.

Let J/Q be fixed such that \vop, p £ Pn are all different. Let B be the matrix in (1) with T replaced by Tμo. Note that the uniqueness part of Lemma 4 is already established now. Let A = diag(λι/oί>,p £ Pn). Then for any μ Λ(BTμ)= (AB)Tμ = (BT^Tμ = B[TμoTμ) = B(TμTμo) = (BTμ)Tμo, or A31 = B\Tμ0 where B\ = BTμ. Now by the uniqueness part of Lemma 1 we have S i = DB for some diagonal D or BTμ = DB. Considering the diagonal elements we see that D = Aμ = diag(λμp,p 6 Pn) Therefore BTμ = AμB for all μ.

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