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Geometry Topology

By Brannan D.A., Esplen M.F., Gray J.J.

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On the other hand we do not survey various known identities involving zonal polynomials. For this purpose the reader is referred to an excellent survey paper by Subrahmaniam (1976). Actually in the discussion of the orthogonally invariant distributions we saw that zonal polynomials satisfy an infinite number of identities. It is a rather frustrating fact that although many identities for zonal polynomials are already known, explicit forms of zonal polynomials are not known. 1 which played an essential role for the subsequent development in Chapter 3 is not complete as it is.

For a symmetric positive semi-definite matrix A let A? , A? = ΓD^Γ where Γ\s orthogo1 nal and D is diagonal in A = ΓDΓ . Now let W,Vbe independently distributed according to 1^(1^,1/), 1^(1^,μ) respectively. Consider (23) £W>V{U(AWA*W)}, where U =(2i( n )>^( n -l,l)> •• >^(i»)/ first we obtain (24) Taking expectation with respect to W = £V{TμU(AV)} = TvTμU{A). We used the cyclic permutation of the matrices since nonzero characteristic roots are invariant. Similarly taking expectation with respect to V first we obtain (25) £Wy{U{AWAhw)} = TμTμU(A).

Lemma 2. Let Abe at X t positive definite matrix. Then μr (4) where 8 > h{p), t > l{p). Without loss of generality let A = d i a g ( α i , . . ,c*f). let / ( A ) = | A | β Proof. yp(A~ι). 2. ,l/at). Note that q -< p implies h(q) < h(p). , l/αj) is h(q). (αi yp(A~i) Hence the degree of l / α t in yp(A~ι) is h(p). Now |A|β^ at)8 and β > h(p). ,at). Clearly it is symmetric and homoge- neous of degree st — \p\. 2. Now = Mq* t(A), where 1 < j \ , . . ,jt-t ^ t are indices not included in ( ι ' i , .

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