A biplot method for multivariate normal populations with by Calvo M., Villarroya A., Oller J.M.

By Calvo M., Villarroya A., Oller J.M.

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The {f (s) vλ | s ∈ S(λ)} form a basis for the module VZ (λ) and for all s < s we have VZ (λ)s is a direct summand of VZ (λ)s as a Z-module. ) (iii) With the notation as above: let k be a field and denote by Vk (λ) = VZ (λ) ⊗Z k, Uk (g) = UZ (g) ⊗Z k, Uk (n− ) = UZ (n− ) ⊗Z k etc. the objects obtained by base change. The {f (s) vλ | s ∈ S(λ)} form a basis for the module Vk (λ). Proof We know that the elements f (s) vλ , s ∈ S(λ), span VZa (λ), see Theorem 1. By [6], the number S(λ) is equal to dim V (λ), which implies the linear independence.

Feigin + Denote by dv,i the dimension of the fiber of the locally trivial bundle Cv,i → Tθ M(r, n)v,iα,β . + doesn’t depend on k ∈ Iv and is equal to Lemma 3 The dimension dv,k + = dv,k 1 dim M(v, w). 2 Proof The set of fixed points of the (C∗ )2 × (C∗ )r -action on M(r, n) is finite and is parametrized by the set of r-tuples D = (D1 , D2 , . . g. [23]). ∗ 2 ∗ r Let p ∈ M(r, n)(C ) ×(C ) be the fixed point corresponding to an r-tuple D. ∗ 2 ∗ r Let R((C ) × (C ) ) = Z[t1±1 , t2±1 , e1±1 , e2±1 , .

We know by (ii) that f (s) = s mon t ct f (t) in SZ (n−,a )/I . If any of the t with nonzero coefficient ct is not an element in S(λ), then we can again apply a straightening law and replace f (t) by a linear combination of smaller monomials. Since there are only a finite number of monomials of the same total degree, by repeating the procedure if necessary, after a finite number of steps we obtain an expression of f (s) in SZ (n−,a )/I as a linear combination of elements f (t) , t ∈ S(λ). It follows that {f (t) | t ∈ S(λ)} is a spanning set for SZ (n−,a )/I , and hence, by the surjection above, we get a spanning set {f (t) vλ | t ∈ S(λ)} for VZa (λ).

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