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By Flaass D.G.

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Consequently, B = mB and taking n to be any maximal ideal of B such that mB ⊆ n we have m = n ∩ A. As an application of this result we have the following theorem. 15. Let G be a finite group acting on V = Cn and let S = S(V ∗ ). Let {F1 , . . , Fn } be a set of algebraically independent elements of S G such that S G is integral over C[F1 , . . , Fn ]. Then the map ϕ : V → Cn given by v → (F1 (v), . . , Fn (v)) is surjective. Proof. Since F1 , . . , Fn are algebraically independent, the homomorphism ϕ∗ : C[X1 , .

If ω is a cube root of unity, show that the matrices (iii) ω(i − 1) 1 1 −i i 2 and ω(i − 1) 1 i 2 −1 i are reflections and that the group they generate is conjugate in GL2 (C) to the group G of the previous exercise. 6. Let G be a reflection group of rank two generated by reflections r and s of orders p and q, respectively. Show that it is possible to choose roots a and b for r and s, respectively, so that their Cartan matrix has the form 1−α θ , where α is a primitive pth root of unity, β is a primitive 1 1−β q th root of unity and θ is an algebraic integer such that α, β, θ ∈ Z(G).

In 1914 Noether [173] showed that if G is finite, then J is finitely generated even when C is replaced by an arbitrary field (see Flatto [98]). It is also the case that J is finitely generated when G is a reductive algebraic group; for more details see the introduction and Chapter 1 of the book by Benson [12] and the references given there. A fundamental concept in this circle of ideas is that of a Noetherian ring, which is a commutative ring R all of whose ideals are finitely generated (as R-modules).

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