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REDUCTIVE ALGEBRAIC GROUPS 30 a finite set of polynomials F1 : : : Fn in AG . We may assume that each Fi is homogeneous of degree mi > 0. 1) for some homogeneous polynomials P i of degree m mi . Now consider the A defined by the formula operator av : A 2 ; ! 1) we get F = av(P1)F1 + + av(Pn)Fn: By induction we can assume that each invariant homogeneous polynomial of degree < m can be expressed as a polynomial in Fi ’s. Since av(Pi ) is homogeneous of degree < m, we are done. 2. (Gordan–Hilbert) The algebra of invariants Pol(Pol d (V ))SL(V ) is finitely generated over k .

It is not difficult to see that the Pl¨ucker equations define set theoretically the Grassmann varieties in their Pl¨ucker embedding (see, for example, [40]). 5 describes the homogeneous ideal of the Grassmannian. As far as I know the only textbook in algebraic geometry which contains a proof of this fact is [48]. We refer to [33] for another proof based on the representation theory. 2 Let be the omega-operator in the polynomial ring k Mat r r ]. (1 ; Dr );r;1 , Dri (iii) the function f = 1 i=0 1 2!

We call such an action a rational action or, better, a regular action. In particular, a linear representation : G GL(V ) = GLn (k ) will be assumed to be given by regular functions on the affine algebraic variety G. Such linear representations are called rational representations. Let an affine algebraic group G act on an affine variety X = Spm(A). This action can be described in terms of the coaction homomorphism ! : A ! O(G) A where O (G) is the coordinate ring of G. It satisfies a bunch of axioms which are “dual” to the usual axioms of an action; we leave their statements to the reader.

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