If are coordinates in in some basis , then each element determines a linear change of the variables ; by carrying out this change of variables in an arbitrary polynomial one obtains a new polynomial, hence induces a transformation automorphism of the ring of all polynomials in the variables over the field. A polynomial that does not change under all such transformations that is, when runs through the whole of is called an invariant of the representation of the group cf. All the invariants form a -algebra and the aim of the theory of invariants is to describe this algebra.
Thus, the invariants of forms are the invariants of the general linear group with respect to its representation in the space of symmetric tensors of fixed rank of the underlying or dual space the coefficients of the original form are the components of this tensor ; the consideration of covariants reduces to the study of the algebra of invariants for a representation in the space of tensors of mixed valency. In this form the problem of the description of invariants is a special case of the following general problem of the theory of linear representations: To decompose a space of tensors of given valency into irreducible invariant subspaces with respect to a given group of linear transformations of the underlying linear space the search for invariants can be reduced to singling out the one-dimensional invariant subspaces.
Already at the first stages of development of the theory of invariants the following fact was discovered, which allowed one to consider the system of invariants as a whole: In all the cases examined it was possible to select a finite number of basic invariants , that is, invariants such that every other invariant of the given representation can be expressed as a polynomial in them:.
In other words, the algebra of invariants proved to be finitely generated. It also became clear that these basic invariants are, in general, not independent that is, the algebra of invariants is not a free algebra : there can exist non-trivial polynomials , called relations or syzygies, that after the substitutions , , vanish identically.
In the set of relations itself it is again possible to find a finite number of basic relations, all the remaining being algebraic consequences of them the relations form an ideal in the ring of polynomials in the variables , and the basic relations are generators of it. In turn, the basic relations themselves are in general not independent; thus, secondary syzygies can be determined, etc.
The chain of syzygies constructed in this way always turns out to be finite. For example, if is the symmetric group of all permutations of the coordinates , then the algebra of invariants is the algebra of all symmetric polynomials in ; the elementary symmetric polynomials are basic invariants, which are algebraically independent in this case there are no syzygies.
Algebraic Geometry Iv: Linear Algebraic Groups Invariant Theory
These facts led to the statement of two fundamental problems in the classical theory of invariants:. The first fundamental theorem of the theory of invariants for invariants of a form of arbitrary degree in an arbitrary finite number of variables was proved by D. Hilbert  see also Hilbert theorem on invariants. He also proved that the second fundamental theorem of the theory of invariants is true in all those cases when the first fundamental theorem holds, and also that in this case the chain of syzygies is always finite.
Hilbert obtained the proof of the fundamental theorems of the classical theory of invariants on the basis of general abstract algebraic results proved by him with this specific aim. These results subsequently formed the foundation of modern commutative algebra the Hilbert basis theorem, the Hilbert syzygies theorem, Hilbert's Nullstellensatz, cf. Hilbert theorem 1 , 3 and 5. The original proof of the first fundamental theorem of the theory of invariants was non-constructive and gave no upper estimate for the degree of basic invariants.
In the 's H. Weyl, by developing an idea of Hilbert and A. Hurwitz  , proved the first fundamental theorem of the theory of invariants for finite-dimensional representations of arbitrary compact Lie groups and for finite-dimensional representations of arbitrary complex semi-simple Lie groups . The book  , which summarizes the developments of the classical theory of invariants, contains a description of the basic invariants and syzygies for the representations of the classical groups as well as for certain other groups. One of the important applications of the methods of the theory of invariants was the description of the Betti numbers of classical compact groups.
The proof of the second fundamental theorem of the theory of invariants discloses the general algebraic nature of this theorem it is a corollary of Hilbert's basis theorem. In attempting to determine whether this is also true with regard to the first fundamental theorem of the theory of invariants, Hilbert stated the following problem Hilbert's 14th problem : Let be a field, let be the algebra of polynomials over in the variables and let be an arbitrary subfield of the field of fractions of containing.
Is then the algebra finitely generated over? An affirmative answer to this question would imply the validity of the first fundamental theorem of the theory of invariants for arbitrary groups. A negative solution to Hilbert's 14th problem was obtained in  , in which an example was given of a representation of a commutative unipotent group for which the algebra of invariants does not have a finite number of generators. In the 's a number of results was obtained on invariants of finite groups, in particular groups generated by reflections see Reflection group ; Coxeter group.
It has been proved  ,  that finite linear complex groups generated by unitary reflections can be characterized as the finite linear groups whose algebras of invariants do not contain syzygies. A new stage of development of the theory of invariants was related to the extension of the circle of problems and to geometric applications of this theory. The modern theory of invariants or the geometric theory of invariants became a part of the general theory of algebraic transformation groups; the theory of algebraic groups constructed in 's is fundamental to it, and the language of algebraic geometry is fundamental to its language.
In contrast with the classical theory of invariants, whose basic object was the ring of polynomials in variables over a field together with the group of automorphisms induced by linear changes of variables, the modern theory of invariants considers an arbitrary finitely-generated -algebra and the algebraic group of its -automorphisms.
Instead of a linear space and a representation , an arbitrary affine algebraic variety is considered together with the algebraic group of its algebraic transformations automorphisms , such that is the ring of regular functions on and the action of on is induced by that of on. The elements of that are fixed under are the invariants; the entire set of them forms a -algebra,. Other notions of the classical theory of invariants can also be generalized.
For example, the comitant, which is a regular mapping from one such variety into another commuting with the group action; if is finitely generated, then one says that the first fundamental theorem of the theory of invariants is valid. It has been proved that is finitely generated if is a geometrically-reductive group see Mumford hypothesis. In many important cases, for example, in applications to the moduli problem, is in fact a group of this type.
If is finitely generated, then there exists an affine algebraic variety for which is the algebra of regular functions; the imbedding induces a morphism. If is geometrically reductive, then classifies the closed orbits of in : is surjective and in each of its fibres there is exactly one closed orbit. A necessary condition for the existence of a quotient variety of with respect to , which is that all the orbits be closed, turns out to be sufficient also, and proves to be this quotient variety.
Hence the role of in the solution of the geometric problems of classifying and constructing quotient varieties becomes clear; apart from this, the study of which was the final aim of the classical theory of invariants is only the beginning stage for the solution of these geometric problems, since knowledge of does not, in general, provide complete information on the orbits of in and must therefore be combined with the consideration of non-closed orbits, their closures and stabilizers so-called orbital decompositions.
Furthermore, the study of actions of algebraic groups on affine algebraic varieties is only the "local part" of the general theory of algebraic transformation groups just as the theory of affine varieties is the "local part" of the general theory of algebraic varieties. In the general case one considers an algebraic regular action of on an arbitrary algebraic variety glued together from affine pieces , so that, e. For a successful application of the procedure, must, in general, be replaced by some invariant subset of it itself need not have an invariant affine covering.
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Log in. Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. One considers a form of degree in variables with undetermined coefficients: After the linear change of variables where are real or complex numbers, it is converted to the form so that the above linear transformation of variables determines a transformation of the coefficients of the form: A polynomial in the coefficients of the form is called a relative invariant of the form if the following relation holds under any non-degenerate linear change of variables: where is the determinant of the linear transformation and is a constant the weight.
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