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Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. For example, if we consider the action of the special linear group ''SLn'' on the space of ''n'' by ''n'' matrices by left multiplication, then the determinant is an invariant of this action because the determinant of ''A X'' equals the determinant of ''X'', when ''A'' is in ''SLn''. ==Introduction== Let ''G'' be a group, and ''V'' a finite-dimensional vector space over a field ''k'' (which in classical invariant theory was usually assumed to be the complex numbers). A representation of ''G'' in ''V'' is a group homomorphism , which induces a group action of ''G'' on ''V''. If ''k()'' is the space of polynomial functions on ''V, then the group action of ''G'' on ''V'' produces an action on ''k()'' by the following formula: : With this action it is natural to consider the subspace of all polynomial functions which are invariant under this group action, in other words the set of polynomials such that ''g.f = f'' for all ''g'' in ''G''. This space of invariant polynomials is denoted ''k()G''. First problem of invariant theory: Is ''k()G'' a finitely generated algebra over ''k''? For example, if ''G=SLn'' and ''V=Mn'' to space of square matrices, and the action of ''G'' on ''V'' is given by left multiplication, then ''k()G'' is isomorphic to a polynomial algebra in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of power of the determinant polynomial. So in this case, ''k()G'' is finitely generated over ''k''. If the answer is yes, then the next question is to find a minimal basis, and ask whether the module of polynomial relations between the basis elements (known as the syzygies) is finitely generated over ''k()''. Invariant theory of finite groups has intimate connections with Galois theory. One of the first major results was the main theorem on the symmetric functions that described the invariants of the symmetric group ''S''''n'' acting on the polynomial ring R(…, ''x''''n'' ) by permutations of the variables. More generally, the Chevalley–Shephard–Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring. Modern research in invariant theory of finite groups emphasizes "effective" results, such as explicit bounds on the degrees of the generators. The case of positive characteristic, ideologically close to modular representation theory, is an area of active study, with links to algebraic topology. Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic forms and determinants. Another subject with strong mutual influence was projective geometry, where invariant theory was expected to play a major role in organizing the material. One of the highlights of this relationship is the symbolic method. Representation theory of semisimple Lie groups has its roots in invariant theory. David Hilbert's work on the question of the finite generation of the algebra of invariants (1890) resulted in the creation of a new mathematical discipline, abstract algebra. A later paper of Hilbert (1893) dealt with the same questions in more constructive and geometric ways, but remained virtually unknown until David Mumford brought these ideas back to life in the 1960s, in a considerably more general and modern form, in his geometric invariant theory. In large measure due to the influence of Mumford, the subject of invariant theory is seen to encompass the theory of actions of linear algebraic groups on affine and projective varieties. A distinct strand of invariant theory, going back to the classical constructive and combinatorial methods of the nineteenth century, has been developed by Gian-Carlo Rota and his school. A prominent example of this circle of ideas is given by the theory of standard monomials. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Invariant theory」の詳細全文を読む スポンサード リンク
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