Invariant Theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties 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.

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 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 presently 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.