Editing Invariant theory (section)

==Introduction==
Let <math>G</math> be a [[group (mathematics)|group]], and <math>V</math> a finite-dimensional [[vector space]] over a [[field (mathematics)|field]] <math>k</math> (which in classical invariant theory was usually assumed to be the [[complex number]]s). A [[group representation|representation]] of <math>G</math> in <math>V</math> is a [[group homomorphism]] <math>\pi:G \to GL(V)</math>, which induces a [[Group action (mathematics)|group action]] of <math>G</math> on <math>V</math>. If <math>k[V]</math> is the [[Ring of polynomial functions|space of polynomial functions on <math>V</math>]], then the group action of <math>G</math> on <math>V</math> produces an action on <math>k[V]</math> by the following formula: 
:<math>(g \cdot f)(x) := f(g^{-1} (x)) \qquad \forall x \in V, g \in G, f\in k[V]. </math> 
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 <math>g\cdot f = f</math> for all <math>g\in G</math>. This space of '''invariant polynomials''' is denoted <math>k[V]^G</math>.

'''First problem of invariant theory''':<ref>{{cite book|last1=Borel|first1=Armand|author-link = Armand Borel|title=Essays in the History of Lie groups and algebraic groups|date=2001|publisher=American mathematical society and London mathematical society|volume=History of Mathematics, Vol. 21|isbn=978-0821802885}}</ref> Is <math>k[V]^G</math> a [[finitely generated algebra]] over <math>k</math>?

For example, if <math>G=SL_n</math> and <math>V=M_n</math> the space of square matrices, and the action of <math>G</math> on <math>V</math> is given by left multiplication, then <math>k[V]^G</math> is isomorphic to a [[Polynomial ring|polynomial algebra]] in one variable, generated by the determinant. In other words, in this case, every invariant polynomial is a linear combination of powers of the determinant polynomial. So in this case, <math>k[V]^G</math> is finitely generated over <math>k</math>.

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 [[Syzygy (mathematics)|syzygies]]) is finitely generated over <math>k[V]</math>.

Invariant theory of [[finite group]]s has intimate connections with [[Galois theory]]. One of the first major results was the main theorem on the [[symmetric function]]s that described the invariants of the [[symmetric group]] <math>S_n</math> acting on the [[polynomial ring]] <math>R[x_1, \ldots, x_n</math>] by [[permutation]]s 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 (algebra)|characteristic]], ideologically close to [[modular representation theory]], is an area of active study, with links to [[algebraic topology]].

Invariant theory of [[infinite group]]s is inextricably linked with the development of [[linear algebra]], especially, the theories of [[quadratic form]]s and [[determinant]]s. 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 group]]s 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 group]]s on [[affine variety|affine]] and [[projective variety|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 monomial]]s.