Geometric Invariant Theory - Background

Background

Invariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classical invariant theory addresses the situation when X = V is a vector space and G is either a finite group, or one of the classical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomial functions R(V) on V by the formula

The polynomial invariants of the G-action on V are those polynomial functions f on V which are fixed under the 'change of variables' due to the action of the group, so that g·f = f for all g in G. They form a commutative algebra A = R(V)G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V //G. In the language of modern algebraic geometry,

Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra A is finitely generated. This is necessary if one wanted the quotient to be an affine algebraic variety. Whether a similar fact holds for arbitrary groups G was the subject of Hilbert's fourteenth problem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course of development of representation theory in the first half of the twentieth century, a large class of groups for which the answer is positive was identified; these are called reductive groups and include all finite groups and all classical groups.

The finite generation of the algebra A is but the first step towards the complete description of A, and the progress in resolving this more delicate question was rather modest. The invariants had classically been described only in a restricted range of situations, and the complexity of this description beyond the first few cases held out little hope for full understanding of the algebras of invariants in general. Furthermore, it may happen that all polynomial invariants f take the same value on a given pair of points u and v in V, yet these points are in different orbits of the G-action. A simple example is provided by the multiplicative group C* of non-zero complex numbers that acts on an n-dimensional complex vector space Cn by scalar multiplication. In this case, every polynomial invariant is a constant, but there are many different orbits of the action. The zero vector forms an orbit by itself, and the non-zero multiples of any non-zero vector form an orbit, so that non-zero orbits are paramatrized by the points of the complex projective space CPn−1. If this happens, one says that "invariants do not separate the orbits", and the algebra A reflects the topological quotient space X /G rather imperfectly. Indeed, the latter space is frequently non-separated. In 1893 Hilbert formulated and proved a criterion for determining those orbits which are not separated from the zero orbit by invariant polynomials. Rather remarkably, unlike his earlier work in invariant theory, which led to the rapid development of abstract algebra, this result of Hilbert remained little known and little used for the next 70 years. Much of the development of invariant theory in the first half of the twentieth century concerned explicit computations with invariants, and at any rate, followed the logic of algebra rather than geometry.