Invariant theory

Hilbert's theorems

proved that if V is a finite-dimensional representation of the complex algebraic group G = SLn(C) then the ring of invariants of G acting on the ring of polynomials R = S(V) is finitely generated. His proof used the Reynolds operator ρ from R to R**G with the properties

• ρ(1) = 1
• ρ(a + b) = ρ(a) + ρ(b)
• ρ(ab) = a ρ(b) whenever a is an invariant. Hilbert constructed the Reynolds operator explicitly using Cayley's omega process Ω, though now it is more common to construct ρ indirectly as follows: for compact groups G, the Reynolds operator is given by taking the average over G, and non-compact reductive groups can be reduced to the case of compact groups using Weyl's unitarian trick.

Given the Reynolds operator, Hilbert's theorem is proved as follows. The ring R is a polynomial ring so is graded by degrees, and the ideal I is defined to be the ideal generated by the homogeneous invariants of positive degrees. By Hilbert's basis theorem the ideal I is finitely generated (as an ideal). Hence, I is finitely generated by finitely many invariants of G (because if we are given any – possibly infinite – subset S that generates a finitely generated ideal I, then I is already generated by some finite subset of S). Let i1,...,i**n be a finite set of invariants of G generating I (as an ideal). The key idea is to show that these generate the ring R**G of invariants. Suppose that x is some homogeneous invariant of degree d 0. Then :x = a1i1 + ... + anin for some a**j in the ring R because x is in the ideal I. We can assume that a**j is homogeneous of degree d − deg i**j for every j (otherwise, we replace a**j by its homogeneous component of degree d − deg i**j; if we do this for every j, the equation x = a1i1 + ... + anin will remain valid). Now, applying the Reynolds operator to x = a1i1 + ... + anin gives :x = ρ(a1)i1 + ... + ρ(a**n)i**n

We are now going to show that x lies in the R-algebra generated by i1,...,i**n.

First, let us do this in the case when the elements ρ(a**k) all have degree less than d. In this case, they are all in the R-algebra generated by i1,...,i**n (by our induction assumption). Therefore, x is also in this R-algebra (since x = ρ(a1)i1 + ... + ρ(an)in).

In the general case, we cannot be sure that the elements ρ(a**k) all have degree less than d. But we can replace each ρ(a**k) by its homogeneous component of degree d − deg i**j. As a result, these modified ρ(a**k) are still G-invariants (because every homogeneous component of a G-invariant is a G-invariant) and have degree less than d (since deg i**k 0). The equation x = ρ(a1)i1 + ... + ρ(an)in still holds for our modified ρ(a**k), so we can again conclude that x lies in the R-algebra generated by i1,...,i**n.

Hence, by induction on the degree, all elements of R**G are in the R-algebra generated by i1,...,i**n.

Hilbert-Nagata theorem

Hilbert's theorem was later generalized as follows. Let a group G act on a finitely-generated algebra A over an arbitrary field k . Suppose that:

• The action is locally finite, i.e. the orbit of each f\in A is contained in a finite dimensional k -subspace of A .
• If U is a finite-dimensional representation of G over k , and H\subset U is a G -invariant hyperplane, then there exists a G -invariant decomposition U = H\oplus L. Then the ring of invariants A^G is finitely generated over k .

This theorem implies that if G is a finite group of size prime to the characteristic of k then the invariants ring A^G is finitely generated over k . Another corollary is that if G is a reductive group that acts on A algebraically, and k has characteristic zero, then A^G is finitely generated over k .

Geometric invariant theory

The modern formulation of geometric invariant theory is due to David Mumford, and emphasizes the construction of a quotient by the group action that should capture invariant information through its coordinate ring. It is a subtle theory, in that success is obtained by excluding some 'bad' orbits and identifying others with 'good' orbits. In a separate development the symbolic method of invariant theory, an apparently heuristic combinatorial notation, has been rehabilitated.

One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.

Notes

References

• Reprinted as
• A recent resource for learning about modular invariants of finite groups.
• An undergraduate level introduction to the classical theory of invariants of binary forms, including the Omega process starting at page 87.
• An older but still useful survey.
• A beautiful introduction to the theory of invariants of finite groups and techniques for computing them using Gröbner bases.

References

1. (2001). "Essays in the History of Lie groups and algebraic groups". American mathematical society and London mathematical society.
2. Wolfson, Paul R.. (2008). "George Boole and the origins of invariant theory". Elsevier BV.