The invariant theory of finite groups

Invariant theory played a prominent role in nineteenth century mathematics and has important applications to the modern disciplines of algebraic geometry and algebraic topology. The recent resurgence of interest in the area is due, in part, to the importance of computational methods and the enormous advances in computing power.

For any representation of a group, G, on a finite dimensional vector space, V, there is an induced action on the symmetric algebra, ${\bf k}[V]={\bf k}[x_1,x_2,\ldots x_n]$. The ring of invariants of the representation, ${\bf k}[V]^G$, is the subring consisting of those polynomials fixed by this action.

If G is finite, the ring of invariants is finitely generated. Constructing generators for the ring of invariants is an important computational problem. For example, suppose that p is a prime number and ${\bf k}$ is a field of characteristic p. Define an algebra automorphism on ${\bf k} [x_1,x_2,\ldots x_n]$ by $\sigma(x_1)=x_1$ and $\sigma(x_i)=x_i+x_{i-1}$ for i>1. If $n\leq p$, then $\sigma$ generates a group of order p and we are looking at an indecomposable representation of ${\bf Z}/p$. For $n\leq 3$, it easy to describe the generators of the ring of invariants. For n>3, the problem is much harder. The first description of a generating set for arbitrary p for n=4 and n=5 appeared in R.J. Shank, S.A.G.B.I. bases for rings of formal modular seminvariants, Comment. Math. Helv. 73 (1998) 548-565. The problem remains open for $n\geq 6$. The solution to the problem at n=4 and n=5 made use of the theory of SAGBI bases, the analog of a Gröbner basis for a subalgebra, and had its origins in a number of computer calculations for small primes.

There are a number of important open structural questions in invariant theory. For the most part these questions address the relationship between properties of the representation and properties of the ring of invariants. If ${\bf k}$ is a field of characteristic zero then ${\bf k}[V]^G$ is always Cohen-Macaulay and is polynomial if and only if G is generated by pseudo-reflections. However, if the characteristic of ${\bf k}$ divides the order of G, i.e., the representation is modular, then both of these results fail leaving us with two important open problems:
(i) Which modular reflection groups have polynomial invariants?
(ii) Which modular representations have Cohen-Macaulay invariants?