Polynomial Invariants of Finite Groups a Survey of Recent Developments

The polynomial invariants of finite groups have been studied for more than a century now and continue to find new applications and generate interesting problems. In this article we will survey some of the recent developments coming primarily from algebraic topology and the rediscovery of old open problems. It has been almost two decades since the Bulletin of the AMS published the marvelous survey article [111] of R. P. Stanley. Since then the invariant theory of finite groups has taken on a central role in many problems of algebraic topology, such as e.g. [22], [2], [101], [65], [105], [84], [106] chapter 11, and the references there. It has received new impetus as a subject of study in its own right, [72]–[81], [3], [43], and several textbooks with varying viewpoints [9], [114], and [106], as well as a reprint of venerable old lecture notes [48], have recently appeared. In this survey article I will try to discuss some of these developments as seen through the eyes of one who came to the subject from algebraic topology. That means that finite groups and finite fields will play a central role, and the modular case, i.e. where the characteristic of the field divides the order of the group, will play (in contrast to [111]) an important part. Generally speaking, invariant theory is concerned with the action of groups on rings and the invariants of the action, e.g. the fixed subring and related objects. Here we will restrict ourselves to the actions of finite linear groups on polynomial rings. To be more specific, fix a field F to serve as ground field. For a finite-dimensional vector space V over F we denote by F[V ] the algebra of homogeneous polynomial functions on V , which we define to be the symmetric algebra on V ∗, the dual of V . In other words, the homogeneous component of F[V ] of degree m, denoted by F[V ]m (see [31] §1.5 or [106] chapter 4 for a discussion of gradings) is S(V ∗), the m-th symmetric power of V ∗. With this grading linear forms have degree one. If you are a topologist, you may occasionally wish to double the degrees. If z1, . . . , zn ∈ V ∗ is a basis, we also denote F[V ] by F[z1, . . . , zn]. The elements of F[z1, . . . , zn] are just homogeneous polynomials in the linear forms z1, . . . , zn with coefficients in F. It is often convenient to think of the elements of the polynomial algebra F[V ] as being functions. There is a problem if F is a Galois field of characteristic p, since it Received by the editors January 3, 1997. 1991 Mathematics Subject Classification. Primary 13A50; Secondary 55S10.