ABSTRACT
Chapter 10 Invariant Theory and Algebraic Topology In chapter 8 we encountered the Dickson algebra: an algebra of universal in variants in the sense that D*(n) is contained in the ring of invariants of any representation g : G £-» GL(n, F ) when F is a Galois field. The Dickson algebra is a feature of invariant theory over finite fields not present over other fields. Another such feature is the Steenrod algebra. It is derived from another aspect of algebra over fields of nonzero characteristic, the Frobenius homo morphism. In section1 10.3 we develop the Steenrod algebra in a manner completely independent of any knowledge of algebraic topology, and indicate some of its elementary applications to invariant theory in section 10.4 and chapter 11. Some less elementary applications appear in sections 10.5 and 10.6. The groups generated by psuedoreflections have played an important role in the interaction between algebraic topology and invariant theory. In section 10.2 we show how the classification of Shephard and Todd of the complex pseudoreflection groups can be extended to cover the classification of the pseudoreflection representations of these groups at the good primes.
