We prove a characteristic free version of Weyl's theorem on polarization. Our result is an exact analogue of Weyl's theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

Abstract. We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three dimensional representation of an elementary abelian p-group is a complete intersection.

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PrefaceAt the time we write this book there are several excellent references available which discuss various aspects of modular invariant theory from various points of view: [103]. In this book, we concentrate our attention on the modular invariant theory of finite groups. We have included various techniques for determining the structure of and generators for modular rings of invariants, while attempting to avoid too much overlap with the existing literature. An important goal has been to illustrate many topics with detailed examples. We have contrasted the differences between the modular and non-modular cases, and provided instances of our guiding philosophies and analogies. We have included a quick survey of the elements of algebraic geometry and commutative algebra as they apply to invariant theory. Readers who are familiar with these topics may safely skip this chapter.We wish to thank our principal collaborators over the years with whom we have had so much pleasure exploring this fascinating subject: Ian Hughes, Gregor Kemper, R. James Shank, John Harris as well as our students and friends,