Modular Invariant Theory

Abstract

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,

The invariant rings of the Sylow groups of GU(3,q2), GU(4,q2), Sp(4,q) and O+(4,q) in the natural characteristic

“…In this case the kernel of φ is trivial since φ(ti) = x The previous corollary can also be seen as a particular case of Proposition 5.2.2 in (Campbell and Wehlau, 2011).…”

Computational Invariant Theory

“…In contrast, our book touches only occasionally on classical invariant theory. In its emphasis on modular invariant theory, our book has something in common with Benson [8], Smith [9], Neusel and Smith [10], and Campbell and Wehlau [14]. It is probably fair to say that most of the material covered in Chaps.…”

“…Moreover, it is specifically the computational aspect of invariant theory that lends itself to applications particularly well. [13], and Campbell and Wehlau [14]. We hope that our book will serve as a useful addition to these.…”

“…An important role in boosting interest in computational invariant theory was also played by Sturmfels's book Algorithms in Invariant Theory [7]. Various other monographs (such as Benson [8], Smith [9], Neusel and Smith [10], Dolgachev [11], Lorenz [12], Neusel [13], and Campbell and Wehlau [14]) and numerous research articles on invariant theory have appeared quite recently, all evidence of a field in ferment.…”

The Invariant Theory of Finite Groups

“…Our goal is to understand F[V ] G . We would like to be able to effectively compute generators for For more on the invariant theory of finite groups, see Campbell and Wehlau [14], Derksen and Kemper [15], Neusel and Smith [32], and Benson [5].…”