Invariant theory of finite groups (Q2771508)

The book gives a comprehensive overview of the invariant theory of finite groups acting linearly on polynomial algebras. Its main features are located around various finiteness results classically known in the non-modular case and their extension to the modular situation. The book is in a certain sense a continuation and extension of \textit{L. Smith}'s book: `Polynomial invariants of finite groups' (Boston 1995; Zbl 0864.13002); second edition 1997. It covers a lot of information and various instructive examples. NEWLINENEWLINENEWLINEThe book is divided into 10 chapters and one appendix. The first chapter, `Invariants, their relatives, and problems', begins with a survey about the basic definitions and the problems centered around finiteness, computations, and special cases of invariants. Chapter 2, `Algebraic finiteness', deals with the finite generation of the rings of invariants. There is a proof of Noether's finiteness theorem in the non-modular case saying that the ring of invariants \(\mathbb F[V]^G\) of a finite group \(G\) such that the characteristic of \(\mathbb F\) does not divide the order \(|G|\) of \(G\) is generated by \(G\)-invariant forms of degree at most \(|G|.\) The counting of invariants with the Poincaré series is the central theme of chapter 3, `Combinatorial finiteness'. There is a proof of Molien's theorem in the non-modular case with a view towards the modular situation. It covers also Göbel's theorem on permutation invariants. Chapter 4, `Noetherian finiteness', is centered around the Noether normalization theorem. This has to do with tools for the construction of invariants. It culminates in the construction of a Dade bases for a system of parameters. Chapter 5, `Homological finiteness', contains results about syzygies, finite free resolutions and Cohen-Macaulayness of rings of invariants. It covers also Gorenstein rings of invariants and examples of non-Cohen-Macaulay rings of invariants. Chapter 6, `Modular invariant theory', is concerned with an introduction to the fast growing subject of modular invariant theory, i.e. the case where the characteristic of the ground field divides \(|G|,\) the order of the group. It covers the Dickson algebra, the transvection groups, and the transfer variety. In the modular case rings of invariants are often not Cohen-Macaulay. So there is a subsection about the rôle of the Koszul complex in invariant theory in order to estimate, respectively to compute, the depth of rings of invariants. Chapter 7, `Special classes of invariants', deals mainly with pseudoreflection representations, low degree representatations of a solvable group. There is a proof of the Sheppard-Todd-Chevalley theorem about polynomial algebras of invariants as well as a discussion in the modular case. It is completed by computations of invariants in small degrees for a certain list of finite subgroups of \(GL(n, \mathbb Z).\) As a technical tool the authors prove a relative Noether bound for the maximum degree of a generator in a minimal algebra generating set for the invariants depending on a subgroup. Chapter 8, `The Steenrod algebra and invariant theory', has its origin in the Frobenius homomorphism of raising linear forms in the polynomial ring over \(\mathbb F_q\) to the \(q\)-th power. So it gives an additional structure on the ring of invariants. In the case of a Galois field as a ground field this is used in order to describe all the possible rings of invariants (of finite groups) of a fixed polynomial ring. Another application is the solution to the Stong-Landweber conjecture by \textit{D. Bourguiba} and \textit{G. Zarati} [see Invent. Math. 128, 589-602 (1997; Zbl 0874.55017)]. Chapter 9, `Invariant ideals' is devoted to the study of closed ideals under the action of the Steenrod algebra. Chapter 10, `Lannes's \(T\)-functor and applications', provides one of the recent results [see \textit{J. Lannes}, Publ. Math., Inst. Hautes Étud. Sci. 75, 135-244 (1992; Zbl 0857.55011)] to invariant theory. The power of the \(T\)-functor lies in the preservation of homological properties. It provides applications to complete intersections and the descent of the Cohen-Macaulay property for stabilizer subgroups. NEWLINENEWLINENEWLINEThe book is completed by an appendix: `Review of commutative algebra'. It covers a step by step review of non-standard gradings, Noetherianness, graded primary decomposition. NEWLINENEWLINENEWLINEThe book as a whole starts with elementary questions about invariant theory, develops the techniques for various classical results (presented with original proofs following recent research results) mainly in the non-modular situation. The presentation of the material is worked out always with a view towards the situation in the modular case. Some of the chapters are dedicated to recent research in particular to the modular invariant theory. So interested readers are invited to start with introductory concepts and will be pushed forward to the front of present day research. While the authors try to develop all prerequisites some technical knowledge about homological algebra etc. could be helpful. As a source for that one might consult the second author's book, (loc. cit.). NEWLINENEWLINENEWLINEThe references consist of 466 items, an rather complete picture about the recent research. The famous example by \textit{M.-J. Bertin} [C. R. Acad. Sci., Paris, Sér. A 264, 653-656 (1967; Zbl 0147.29503)], known as the first example of a unique factorization domain that is not Cohen-Macaulay, is referred several times for its non-Cohen-Macaulayness property. Its factoriality does not play any rôle in the book and is nowhere mentioned. NEWLINENEWLINENEWLINEThe divisor class theory of rings of invariants is not part of the book. The interested reader might find a presentation in the book by \textit{D. J. Benson}, `Polynomial invariants of finite groups' (1993; Zbl 0864.13001), respectively by \textit{P. Samuel}, `Lectures on unique factorization domains' (1964; Zbl 0184.06601). By the authors intention (following their classical theme of invariant theory) the typographical \TeX-layout is based on the traditional Garamond type face. For the revierwer's point of view the book will be a basic reference as well as a source for further research on the subject.