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Multiplicative invariant theory. (English) Zbl 1078.13003
Encyclopaedia of Mathematical Sciences 135. Invariant Theory and Algebraic Transformation Groups 6. Berlin: Springer (ISBN 3-540-24323-2/hbk). xi, 177 p. (2005).
Let \(G\) denote a group. The author considers multiplicative actions arising from a representation \( G \to \text{GL}(L)\) of \(G\) on a lattice \(L.\) That is, \(L\) is a free \(\mathbb Z\)-module of finite rank on which \(G\) acts by automorphisms, or in other words \(L\) is a \(G\)-lattice. Let \(k\) denote a base ring. Then the \(G\)-action on \(L\) extends uniquely to an action by \(k\)-algebra automorphisms on the group algebra \(k[L]\) over \(k.\) Then multiplicative invariant theory is concerned with the study of \[ k[L]^G = \{ f \in k[L] \mid g(f) = f \text{ for all } g \in G\}. \] This notion was introduced by D. R. Farkas [Enseign. Math. 30, 141–157 (1984; Zbl 0546.14003)]. It is in contrast to the polynomial invariant theory, where the \(G\)-action \(G \to \text{GL}(V)\) on a \(k\)-vector space \(V\) is extended to the symmetric algebra \(S(V).\) The subalgebra \(S(V)^G\) of all \(G\)-invariant polynomials in \(S(V)\) is usually called the algebra of polynomial invariants. There are several textbooks about polynomial invariant theory [see e.g. D. J. Benson, “Polynomial invariants of finite groups”, Lond. Math. Society Lect. Note Series. 190 (Cambridge 1993; Zbl 0864.13001), H. Derksen and G. Kemper, “Computational invariant theory”, Encyclopaedia of Mathematical Sciences (Berlin 2002; Zbl 1011.13003), and L. Smith, “Polynomial invariants of finite groups”, Res. Notes Math. (Boston, Mass. 1995; Zbl 0864.13002)]. The book under review is the first systematic treatment of multiplicative invariant theory in the form of a textbook written by an author who has contributed several research articles during to the subject the last years.
The book is devided into ten chapters. Chapter 1, “Groups acting on lattices”, and chapter 2, “Permutation lattices and flasque equivalence”, are devoted to group actions on lattices in order to collect all representation theoretic techniques and results needed for further investigations. It covers also an important equivalence relation between \(G\)-lattices (flasque equivalence) needed in chapter 9, “Multiplicative invariant fields”. The main theme in chapter 9 is the rationality problem for invariant fields, also known as Noether’s problem. Chapter 3, “Multiplicative actions”, is devoted to the basics, as finite generation of \(k[L]^G\), the existence of a \(\mathbb Z\)-structure, and the fact that it suffices to investigate the case of finite group actions. It is also shown that multiplicative invariant algebras of weight lattices under the action of the Weyl group are polynomial algebras (Bourbaki’s theorem). The converse of this statement is shown in chapter 7, “Regularity”, where the regularity of \(k[L]^G\) is studied. Note that Bourbaki’s theorem and its converse are the multiplicative analogs of the results of Shephard-Todd and Chevalley in the case of polynomial invariants. In chapter 4, “Class group”, a formula for the class group CL\((k[L]^G)\) is given. This answers the problem when \(k[L]^G\) is a unique factorization domain. The Picard group, a subgroup of the class group, is calculated in chapter 5, “Picard group”. In contrast to the case of polynomial invariant theory, where the Picard group is trivial, this is not true in multiplicative invariant theory. In chapter 6, “Multiplicative invariants of reflection groups”, there is – motivated by the Shepphard-Todd-Chevalley theorem – a complete description of \(k[L]^G\) for finite reflection groups \(G.\) It turns out that all of them are affine normal semigroup algebras and therefore Cohen-Macaulay rings. This class of examples leads the author to the investigation of the Cohen-Macaulayness of \(k[L]^G\) in chapter 8, “The Cohen-Macaulay property”. As a technical tool there is a detailed discussion of the the Ellingsrud-Skjelbred spectral sequence for local cohomology. In this chapter there are some directions for further research on the subject. This is continued in chapter 10, “Problems”, where the author addresses problems about the Cohen-Macaulay-property, semigroup algebras, computational issues, dimension estimates, and rationality problems. The book concludes with references, covering most recent research and related subjects in 227 items. The book contains numerous examples, illustrating the theory, among them a complete list of all multiplicative invariant algebras for lattices of rank 2.
The book is recommended for graduate and postgraduate students as well as researchers in representation theory, commutative algebra, and invariant theory. It opens a fresh view to research problems on these fields related to multiplicative invariants.

13A50 Actions of groups on commutative rings; invariant theory
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D45 Local cohomology and commutative rings
20C10 Integral representations of finite groups
12F20 Transcendental field extensions
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