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Invariant theory. (Russian) Zbl 0735.14010
As is well-known, invariant theory is an old subject that ”suffered” periods of relative inactivity which ended in spectacular breakpoints. To cite just two, the Hilbert finiteness theorem, opening a new era in commutative algebra, and D. Mumford’s book [“Geometric invariant theory” (1965; Zbl 0147.39304)], applying the algebro-geometric intuition and tools developed by Grothendieck and others to the area. In particular, Mumford’s ideas provoked an explosion of activity, leading to a huge amount of important results [cf. also the appendix to the second edition of the cited book: D. Mumford and J. Fogarty, “Geometric invariant theory” (1982; Zbl 0504.14008)].
The paper under review, written by two of the main contributors in this last period, is a beautiful survey of the matter, flavouring each introduced notion by examples, applications or historical remarks, and supplemented by an impressive list of references. It should be considered as a book, which is probably the format it would have if translated. Let us give a rough and incomplete look to the material presented.
In chapter 0, the authors discuss a historical brief and many instructive examples of the “classical” invariant theory, throwing into relief their geometrical interpretation. This includes for example the unimodular and orthogonal invariant of quadratic forms, the invariant of (low degree) binary and cubic forms, etc.
In chapter 1, the basic notions concerning the actions of algebraic groups are presented. — Chapter 2 is devoted to rational invariants. It is discussed, for example, the problem of rationality of the field of invariants, in particular a theorem of Katsilo on the rationality of the field of invariants of projective representations of \(SL_ 2\). – Chapter 3 is concerned with regular invariants and covariants: their relation with rational ones; Hilbert theorem, and consequently constructive invariant theory; Hilbert’s 14th problem, the Nagata counterexample and Grosshans subgroups, Poincaré series, …
From chapter 4 on, the authors begin to treat the modern algebro- geometric interpretation of invariant theory. In this chapter, they consider the set-theoretical quotient of a manifold by the rational action of an algebraic group, to motivate the discussion of the geometrical and categorical quotients. — Chapter 5 is devoted to the study of the nilpotent cone, the famous Hilbert-Mumford theorem and some of its main consequences. – In chapter 6, Luna’s celebrated slice étale theorem is presented, allowing to study the geometric structure of the fibers of the “categorical” quotient. It also includes discussions of Luna’s stratification, Dixmier sheets, …– In chapter 7, the notion of “stabilizer in general position” is discussed. The classification of linear representations of simple algebraic groups with “good” properties (freeness of the ring of invariants, same dimension for all the orbits, only finite number of orbits in each fiber of the quotient map, non-triviality of the stabilizer in general position) is recalled in chapter 8. (The corresponding tables are included at the end.) – The striking coincidence of all this tables motivates the so called “Russian” conjecture (due in fact to one of the authors), which is recalled and discussed here.
Finally, the classical invariant theory [as by H. Weyl, in his book: “The classical groups, their invariants and representations” (1939; Zbl 0020.20601); see also G. Gurevich, “Foundations of the theory of algebraic invariants” (Moscow 1948)] is treated in chapter 9.

MSC:
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
14-03 History of algebraic geometry
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
20G05 Representation theory for linear algebraic groups
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