Polynomial invariants of finite groups. A survey of recent developments. (English) Zbl 0904.13004
Almost two decades ago R. P. Stanley published his important survey article “Invariants of finite groups and their application to combinatorics”, Bull. Am. Math. Soc., New Ser. 1, 475-511 (1979; Zbl 0497.20002). Since that time invariant theory of finite groups has taken a central rôle in different branches of algebra, in particular in algebraic topology as follows e.g. from the author’s book [L. Smith, “Polynomial invariants of finite groups” (1995; Zbl 0864.13002)]. The main subject of the present article is a review of the recent developments with respect to a particular interest in algebraic topology. A central rôle is played – in contrast to R. P. Stanley’s article (loc. cit.) – by the modular case, i.e. where the characteristic of the ground field divides the order of the group. The present survey is divided into six chapters:
1. The transfer and the classical finiteness theorems,
2. Orbit Chern classes and finiteness theorems,
3. Noether’s bound: a forgotten problem,
4. The Dickson algebra and modular invariant theory,
5. The Steenrod algebra and modular invariant theory, and
6. All together now: the depth conjecture.
The first chapter reviews Hilbert’s finiteness results, Molien series, and Cohen-Macaulayness of rings of invariants by the transfer principle, a slight extension of the Reynolds operator to the modular case. The second chapter is devoted to orbit Chern classes, an idea how to produce invariants in the modular case, and finiteness results whenever the characteristic divides the order of the group, including E. Noether’s finiteness theorem. In the third chapter there are various observations concerning bounds on the maximal degree of generators of the ring of invariants. This is stimulated – among others – by the work of B. J. Schmid [in: Topics in invariant theory, Sémin. Algèbre P. Dubreil et M.-P. Malliavin, Paris 1989-1990, Lect. Notes Math. 1478, 35-66 (1991; Zbl 0770.20004)]. Moreover it covers recent approaches of constructive aspects for invariants. Apart from Noether’s finiteness theorem and Hilbert’s syzygy theorem ‘everything’ can go wrong in the modular case as shown by examples in chapter 4.
There are compensations that make invariant theory in the modular case interesting and exciting. The first subject is the Dickson algebra introduced in chapter 5, an algebra of universal invariants (consisting of polynomials present in all rings of invariants) and providing a universal system of parameters of the ring of invariants. Its power is demonstrated by a short proof of Noether’s finiteness theorem in the modular case. Another modular feature is the Steenrod algebra introduced in chapter 5. It was created in algebraic topology and describes a way how to organize information derived from the Frobenius morphism. It provides a method for constructing new invariants from old ones. Based on these two new features P. S. Landweber and R. E. Stong [in: Number Theory, Semin. New York 1984/85, Lect. Notes Math. 1240, 259-274 (1987; Zbl 0623.55007)] stated a conjecture about the depth of rings of invariants in the modular case. In the final chapter 6 the author describes a recent theorem by D. Bourguiba and S.Zarati [Invent. Math. 128, No. 3, 589-602 (1997; Zbl 0874.55017)] about Steenrod operations that includes the depth conjecture as a particular case.
This survey article covers the recent research about modular invariant theory of finite groups. It is a complement to the author’s encyclopedic book on the subject (loc. cit.). The article underlines the central rôle of the ring of invariants in several branches of algebra and as an interesting subject of recent research.
1. The transfer and the classical finiteness theorems,
2. Orbit Chern classes and finiteness theorems,
3. Noether’s bound: a forgotten problem,
4. The Dickson algebra and modular invariant theory,
5. The Steenrod algebra and modular invariant theory, and
6. All together now: the depth conjecture.
The first chapter reviews Hilbert’s finiteness results, Molien series, and Cohen-Macaulayness of rings of invariants by the transfer principle, a slight extension of the Reynolds operator to the modular case. The second chapter is devoted to orbit Chern classes, an idea how to produce invariants in the modular case, and finiteness results whenever the characteristic divides the order of the group, including E. Noether’s finiteness theorem. In the third chapter there are various observations concerning bounds on the maximal degree of generators of the ring of invariants. This is stimulated – among others – by the work of B. J. Schmid [in: Topics in invariant theory, Sémin. Algèbre P. Dubreil et M.-P. Malliavin, Paris 1989-1990, Lect. Notes Math. 1478, 35-66 (1991; Zbl 0770.20004)]. Moreover it covers recent approaches of constructive aspects for invariants. Apart from Noether’s finiteness theorem and Hilbert’s syzygy theorem ‘everything’ can go wrong in the modular case as shown by examples in chapter 4.
There are compensations that make invariant theory in the modular case interesting and exciting. The first subject is the Dickson algebra introduced in chapter 5, an algebra of universal invariants (consisting of polynomials present in all rings of invariants) and providing a universal system of parameters of the ring of invariants. Its power is demonstrated by a short proof of Noether’s finiteness theorem in the modular case. Another modular feature is the Steenrod algebra introduced in chapter 5. It was created in algebraic topology and describes a way how to organize information derived from the Frobenius morphism. It provides a method for constructing new invariants from old ones. Based on these two new features P. S. Landweber and R. E. Stong [in: Number Theory, Semin. New York 1984/85, Lect. Notes Math. 1240, 259-274 (1987; Zbl 0623.55007)] stated a conjecture about the depth of rings of invariants in the modular case. In the final chapter 6 the author describes a recent theorem by D. Bourguiba and S.Zarati [Invent. Math. 128, No. 3, 589-602 (1997; Zbl 0874.55017)] about Steenrod operations that includes the depth conjecture as a particular case.
This survey article covers the recent research about modular invariant theory of finite groups. It is a complement to the author’s encyclopedic book on the subject (loc. cit.). The article underlines the central rôle of the ring of invariants in several branches of algebra and as an interesting subject of recent research.
Reviewer: P.Schenzel (Halle)
Keywords:
finite group; ring of invariants; Dickson algebra; modular invariant theory; Steenrod algebra; finiteness theorem; Hilbert’s syzygy theorem; depth conjecture; Frobenius morphismReferences:
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