Invariant Theory

This book presents an introduction to invariant theory of finite groups acting on polynomial algebras of characteristic 0. That is, having a vector space V over a field F of characteristic 0, a finite group G and a representation ρ of G in GL(V), we have an action of G on the polynomial algebra F [V]. The main object of study is its subalgebra F[V]G given by polynomials that are fixed by all elements of G. The book is divided into five parts and one appendix. The first part collects together basic notions of groups, linear representations, associative rings and algebras. The second part presents the so-called Göbel’s bound for generators of F[V]G provided ρ is a permutation representation. This result (taken from a quite recent thesis of M. Göbel) is used in the next part to prove the classical theorem stating that F[V]G is a finitely generated ring. This theorem is then improved and it is shown how to find generators of F[V]G as images of polarized elementary symmetric polynomials. Part 4 explains concepts from commutative algebra and module theory and uses them to provide a different attitude to results obtained earlier. The reader can compare the efficiency of these proofs with the algorithmic approach from parts 2 and 3. The last part of the book uses some counting with power series (Poincaré series) to prove the Shephard-Todd-Chevalley theorem, which characterizes finite subgroups of GL(V) generated by pseudoreflections as those finite groups G of GL(V) having F[V]G isomorphic to a polynomial algebra. An appendix on rational invariants concludes the book.

Although I think that a couple of proofs could be written with more details (for example, in the proof of proposition 6.7, it is shown that the transfers of all variables lie in the image of some restriction of the Noether map but I would need a hint to finish the proof since these transfers do not generate the whole F[V]G), a large part of the book is written in a friendly style and all notions are carefully explained and immediately demonstrated in concrete examples. Each chapter contains a lot of exercises. Moreover, one has to appreciate the many applications of invariant theory to other parts of mathematics, physics and engineering the author has inserted in appropriate places during the exposition.

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