Introduction: Reflection groups and invariant theory

Abstract

This is a graduate level book on the connections between finitegroups G generated by reflections (or pseudo reflections) andinvariant theory. Here a (Euclidean) reflection is an invertiblelinear transformation of a finite dimensional vector space Vthat fixes a certain hyperplane of V and which reflects vectorsperpendicular to the hyperplane, each such reflection r beingdetermined by a vector v(r) \in V perpendicular to thehyperplane.\par There are three main sections. In the first theauthor gives a clear and detailed account of the main resultsconcerning finite Euclidean reflection groups. This includes theelegant theory revolving around the notion of a root system(which is, roughly speaking, a set of vectors { \pm v(r)\vert\r\ \text {a reflection in}\ G} invariant under G). The mainresult here is the classification of all such groups via theclassification of the associated root systems. Also, a discussionof the Coxeter group structure that each such G has is given.Here a Coxeter group is a group having a presentation of the form\langle a\sb 1,a\sb 2,\dots, a\sb n\vert\ a\sb i\sp 2,{\rm prod}(a\sb i,a\sb j,m\sb {ij})={\rm prod}(a\sb j,a\sb i,m\sb {ij})\ranglewhere, for i e j, {\rm prod}(a\sb i,a\sb j,m\sb {ij}) isthe word a\sb ia\sb ja\sb i \dots of lengthm\sb {ij}=m\sb {ji} \in {2,3,\dots }. Various combinatorialobjects and functions are studied, for example, length functionsof elements of G and certain graded algebras.\par Finitereflection groups arise naturally when one studies finitesubgroups of {\rm GL}(V); they are also involved in theclassification of irreducible Lie algebras, where a subclasssatisfying a certain integrality condition occurs. These latterare the crystallographic reflection groups that fix a lattice inV. They are also classified in the present work.\par The secondsection concerns reflection groups and invariant theory. Theresults of this section are all a part of the revival of interestin invariant theory that occurred in the last century andcontinues today. Here the main result is the fact that a finitegroup is a group generated by pseudo reflections if and only ifits ring of invariants is a polynomial algebra. This resultnaturally introduces the pseudo reflection groups studied byShephard and Todd and which are a feature of this work.\par Thethird section of this book develops the theory concerningconjugacy classes, subgroups, eigenvalues and eigenspaces of suchfinite reflection groups. It is more demanding of thereader.\par This book is a very accessible introduction to awonderful part of mathematics that has many applications. Theseapplications have been adequately outlined in the reviews of themore than ten other books that deal with these objects, butinclude Lie theory, algebraic groups, combinatorics, finitegeometry and the homology of classifying spaces. Dually, however,it can be pointed out that the approach taken in this bookincludes examples of the use of Galois theory, Cohen Macaulayproperties, Hopf algebras, representation theory and basicalgebraic geometry. This is a nice way to see applications ofthese areas.\par The chapter headings are as follows, and give agood indication of the scope and the development of the book: 1.Euclidean reflection groups. 2. Root systems. 3. Fundamentalsystems. 4. Length. 5. Parabolic subgroups. 6. Reflection groupsand Coxeter systems. 7. Bilinear forms of Coxeter systems. 8.Classification of Coxeter systems and reflection groups. 9. Weylgroups. 10. The classification of crystallographic root systems.11. Affine Weyl groups. 12. Subroot systems. 13. Formalidentities. 14. Pseudo reflections. 15. Classification of pseudoreflection groups. 16. The ring of invariants. 17. Poincareseries. 18. Nonmodular invariants of pseudo reflection groups.19. Modular invariants of pseudo reflection groups. 20. Skewinvariants. 21. The Jacobian. 22. The extended ring ofinvariants. 23. Poincare series for the ring of covariants. 24.Representations of pseudo reflection groups. 25. Harmonicelements. 26. Harmonics and reflection groups. 27. Involutions.28. Elementary equivalences. 29. Coxeter elements. 30. Minimaldecompositions. 31. Eigenvalues for reflection groups. 32.Eigenvalues for regular elements. 33. Ring of invariants andeigenvalues. 34. Properties of regular elements. There are fourappendices: Rings and modules; Group actions and representationtheory; Quadratic forms; Lie algebras.\par The book wouldcertainly be a good choice to teach the ideas in the above list,as the book flows very well. Each chapter is well motivated andsummarised in its introduction. The use of the book as a textbooksuffers, however, from a total lack of exercises.\par Thereferences are adequate for the task, but do not includeindications of what is going on in this area at present (the mostrecent reference is dated 1997). There are a few minortypographical errors