Titlebook: Reflection Groups and Invariant Theory; Richard Kane,Jonathan Borwein,Peter Borwein Textbook 2001 Springer Science+Business Media New York

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Bilinear forms of Coxeter systemstion between finite reflection groups and Coxeter systems developed in §6–2. The main result of this chapter is that the bilinear form associated to a Coxeter system is always positive definite. In Chapter 8, we shall use the positive definiteness of this bilinear form to classify both finite Coxete

戰(zhàn)役

Classification of Coxeter systems and reflection groupsevery reflection group has a canonical associated Coxeter system. So the classifications are related. In Chapter 7 we introduced the bilinear form of a Coxeter system. Most of this chapter is occupied with determining necessary conditions for the bilinear form .: V × V → ? of a finite Coxeter system

Absenteeism

Synchronism

The Classification of crystallographic root systemsl groups. As we have already mentioned in the introduction to Part III, the study of Weyl groups and crystallographic root systems uses the results about reflection groups from Chapters 1 through 8. In particular, the classification of Weyl groups and crystallographic root systems will turn out to b

Stricture

Affine Weyl groupsstill possesses a structure analogous to that of the Weyl group. Notably, it has a Coxeter group structure. This group is called the affine Weyl group. Affine Weyl groups have a number of uses. They will be used in Chapter 12 to analyze subroot systems of crystallographic root systems. They are even

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bifurcate

Pseudo-reflectionswell as the next, is preliminary to the study of invariant theory, since it is invariant theory that motivates the introduction of pseudo-reflections. Most of our discussion of invariant theory naturally takes place in the context of pseudo-reflection groups. However, it will take several chapters b

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