Abstract
This chapter is the first in which we begin to explore the relation between the algebraic structure of a reflection group and its underlying geometry. The main purpose of this chapter is to introduce the concept of length in a reflection group and to explain how length is related to the action of the reflection group on its root system and on its Weyl chambers. The main application of length will come in Chapter 6, when we prove that a finite Euclidean reflection group is a Coxeter group. In particular, the relationship between the algebra and the geometry of Euclidean reflection groups will become more apparent in that chapter.
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© 2001 Springer Science+Business Media New York
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Kane, R., Borwein, J., Borwein, P. (2001). Length. In: Borwein, J., Borwein, P. (eds) Reflection Groups and Invariant Theory. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3542-0_5
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DOI: https://doi.org/10.1007/978-1-4757-3542-0_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-3194-8
Online ISBN: 978-1-4757-3542-0
eBook Packages: Springer Book Archive