Connection With Reflection Groups
For more details on this topic, see Reflection group.Coxeter groups are deeply connected with reflection groups. Simply put, Coxeter groups are abstract groups (given via a presentation), while reflection groups are concrete groups (given as subgroups of linear groups or various generalizations). Coxeter groups grew out of the study of reflection groups — they are an abstraction: a reflection group is a subgroup of a linear group generated by reflections (which have order 2), while a Coxeter group is an abstract group generated by involutions (elements of order 2, abstracting from reflections), and whose relations have a certain form (, corresponding to hyperplanes meeting at an angle of, with being of order k abstracting from a rotation by ).
The abstract group of a reflection group is a Coxeter group, while conversely a reflection group can be seen as a linear representation of a Coxeter group. For finite reflection groups, this yields an exact correspondence: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space. For infinite Coxeter groups, however, a Coxeter group may not admit a representation as a reflection group.
Historically, (Coxeter 1934) proved that every reflection group is a Coxeter group (i.e., has a presentation where all relations are of the form or ), and indeed this paper introduced the notion of a Coxeter group, while (Coxeter 1935) proved that every finite Coxeter group had a representation as a reflection group, and classified finite Coxeter groups.
Read more about this topic: Coxeter Group
Famous quotes containing the words connection with, connection, reflection and/or groups:
“Morality becomes hypocrisy if it means accepting mothers’ suffering or dying in connection with unwanted pregnancies and illegal abortions and unwanted children.”
—Gro Harlem Brundtland (b. 1939)
“The Transcendentalist adopts the whole connection of spiritual doctrine. He believes in miracle, in the perpetual openness of the human mind to new influx of light and power; he believes in inspiration, and in ecstacy.”
—Ralph Waldo Emerson (1803–1882)
“It is conceivable at least that a late generation, such as we presumably are, has particular need of the sketch, in order not to be strangled to death by inherited conceptions which preclude new births.... The sketch has direction, but no ending; the sketch as reflection of a view of life that is no longer conclusive, or is not yet conclusive.”
—Max Frisch (1911–1991)
“If we can learn ... to look at the ways in which various groups appropriate and use the mass-produced art of our culture ... we may well begin to understand that although the ideological power of contemporary cultural forms is enormous, indeed sometimes even frightening, that power is not yet all-pervasive, totally vigilant, or complete.”
—Janice A. Radway (b. 1949)