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:
“We say that the hour of death cannot be forecast, but when we say this we imagine that hour as placed in an obscure and distant future. It never occurs to us that it has any connection with the day already begun or that death could arrive this same afternoon, this afternoon which is so certain and which has every hour filled in advance.”
—Marcel Proust (1871–1922)
“We will have to give up the hope that, if we try hard, we somehow will always do right by our children. The connection is imperfect. We will sometimes do wrong.”
—Judith Viorst (20th century)
“What chiefly distinguishes the daily press of the United States from the press of all other countries is not its lack of truthfulness or even its lack of dignity and honor, for these deficiencies are common to the newspapers everywhere, but its incurable fear of ideas, its constant effort to evade the discussion of fundamentals by translating all issues into a few elemental fears, its incessant reduction of all reflection to mere emotion. It is, in the true sense, never well-informed.”
—H.L. (Henry Lewis)
“In America every woman has her set of girl-friends; some are cousins, the rest are gained at school. These form a permanent committee who sit on each other’s affairs, who “come out” together, marry and divorce together, and who end as those groups of bustling, heartless well-informed club-women who govern society. Against them the Couple of Ehepaar is helpless and Man in their eyes but a biological interlude.”
—Cyril Connolly (1903–1974)