Difference between revisions of "Chamber"
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− | + | ''in a finite-dimensional real affine space $ E $, | |
+ | with respect to a locally finite set $ \mathfrak F $ | ||
+ | of hyperplanes in $ E $'' | ||
− | + | A connected component of the set $ E \setminus \cup _ {H \in \mathfrak F } H $. | |
+ | A chamber is an open convex subset of $ E $. | ||
+ | |||
+ | Let $ \mathfrak F $ | ||
+ | be a set of hyperplanes in $ E $ | ||
+ | such that the group $ W $ | ||
+ | of motions of $ E $ | ||
+ | generated by the orthogonal reflections with respect to the hyperplanes of $ \mathfrak F $ | ||
+ | is a [[Discrete group of transformations|discrete group of transformations]] of $ E $, | ||
+ | and such that moreover the system $ \mathfrak F $ | ||
+ | is invariant with respect to $ W $. | ||
+ | In this case one speaks about a chamber relative to $ W $. | ||
+ | The group $ W $ | ||
+ | acts simply transitively on the set of all chambers and is generated by the set $ S $ | ||
+ | of orthogonal reflections with respect to hyperplanes of $ \mathfrak F $ | ||
+ | containing the $ ( \mathop{\rm dim} E - 1 ) $- | ||
+ | dimensional faces of any fixed chamber $ C $; | ||
+ | moreover, the pair $ ( W , S ) $ | ||
+ | is a Coxeter system and the closure of $ C $ | ||
+ | is a fundamental domain of $ W $. | ||
+ | The structure of $ C $( | ||
+ | the description of the dihedral angles between the walls) completely determines the structure of $ W $ | ||
+ | as an abstract group. The study of this structure is an important step in obtaining a complete classification of the discrete groups generated by the reflections in $ E $( | ||
+ | see [[Coxeter group|Coxeter group]]). Along with this classification, a complete description is obtained of the structure of chambers for such groups $ W $. | ||
+ | |||
+ | If $ W $ | ||
+ | is the Weyl group of a root system of a semi-simple Lie algebra, a chamber relative to $ W $ | ||
+ | is called a Weyl chamber of $ W $. | ||
The notion of a chamber can also be defined for hyperplanes and discrete groups generated by reflections in Lobachevskii space or on a sphere [[#References|[2]]]. | The notion of a chamber can also be defined for hyperplanes and discrete groups generated by reflections in Lobachevskii space or on a sphere [[#References|[2]]]. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.B. Vinberg, "Discrete linear groups that are generated by reflections" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' : 5 (1971) pp. 1072–1112 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.B. Vinberg, "Discrete linear groups that are generated by reflections" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' : 5 (1971) pp. 1072–1112 (In Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | A Coxeter system | + | A Coxeter system $ ( W , S ) $ |
+ | consists of a group $ W $ | ||
+ | and a subset $ S $ | ||
+ | of $ W $ | ||
+ | which generates $ W $ | ||
+ | such that $ 1 \notin S $ | ||
+ | and, $ s ^ {2} = 1 $ | ||
+ | for all $ s \in S $ | ||
+ | and, moreover, for all $ s , s ^ \prime \in S $ | ||
+ | one has the condition | ||
− | (c) for all | + | (c) for all $ s , s ^ \prime $ |
+ | let $ m ( s , s ^ \prime ) $ | ||
+ | be the order of the group element $ s s ^ \prime $; | ||
+ | let $ I $ | ||
+ | be the set of pairs $ ( s , s ^ \prime ) $ | ||
+ | such that $ m ( s , s ^ \prime ) $ | ||
+ | is finite. Then the generating set $ S $ | ||
+ | and the relations $ s ^ {2} = 1 $ | ||
+ | and $ ( s s ^ \prime ) ^ {m ( s , s ^ \prime ) } = 1 $ | ||
+ | for $ ( s , s ^ \prime ) \in I $ | ||
+ | form a presentation of $ W $. | ||
− | For example, let | + | For example, let $ \sigma _ {i} $ |
+ | be the permutation $ ( i i + 1 ) $ | ||
+ | in the group of permutations on $ n $ | ||
+ | letters $ S _ {n} $, | ||
+ | then $ ( S _ {n} , \{ \sigma _ {1} \dots \sigma _ {n-} 1 \} ) $ | ||
+ | is a Coxeter system. | ||
Recently, chambers and chamber systems were defined in a more abstract setting for synthetic geometry by J. Tits [[#References|[a1]]]. | Recently, chambers and chamber systems were defined in a more abstract setting for synthetic geometry by J. Tits [[#References|[a1]]]. |
Latest revision as of 16:43, 4 June 2020
in a finite-dimensional real affine space $ E $,
with respect to a locally finite set $ \mathfrak F $
of hyperplanes in $ E $
A connected component of the set $ E \setminus \cup _ {H \in \mathfrak F } H $. A chamber is an open convex subset of $ E $.
Let $ \mathfrak F $ be a set of hyperplanes in $ E $ such that the group $ W $ of motions of $ E $ generated by the orthogonal reflections with respect to the hyperplanes of $ \mathfrak F $ is a discrete group of transformations of $ E $, and such that moreover the system $ \mathfrak F $ is invariant with respect to $ W $. In this case one speaks about a chamber relative to $ W $. The group $ W $ acts simply transitively on the set of all chambers and is generated by the set $ S $ of orthogonal reflections with respect to hyperplanes of $ \mathfrak F $ containing the $ ( \mathop{\rm dim} E - 1 ) $- dimensional faces of any fixed chamber $ C $; moreover, the pair $ ( W , S ) $ is a Coxeter system and the closure of $ C $ is a fundamental domain of $ W $. The structure of $ C $( the description of the dihedral angles between the walls) completely determines the structure of $ W $ as an abstract group. The study of this structure is an important step in obtaining a complete classification of the discrete groups generated by the reflections in $ E $( see Coxeter group). Along with this classification, a complete description is obtained of the structure of chambers for such groups $ W $.
If $ W $ is the Weyl group of a root system of a semi-simple Lie algebra, a chamber relative to $ W $ is called a Weyl chamber of $ W $.
The notion of a chamber can also be defined for hyperplanes and discrete groups generated by reflections in Lobachevskii space or on a sphere [2].
References
[1] | N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) |
[2] | E.B. Vinberg, "Discrete linear groups that are generated by reflections" Izv. Akad. Nauk SSSR Ser. Mat. , 35 : 5 (1971) pp. 1072–1112 (In Russian) |
Comments
A Coxeter system $ ( W , S ) $ consists of a group $ W $ and a subset $ S $ of $ W $ which generates $ W $ such that $ 1 \notin S $ and, $ s ^ {2} = 1 $ for all $ s \in S $ and, moreover, for all $ s , s ^ \prime \in S $ one has the condition
(c) for all $ s , s ^ \prime $ let $ m ( s , s ^ \prime ) $ be the order of the group element $ s s ^ \prime $; let $ I $ be the set of pairs $ ( s , s ^ \prime ) $ such that $ m ( s , s ^ \prime ) $ is finite. Then the generating set $ S $ and the relations $ s ^ {2} = 1 $ and $ ( s s ^ \prime ) ^ {m ( s , s ^ \prime ) } = 1 $ for $ ( s , s ^ \prime ) \in I $ form a presentation of $ W $.
For example, let $ \sigma _ {i} $ be the permutation $ ( i i + 1 ) $ in the group of permutations on $ n $ letters $ S _ {n} $, then $ ( S _ {n} , \{ \sigma _ {1} \dots \sigma _ {n-} 1 \} ) $ is a Coxeter system.
Recently, chambers and chamber systems were defined in a more abstract setting for synthetic geometry by J. Tits [a1].
References
[a1] | J. Tits, "A local approach to buildings" C. Davis (ed.) B. Grünbaum (ed.) F.A. Sherk (ed.) , The geometric vein (Coxeter-Festschrift) , Springer (1981) pp. 519–547 |
[a2] | N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1968) pp. Chapt. 4. Groupes de Coxeter et systèmes de Tits |
Chamber. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chamber&oldid=12004