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''in a finite-dimensional real affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214301.png" />, with respect to a locally finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214302.png" /> of hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214303.png" />''
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A connected component of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214304.png" />. A chamber is an open convex subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214305.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214306.png" /> be a set of hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214307.png" /> such that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214308.png" /> of motions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c0214309.png" /> generated by the orthogonal reflections with respect to the hyperplanes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143010.png" /> is a [[Discrete group of transformations|discrete group of transformations]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143011.png" />, and such that moreover the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143012.png" /> is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143013.png" />. In this case one speaks about a chamber relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143014.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143015.png" /> acts simply transitively on the set of all chambers and is generated by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143016.png" /> of orthogonal reflections with respect to hyperplanes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143017.png" /> containing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143018.png" />-dimensional faces of any fixed chamber <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143019.png" />; moreover, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143020.png" /> is a Coxeter system and the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143021.png" /> is a fundamental domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143022.png" />. The structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143023.png" /> (the description of the dihedral angles between the walls) completely determines the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143025.png" /> (see [[Coxeter group|Coxeter group]]). Along with this classification, a complete description is obtained of the structure of chambers for such groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143026.png" />.
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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 $''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143027.png" /> is the Weyl group of a root system of a semi-simple Lie algebra, a chamber relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143028.png" /> is called a Weyl chamber of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143029.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143030.png" /> consists of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143031.png" /> and a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143033.png" /> which generates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143035.png" /> and, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143037.png" /> and, moreover, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143038.png" /> one has the condition
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143039.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143040.png" /> be the order of the group element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143041.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143042.png" /> be the set of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143044.png" /> is finite. Then the generating set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143045.png" /> and the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143047.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143048.png" /> form a presentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143049.png" />.
+
(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143050.png" /> be the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143051.png" /> in the group of permutations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143052.png" /> letters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143053.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021430/c02143054.png" /> is a Coxeter system.
+
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
How to Cite This Entry:
Chamber. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chamber&oldid=12004
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article