Chamber
in a finite-dimensional real affine space 
, with respect to a locally finite set 
of hyperplanes in 
A connected component of the set 
. A chamber is an open convex subset of 
.
Let 
be a set of hyperplanes in 
such that the group 
of motions of 
generated by the orthogonal reflections with respect to the hyperplanes of 
is a discrete group of transformations of 
, and such that moreover the system 
is invariant with respect to 
. In this case one speaks about a chamber relative to 
. The group 
acts simply transitively on the set of all chambers and is generated by the set 
of orthogonal reflections with respect to hyperplanes of 
containing the 
-dimensional faces of any fixed chamber 
; moreover, the pair 
is a Coxeter system and the closure of 
is a fundamental domain of 
. The structure of 
(the description of the dihedral angles between the walls) completely determines the structure of 
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 
(see Coxeter group). Along with this classification, a complete description is obtained of the structure of chambers for such groups 
.
If 
is the Weyl group of a root system of a semi-simple Lie algebra, a chamber relative to 
is called a Weyl chamber of 
.
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 
consists of a group 
and a subset 
of 
which generates 
such that 
and, 
for all 
and, moreover, for all 
one has the condition
(c) for all 
let 
be the order of the group element 
; let 
be the set of pairs 
such that 
is finite. Then the generating set 
and the relations 
and 
for 
form a presentation of 
.
For example, let 
be the permutation 
in the group of permutations on 
letters 
, then 
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=46303