Chamber

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[edit]

[1] |  N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)

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[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[edit]

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[edit]

[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

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[a2] |  N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1968) pp. Chapt. 4. Groupes de Coxeter et systèmes de Tits