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in a finite-dimensional real affine space , 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=46303
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article