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Chamber

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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
How to Cite This Entry:
Chamber. V.L. Popov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Chamber&oldid=12004
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098