Chamber
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 |
Chamber. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chamber&oldid=46303