# Difference between revisions of "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

 [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)

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