Coxeter complex

In mathematics, the Coxeter complex, named after H. S. M. Coxeter, is a geometrical structure (a simplicial complex) associated to a Coxeter group. Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building.
Construction.
The canonical linear representation.
The first ingredient in the construction of the Coxeter complex associated to a Coxeter group "W" is a certain representation of "W", called the canonical representation of "W".
Let formula_1 be a Coxeter system associated to "W", with Coxeter matrix formula_2. The canonical representation is given by a vector space "V" with basis of formal symbols formula_3, which is equipped with the symmetric bilinear form formula_4. The action of "W" on this vector space "V" is then given by formula_5, as motivated by the expression for reflections in root systems.
This representation has several foundational properties in the theory of Coxeter groups; for instance, the bilinear form "B" is positive definite if and only if "W" is finite. It is (always) a faithful representation of "W".
Chambers and the Tits cone.
One can think of this representation as expressing "W" as some sort of reflection group, with the caveat that "B" might not be positive definite. It becomes important then to distinguish the representation "V" from its dual "V"*. The vectors formula_6 lie in "V", and have corresponding dual vectors formula_7 in "V"*, given by:
where the angled brackets indicate the natural pairing of a dual vector in "V"* with a vector of "V", and "B" is the bilinear form as above.
Now "W" acts on "V"*, and the action satisfies the formula
for formula_10 and any "f" in "V"*. This expresses "s" as a reflection in the hyperplane formula_11. One has the fundamental chamber formula_12, this has faces the so-called walls, formula_13. The other chambers can be obtained from formula_14 by translation: they are the formula_15 for formula_16.
Given a fundamental chamber formula_17, the Tits cone is defined to be formula_18. This need not be the whole of "V"*. Of major importance is the fact that the Tits cone "X" is convex. The action of "W" on the Tits cone "X" has fundamental domain the fundamental chamber formula_17.
The Coxeter complex.
Once one has defined the Tits cone "X", the Coxeter complex formula_20 of "W" with respect to "S" can be defined as the quotient of "X", with the origin removed, by the positive reals (ℝ+, ×):
Examples.
Finite dihedral groups.
The dihedral groups formula_22 (of order 2"n") are Coxeter groups, of corresponding type formula_23. These have the presentation formula_24.
The canonical linear representation of formula_23 is the usual reflection representation of the dihedral group, as acting on a "n"-gon in the plane (so formula_26 in this case). For instance, in the case "n" = 3, we get the Coxeter group of type formula_27, acting on an equilateral triangle in the plane. Each reflection "s" has an associated hyperplane "H""s" in the dual vector space (which can be canonically identified with the vector space itself using the bilinear form "B", which is an inner product in this case as remarked above), these are the walls. They cut out chambers, as seen below:
The Coxeter complex is then the corresponding 2"n"-gon, as in the image above. This is a simplicial complex of dimension 1, and it can be colored by cotype.
The infinite dihedral group.
Another motivating example is the infinite dihedral group formula_28. This can be seen as the group of symmetries of the real line that preserves the set of points with integer coordinates; it is generated by the reflections in formula_29 and formula_30. This group has the Coxeter presentation formula_31.
In this case, it is no longer possible to identify "V" with the dual space "V"*, as "B" is not positive definite. It is then better to work solely with "V"*, which is where the hyperplanes are defined. This then gives the following picture:
In this case, the Tits cone is not the whole plane, but only the upper half plane. Quotienting out by the positive reals then yields another copy of the real line, with marked points at the integers. This is the Coxeter complex of the infinite dihedral group.
Alternative construction of the Coxeter complex.
Another description of the Coxeter complex uses standard cosets of the Coxeter group "W". A standard coset is a coset of the form formula_32, where formula_33 for some subset "J" of "S". For instance, formula_34 and formula_35.
The Coxeter complex formula_36 is then the poset of standard cosets, ordered by reverse inclusion. This has a canonical structure of a simplicial complex, as do all posets that satisfy:
Properties.
The Coxeter complex associated to formula_1 has dimension formula_39. It is homeomorphic to a formula_40-sphere if "W" is finite and is contractible if "W" is infinite.