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 system  is a certain representation of , called the canonical representation of .

Let  be a Coxeter system with Coxeter matrix . The canonical representation is given by a vector space  with basis of formal symbols , which is equipped with the symmetric bilinear form . In particular, . The action of  on  is then given by .

This representation has several foundational properties in the theory of Coxeter groups; for instance,  is positive definite if and only if  is finite. It is a faithful representation of .

Chambers and the Tits cone
This representation describes  as a reflection group, with the caveat that  might not be positive definite. It becomes important then to distinguish the representation  from its dual . The vectors  lie in  and have corresponding dual vectors  in  given by

 

where the angled brackets indicate the natural pairing between  and . 

Now  acts on  and the action is given by

 

for  and any . Then  is a reflection in the hyperplane . One has the fundamental chamber ; this has faces the so-called walls, . The other chambers can be obtained from  by translation: they are the  for .

The Tits cone is . This need not be the whole of . Of major importance is the fact that  is convex. The closure  of  is a fundamental domain for the action of  on .

The Coxeter complex
The Coxeter complex  of  with respect to  is 
, where  is the multiplicative group of positive reals.

Examples

Finite dihedral groups
The dihedral groups  (of order 2n) are Coxeter groups, of corresponding type . These have the presentation .

The canonical linear representation of  is the usual reflection representation of the dihedral group, as acting on an -gon in the plane (so  in this case). For instance, in the case  we get the Coxeter group of type , acting on an equilateral triangle in the plane. Each reflection  has an associated hyperplane  in the dual vector space (which can be canonically identified with the vector space itself using the bilinear form , 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 -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 . 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  and . This group has the Coxeter presentation .

In this case, it is no longer possible to identify  with its dual space , as  is degenerate. It is then better to work solely with , 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. Taking the quotient 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 . A standard coset is a coset of the form , where  for some proper subset  of . For instance,  and .

The Coxeter complex  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:
Any two elements have a greatest lower bound.
The poset of elements less than or equal to any given element is isomorphic to the poset of subsets of  for some integer n.

Properties
The Coxeter complex associated to  has dimension . It is homeomorphic to a -sphere if W is finite and is contractible if W is infinite.

Every apartment of a spherical Tits building is a Coxeter complex.

See also
Buildings
Weyl group
Root system

References

Sources
 Peter Abramenko and Kenneth S. Brown, Buildings, Theory and Applications. Springer, 2008.

Group theory

Algebraic combinatorics
Geometric group theory
Mathematical structures