Coxeter complexes are the basic objects that allow the construction of buildings; they form the apartments of a building.
The canonical representation is given by a vector space
with basis of formal symbols
, which is equipped with the symmetric bilinear form
This representation has several foundational properties in the theory of Coxeter groups; for instance,
as a reflection group, with the caveat that
It becomes important then to distinguish the representation
given by where the angled brackets indicate the natural pairing between
; this has faces the so-called walls,
Of major importance is the fact that
is a fundamental domain for the action of
is the multiplicative group of positive reals.
(of order 2n) are Coxeter groups, of corresponding type
is the usual reflection representation of the dihedral group, as acting on an
we get the Coxeter group of type
, acting on an equilateral triangle in the plane.
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.
This is a simplicial complex of dimension 1, and it can be colored by cotype.
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
This group has the Coxeter presentation
In this case, it is no longer possible to identify
In this case, the Tits cone is not the whole plane, but only the open upper half plane with the origin.
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.
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: The Coxeter complex associated to
-sphere if W is finite and is contractible if W is infinite.
Every apartment of a spherical Tits building is a Coxeter complex.