A Coxeter complex is a simplicial complex naturally associated with a Coxeter system $(W,S)$, where $W$ is a Coxeter group generated by the set of involutive generators $S$. The complex encodes the combinatorial and geometric structure of the reflections that generate $W$ and plays a central role in the theory of reflection groups, buildings, and geometric group theory.
Definition
For a Coxeter system $(W,S)$ with $|S| = n$, the Coxeter complex $\Sigma(W,S)$ is the abstract simplicial complex whose simplices are the cosets
$$ wW_J \qquad (w\in W,; J\subseteq S), $$
where $W_J$ denotes the standard parabolic subgroup generated by $J$. The inclusion relation
$$ wW_J \subseteq wW_{J'} \iff J' \subseteq J $$
gives the face relation of the complex. Maximal simplices (called chambers) correspond to the cosets $wW_{\emptyset}= {w}$, and therefore are in bijection with the elements of $W$.
Geometric Realisation
- Finite Coxeter groups: When $W$ is finite, $\Sigma(W,S)$ can be realised as a triangulation of the unit sphere $S^{n-1}$. Each chamber is an $n$-simplex, and the walls (codimension‑1 faces) correspond to the reflecting hyperplanes of $W$.
- Affine (and more general infinite) Coxeter groups: For affine Coxeter groups the complex can be realised as a tessellation of Euclidean space by simplices, while for arbitrary infinite $W$ the abstract complex is infinite and contractible.
Structural Properties
| Property | Description |
|---|---|
| Thinness | Every codimension‑1 face is contained in exactly two chambers. |
| Strongly flag-transitive action | The group $W$ acts simplicially, freely and transitively on chambers. |
| Spherical building | For finite $W$, $\Sigma(W,S)$ is a spherical building of type $(W,S)$. |
| Cohomology | When $W$ is finite, $\Sigma(W,S)$ is homeomorphic to an $(n-1)$-sphere, so its reduced homology is concentrated in top dimension. |
| Contractibility | For infinite $W$, the complex is contractible. |
Examples
- Type $A_{n-1}$ (symmetric group $S_n$): The Coxeter complex is the barycentric subdivision of the boundary of an $(n-1)$-simplex; chambers correspond to total orderings of ${1,\dots ,n}$.
- Type $B_n$ / $C_n$ (signed permutation group): The complex triangulates the sphere by chambers representing signed permutations.
- Affine type $\tilde{A}_{n-1}$: The complex is a tessellation of Euclidean $(n-1)$-space by simplices whose vertices lie on the affine hyperplane $\sum x_i = 0$.
Applications
- Buildings: Coxeter complexes provide the apartments of spherical and affine buildings, serving as local models for these highly symmetric geometric structures.
- Representation theory: The combinatorics of the complex underlies the Solomon–Tits theorem and the study of Hecke algebras.
- Topology of groups: The complex is used to construct $K(\pi,1)$ spaces for Artin groups, via the Deligne complex.
- Algebraic combinatorics: Flags of the complex correspond to chains in the Coxeter lattice; enumeration of chambers yields the order of $W$.
References
- J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.
- M. W. Davis, The Geometry and Topology of Coxeter Groups, Princeton University Press, 2008.
- J. Tits, “Buildings of Spherical Type and Finite BN‑Pairs,” Lecture Notes in Mathematics, vol. 386, Springer, 1974.
The Coxeter complex thus furnishes a canonical simplicial model on which a Coxeter group acts with maximal symmetry, linking algebraic properties of the group to geometric and topological structures.