Coxeter complex

The Coxeter complex of a Coxeter system $(W,S)$ is the abstract simplicial complex whose faces are the cosets $w{W}_J$ of proper standard parabolic subgroups, ordered by reverse inclusion. Chambers correspond to elements $w\in W$; two chambers sharing a panel of type $s$ are $w$ and $ws$. The face $w{W}_J$ has type $S\setminus J$ and dimension $|S\setminus J|-1$ when $S$ is finite.

For a finite Coxeter group, the Geometric representation of a Coxeter group is a real reflection action with positive-definite Coxeter Gram matrix. Its reflecting hyperplanes cut the unit sphere into the chambers of this simplicial complex. For $S_3$ with its two simple transpositions, the complex is a circle subdivided into six edges. For an infinite Coxeter group, the geometric chamber construction instead uses the Tits cone; a decomposition of the entire unit sphere is not generally available. The Davis complex uses spherical standard parabolic subgroups to give a different, contractible complex.