= Coxeter complex
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The Coxeter complex of a <Coxeter system> $(W,S)$ is the abstract simplicial complex whose faces are the <cosets> $wW_J$ 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 $wW_J$ 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.
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