Davis complex ($\Sigma (W,S)$)

The Davis complex is the geometric realization of the poset of cosets $w{W}_T$ for spherical subsets $T\subseteq S$. Equivalently, it is the basic construction of a Coxeter group obtained by gluing copies of the Davis chamber along mirrors. It is a contractible proper $W$-CW complex.

For a mirrored chamber $K$, the basic construction iswhere $(w,x)\sim (w^{\prime },x)$ when $w^{-1}{w}^{\prime }$ belongs to the subgroup generated by the mirrors containing $x$.