Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 4 / a

Here $W=⟨s,t:s^2=t^2=(st{)}^2=1⟩\cong C_2\times C_2$. In the Davis complex described by the basic construction of a Coxeter group, the fundamental chamber $K$ is the order complex ofit consists of two triangles sharing the edge from $\varnothing$ to $\{s,t\}$. Four labelled copies $wK$ are glued along their $s$- and $t$-mirrors. The result is a square subdivided from its centre to the midpoints and vertices of its boundary, with eight triangular chambers.

For the poset description, the spherical cosets consist of four singleton cosets $w{W}_{\varnothing }$, four cosets of rank-one parabolics, and the single coset $W_{\{s,t\}}=W$. Their flag realization has one vertex at each original square vertex, one at each edge midpoint, and one at the square centre; its flags are precisely the same eight triangles. Thus the two requested drawings are the chamber-gluing picture and the barycentric subdivision of a square, respectively.