Suppose $\sigma^*(w)(C)$ meets $\sigma^*(v)(C)$. Applying the homeomorphism $\sigma^*(v^{-1})$ shows that $C$ meets $\sigma^*(v^{-1}w)(C)$. If $v^{-1}w\ne1$, choose one of its left descents. Part b places its image of $C$ in the corresponding negative half-space, while $C$ lies in the positive half-space. This is impossible, so $v=w$. The union
$$
\mathcal C=\bigsqcup_{w\in W}\sigma^*(w)(C)
$$
is disjoint.

When $W$ is finite, the closures $\sigma^*(w)(D)$ are the simplicial chambers cut out by the reflecting hyperplanes. Intersecting them with a sphere centred at the origin gives simplices. A face of type $J\subseteq I$ has stabilizer the <standard parabolic subgroup> $W_J$, so its translates are indexed by cosets $wW_J$, with reverse inclusion of cosets encoding incidence. This is precisely the <Coxeter complex> of $(W,I)$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 111 / 3 / c / Solution

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For $J\subseteq I$, the standard parabolic subgroup is $W_J=\langle s_j:j\in J\rangle$. Its cosets index faces of a fixed type in the <Coxeter complex>.

Standard parabolic subgroup

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The Coxeter complex has faces represented by cosets $wW_J$ of <standard parabolic subgroup>[standard parabolic subgroups], ordered by reverse inclusion. Geometrically, its maximal simplices are the chambers of the Coxeter arrangement intersected with a sphere.

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 Incoming links: Coxeter complex