Geodesic chamber paths

PDF p. 13: §3.4, last paragraph before Proposition 3.29

Fix a finite-rank Coxeter system and a subset $J$ of its vertex set. In the relative arrangement $(\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}(J{)}^{\circ },{\mathcal{H}}^J)$, a chamber path is a finite nonempty sequence of chambers with consecutive chambers adjacent. Its length is its number of consecutive pairs. At each adjacency the path crosses the unique relative hyperplane supporting the shared wall; repeated crossings are counted with multiplicity.

A path is geodesic if its length is minimal among chamber paths with the same source and target. Equivalently, it crosses each hyperplane of ${\mathcal{H}}^J$ at most once. Hyperplanes are counted as distinct sets, even when different roots define the same relative hyperplane. A one-chamber path has length zero and is geodesic.

Chamber paths, concatenation and wall words

Tits cone intersections, relative roots and arrangements

Chambers, walls and adjacency

Local finiteness of the interior relative arrangement

Basic Tits-cone properties

Finite galleries and separating hyperplanes

The relative cone and its hyperplanes are specified by Tits cone intersections, relative roots and arrangements, and the path, endpoint and adjacency conventions by Chamber paths, concatenation and wall words and Chambers, walls and adjacency. The Tits cone is convex by Basic Tits-cone properties; intersecting it with the relative linear space and taking the interior gives an open convex set. The relative arrangement is locally finite there by Local finiteness of the interior relative arrangement. Hyperplanes missing the interior do not affect its chambers or crossings.

The chambers are open in the relative space, so their affine dimension is that of the relative space; the wall convention therefore agrees with the gallery interface. By Finite galleries and separating hyperplanes, each adjacency has a unique supporting hyperplane, any two chambers are joined by a finite gallery, and a gallery is shortest precisely when it crosses each separating hyperplane once and no other hyperplane. In any gallery, a separating hyperplane must be crossed an odd number of times and a nonseparating one an even number of times. Hence crossing every hyperplane at most once is equivalent to that shortest-gallery condition. The one-chamber path has no crossings and has the least possible length, zero.

The article reference attributes the terminology and equivalence. The justification uses the declared mathematical suppliers.