Coxeter Complexes
A Coxeter complex is the simplicial complex of cosets of standard parabolic subgroups in a Coxeter group. It is the apartment in the theory of buildings — the model fragment from which the full building is glued.
Definition
Let \((W, S)\) be a Coxeter system. The Coxeter complex \(\Sigma(W, S)\) is the abstract simplicial complex whose:
- vertices are cosets \(w W_{S \setminus \{s\}}\) for \(s \in S, w \in W\) (one orbit per simple reflection);
- simplices are cosets \(w W_I\) for proper subsets \(I \subsetneq S\), ordered by reverse inclusion.
The top-dimensional simplices (chambers) correspond to cosets \(w W_\emptyset = \{w\}\) and are in bijection with \(W\).
Geometric realisation
Spherical case (\(W\) finite)
If \(W\) is finite, \(\Sigma(W, S)\) has a natural realisation as a sphere triangulation: \(W\) acts on a Euclidean space \(V\) by the reflection representation, with the unit sphere \(S^{r-1} \subseteq V\) (\(r\) = rank) triangulated by the hyperplane arrangement. Each chamber is a spherical simplex.
Affine case (\(W\) affine)
For an affine Weyl group \(W_{\mathrm{aff}}\) acting on a Euclidean space \(\mathbb{E}\) (= \(\mathbb{R}^r\)), the Coxeter complex is the Euclidean apartment — a tiling of \(\mathbb{E}\) by congruent simplices (alcoves) under the action of \(W_{\mathrm{aff}}\).
Hyperbolic case
For Coxeter groups with no finite/affine reduction, the realisation is a tessellation of hyperbolic space.
Examples
\(W = S_3\) (type \(A_2\))
\(\Sigma(S_3, S)\) is the boundary of a triangle: 6 chambers (= permutations), arranged as a hexagonal cycle when realised as the triangulation of \(S^1\) by 6 arcs.
\(W = S_4\) (type \(A_3\))
\(\Sigma(S_4, S)\) triangulates a 2-sphere: 24 chambers, 36 edges, 14 vertices. It is the "permutahedron" boundary realised spherically.
Affine \(\tilde A_1\)
\(W_{\mathrm{aff}} = \langle s_0, s_1 : s_0^2 = s_1^2 = 1 \rangle = \mathbb{Z} \rtimes \mathbb{Z}/2\), the infinite dihedral group. The Coxeter complex is the real line tiled by unit intervals; chambers = intervals = 0-simplices' edges between adjacent vertices.
Affine \(\tilde A_2\)
The Euclidean plane tiled by equilateral triangles — the triangular lattice apartment. This is the prototype for affine apartments in type \(A_n\).
Structure
Galleries
A gallery from chamber \(C\) to chamber \(D\) is a sequence \(C = C_0, C_1, \ldots, C_n = D\) of chambers with \(C_i, C_{i+1}\) sharing a codimension-1 face. The minimum length is the gallery distance \(d(C, D)\), equal to the Coxeter group distance.
Walls and roots
Each codimension-1 simplex (panel) is contained in exactly two chambers. The hyperplane stabilising it (= wall) splits the Coxeter complex into two halves (roots in the building sense). Each wall corresponds to a reflection in \(W\).
Apartments are thin
A simplicial complex is thin if every panel is contained in exactly two chambers. Coxeter complexes are thin; this is the characteristic property that distinguishes them from buildings (which are "thick" except in apartments).
From Coxeter complex to building
A building \(\mathcal{B}\) of type \((W, S)\) is a simplicial complex obtained by gluing copies of \(\Sigma(W, S)\) (apartments) according to a system of charts:
- every two chambers lie in a common apartment;
- the gluings between apartments are isomorphisms.
The apartments are the Coxeter complexes; the building itself is "thick" — most panels are contained in many chambers.
Role for reductive groups
For a connected reductive group \(G\) over a field \(k\):
- Spherical building \(\Delta(G, k)\): built from the Coxeter complex of the finite Weyl group of \(G\). Vertices = maximal parabolics of \(G\); chambers = Borels.
- Affine (Bruhat-Tits) building: for \(G\) over a local field, built from the affine Coxeter complex of \(G\). Vertices = maximal parahoric subgroups; chambers = Iwahori subgroups.
The geometry of these buildings encodes everything from Bruhat decomposition (spherical case) to the local Langlands correspondence and the geometric Satake equivalence (affine case).
References
- Tits, Buildings of Spherical Type and Finite BN-Pairs (LNM 386, 1974), Ch. 2.
- Brown, Buildings (1989), Ch. III.
- Abramenko–Brown, Buildings: Theory and Applications (2008), Ch. 3.