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Finite Reflection Arrangements and Spherical Coxeter Complexes

1 · Prerequisites

2 · Summary

Finite root hyperplanes divide Euclidean space into simplicial chambers. Their spherical sections give the Coxeter complex, with faces indexed by cosets and stabilizers proved rather than assumed.

The page is authored as three draft items, in dependency order. The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset identifies $V$ with $V^{*}$ by the positive definite form $B$ only here, and defines the finite arrangement $A$ of root hyperplanes, its chambers and closed chambers, the closed and open faces $\overline{C_{I}}$ and $C_{I}$ of the fundamental chamber, the spherical chamber complex on the unit sphere $S^{∣ S ∣ - 1}$ , and the coset face poset ${w W_{I} : I ⊊ S}$ ordered by reverse inclusion; clause (4) records explicitly that the definition asserts neither the tiling nor the face or triangulation identifications, which are the content of its recorded justifier.

The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere proves them: the Tits-cone criterion gives $U = V^{*}$ , the complement of the walls is the disjoint union of the open chambers $w C^{\circ}$ with closures $w C$ , the $B$ -dual basis describes each face $\overline{C_{I}}$ as the cone on ${v_{s} : s \notin I}$ , the dihedral angle between adjacent walls is $π / m (s, t)$ , the assignment $w W_{I} \mapsto w \overline{C_{I}}$ is a bijection onto the proper faces with the intersection formula $w \overline{C_{I}} \cap v \overline{C_{J}} = w \overline{C_{I \cup J \cup S (v^{- 1} w)}}$ , point stabilizers are $w W_{I} w^{- 1}$ , and the abstract coset complex is identified with a triangulation of the sphere $S^{∣ S ∣ - 1}$ by the radial normalization of an explicit simplexwise affine map. The empty case $S = \emptyset$ is separated first.

The longest element as the opposition of the chamber, and longest elements of finite parabolics proves for a general finite Coxeter system the unique $w_{0}$ with $w_{0} \cdot C = - C$ , equivalently $N (w_{0}) = Φ_{+}$ , its length $ℓ (w_{0}) = ∣ Φ_{+} ∣ = ∣ T ∣$ , the length-complement identities, $w_{0}^{2} = 1$ and the permutation $w_{0} s w_{0} = σ (s)$ of $S$ ; and for every finite standard parabolic $W_{I}$ the analogous longest element with $ℓ (w_{0} (I)) = ∣ Φ_{I} \cap V_{I}^{+} ∣$ .

Prerequisites and reading

Required earlier pages: finite-coxeter-diagrams-and-complete-classification, finite-lattice-projections-and-coxeter-chain-labels, further-trigonometric-identities-and-inverses. The companion finite-reflection-arrangements-and-spherical-coxeter-complexes-examples tests these constructions and conventions. Exact item dependencies and source reading limits are recorded in research/coxeter-scaffold/inventory.json and research/plan-coxeter-groups-track.md.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset

Definition

Let $(W, S)$ be a Coxeter system of finite type with $S$ finite, $n : = ∣ S ∣$ , and length function $ℓ$ ; thus $W$ is a finite group (Coxeter diagrams: edges, labels, components and finite type, clause (4), Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups). On $V : = R^{S}$ let $B$ be the Coxeter form, let $r_{a}$ be the reflection with normal $a$ (defined when $B (a, a) \neq 0$ ), and let $ρ : W \to G L (V)$ be the canonical reflection homomorphism with root system $Φ = {ρ (w) e_{s} : w \in W, s \in S} \subset V$ and reflection set $T = {w s w^{- 1} : w \in W, s \in S} \subset W$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Let $V^{*}$ be the algebraic dual with its dual action, its closed chamber $C$ , its interior $C^{\circ}$ and its root hyperplanes $H_{α}$ (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling).

Since $W$ is finite, $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite, clause (1), Disconnected diagrams, direct products, and comparison of invariant forms, clause (4)); hence $b : V \to V^{*}$ , $b (v) : = B (v, \cdot)$ , is a linear isomorphism with $b (ρ (w) v) = w \cdot b (v)$ for all $w \in W$ and $v \in V$ , because $B$ is $ρ$ -invariant (Descent of the reflection representation, unit root norms, and conjugation of reflections, Invertible linear maps, linear isomorphisms, and inverse linear maps). Identify $V$ with $V^{*}$ by $b$ only now, and transfer $C$ , $C^{\circ}$ and the $H_{α}$ along $b^{- 1}$ ; with the same letters

$C = {v \in V : B (v, e_{s}) \geq 0 for every s \in S}, C^{\circ} = {v \in V : B (v, e_{s}) > 0 for every s \in S}, H_{α} = {v \in V : B (v, α) = 0} .$

Give $V$ the metric topology of the inner product $B$ (Real and complex inner-product spaces and their induced length, Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement); it is canonical because $B$ is determined by $(W, S)$ , and $b$ is an isometry for $B$ and its transferred form, so topological notions may be moved across the identification.

(1) The finite reflection arrangement. Put $A : = {H_{α} : α \in Φ} .$ Each $α \in Φ$ satisfies $B (α, α) = 1 \neq 0$ (Descent of the reflection representation, unit root norms, and conjugation of reflections, clause (3)), so $H_{α}$ is the kernel of the nonzero linear functional $B (\cdot, α)$ and in particular contains $0$ ; $A$ is finite because $Φ$ is the image of the finite set $W \times S$ under $(w, s) \mapsto ρ (w) e_{s}$ ; and $A$ is permuted by $ρ (W)$ , since $ρ (w) H_{α} = H_{ρ (w) α}$ for all $w \in W$ , $α \in Φ$ (the form $B$ is $ρ$ -invariant). A chamber of $A$ is a connected component of $V ∖ ⋃_{α \in Φ} H_{α}$ , and a closed chamber is the closure of a chamber (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, Connected components, quasicomponents, and totally disconnected spaces, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement). For $I \subseteq S$ put $\overline{C_{I}} : = {v \in C : B (v, e_{s}) = 0 for s \in I}, C_{I} : = {v \in C : B (v, e_{s}) = 0 for s \in I and B (v, e_{s}) > 0 for s \notin I},$ the closed face and the open face of $C$ of type $I$ ; for $w \in W$ put $w C : = ρ (w) C$ , $w \overline{C_{I}} : = ρ (w) \overline{C_{I}}$ and $w C_{I} : = ρ (w) C_{I}$ .

(2) The spherical chamber complex. The form $B$ is an inner product on $V$ ; let $∥ v ∥_{B} : = B (v, v)^{1 / 2}$ and let $S^{n - 1} : = {v \in V : ∥ v ∥_{B} = 1}$ be the unit sphere, with the subspace topology (The induced length is a norm, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, Euclidean spheres and closed balls as subspaces of $R^{n}$ ); the exponent is notation for the unit sphere of the $n$ -dimensional inner-product space $(V, B)$ , and no dimension theory of spheres is claimed here. The spherical chamber belonging to $w \in W$ is $w C \cap S^{n - 1}$ , and its spherical faces are the sets $w \overline{C_{I}} \cap S^{n - 1}$ .

(3) The coset face poset. For $I \subseteq S$ put $W_{I} : = ⟨ s : s \in I ⟩ \leq W$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups) and let $w W_{I} : = {w u : u \in W_{I}}$ be a left coset (Left and right cosets $g H$ and $H g$ of a subgroup). The coset face poset is the set ${w W_{I} : w \in W, I ⊊ S}$ of cosets of proper standard parabolics, ordered by reverse inclusion as subsets of $W$ : $σ ⪯ τ$ if and only if $σ \supseteq τ$ .

(4) Abstentions. This definition asserts neither that the sets $w C^{\circ}$ are the connected components of $V ∖ ⋃_{α} H_{α}$ nor that they cover it, neither that the cosets $w W_{I}$ index the faces $w \overline{C_{I}}$ nor that the spherical faces triangulate $S^{n - 1}$ ; those assertions are proved in The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere ↗, the recorded justifier of this definition. Separately, the $B$ -dual family of the basis $(e_{s})$ is provided by The dual family $(b^{*})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{*} (c) = δ_{b c}$ , Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis and The dual family of a finite basis is a basis of the dual space, with the same dimension, and no Choice is used: $S$ and $W$ are finite and all objects here are finite-dimensional or set-theoretic.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-08Open item page →

The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere

Statement

Let $(W, S)$ be a Coxeter system of finite type with $S$ finite, $n : = ∣ S ∣$ , and let the arrangement $A$ , the chamber $C$ , its interior $C^{\circ}$ , the closed and open faces $\overline{C_{I}}$ and $C_{I}$ , the translated objects $w C$ , $w \overline{C_{I}}$ , $w C_{I}$ , the unit sphere $S^{n - 1}$ and the coset face poset be as in The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset. Let $ρ$ , $Φ = Φ_{+} ⊔ Φ_{-}$ , $V_{+}$ , $T$ , $ℓ$ be as in The canonical reflection homomorphism, roots, reflections, and the positive cone, Root sign coherence and the action of simple reflections on positive roots and Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups; let $S (u)$ denote the support of $u \in W$ , the set of letters occurring in a reduced expression of $u$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, clause (1)); and let $U = ⋃_{w \in W} w C \subseteq V^{*}$ be the Tits cone of the dual action with its negative-root sets $Neg (f) = {α \in Φ_{+} : f (α) < 0}$ (The Tits cone, its interior, and the negative-root set of a functional). Then:

(1) The chamber tiling. $U = V^{*}$ , and under the identification of the definition $V = ⋃_{w \in W} w C$ ; moreover $V ∖ ⋃_{α \in Φ} H_{α} = ⨆_{w \in W} w C^{\circ},$ the connected components of $V ∖ ⋃_{α} H_{α}$ are exactly the sets $w C^{\circ}$ $(w \in W)$ with closures $\overline{w C^{\circ}} = w C$ , so the closed chambers of $A$ are exactly the sets $w C$ ; the map $w \mapsto w C^{\circ}$ is a bijection from $W$ onto the set of chambers, distinct closed chambers have disjoint interiors, and every $W$ -orbit in $V$ meets $C$ in exactly one point.

(2) Simplicial chambers and their vertices. Let $v_{s} \in V$ be the $B$ -dual basis of $(e_{s})$ , i.e. $B (v_{s}, e_{t}) = δ_{s t}$ (The dual family $(b^{*})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{*} (c) = δ_{b c}$ , The dual family of a finite basis is a basis of the dual space, with the same dimension). Then $\overline{C_{I}} = {\sum_{s \notin I} λ_{s} v_{s} : λ_{s} \geq 0}, C_{I} = {\sum_{s \notin I} λ_{s} v_{s} : λ_{s} > 0}$ for every $I \subseteq S$ ; the closed chamber $C$ is the simplicial cone with extreme rays $R_{\geq 0} v_{s}$ $(s \in S)$ ; the vertices of the spherical simplex $C \cap S^{n - 1}$ are exactly the points $v_{s} / ∥ v_{s} ∥_{B}$ , in the precise sense that $C \cap S^{n - 1} \cap ⋂_{t \neq s} H_{e_{t}} = {v_{s} / ∥ v_{s} ∥_{B}}$ for every $s$ ; and the dihedral angle between the walls $H_{e_{s}}$ and $H_{e_{t}}$ is $π / m (s, t)$ in the following exact sense: the tangent sector ${v : B (v, e_{s}) \geq 0, B (v, e_{t}) \geq 0}$ of $C$ along the codimension-two face $C \cap H_{e_{s}} \cap H_{e_{t}}$ projects under the orthogonal projection $V \to R e_{s} + R e_{t}$ onto a sector in the two-plane $R e_{s} + R e_{t}$ bounded by its two lines $H_{e_{s}}$ and $H_{e_{t}}$ , and that sector has angle $π / m (s, t)$ . More generally each $w \overline{C_{I}}$ is a simplicial cone with the linearly independent generators $ρ (w) v_{s}$ $(s \notin I)$ . The walls of the chamber $w C$ are the root hyperplanes $w H_{e_{s}} = H_{ρ (w) e_{s}}$ .

(3) The face identification and the stabilisers. The assignment $w W_{I} \mapsto w \overline{C_{I}}$ is a well-defined bijection from the coset face poset onto the set of proper faces ${w \overline{C_{I}} : w \in W, I ⊊ S}$ (the remaining sets $w \overline{C_{S}}$ are all equal to ${0}$ , the common face of all chambers), and for all $w, v \in W$ and $I, J \subseteq S$ $w W_{I} = v W_{J} ⟺ w \overline{C_{I}} = v \overline{C_{J}}, w W_{I} \subseteq v W_{J} ⟺ v \overline{C_{J}} \subseteq w \overline{C_{I}},$ $w \overline{C_{I}} \cap v \overline{C_{J}} = w \overline{C_{I \cup J \cup S (v^{- 1} w)}}, w C_{I} \cap v C_{J} \neq \emptyset ⟺ w W_{I} = v W_{J},$ so the relative interiors of the faces partition $V$ ; moreover $V ∖ {0} = ⨆ w C_{I}$ , the disjoint union running over the cosets $w W_{I}$ with $I ⊊ S$ . For every $x \in w C_{I}$ one has ${Stab}_{W} (x) = w W_{I} w^{- 1}$ , and the setwise stabiliser ${w^{'} \in W : w^{'} w \overline{C_{I}} = w \overline{C_{I}}}$ of the face $w \overline{C_{I}}$ is the same subgroup $w W_{I} w^{- 1}$ .

(4) The spherical triangulation. Assume $n \geq 1$ . Then $S^{n - 1} = ⨆ (w C_{I} \cap S^{n - 1})$ , the union running over the cosets $w W_{I}$ with $I ⊊ S$ , and $Σ : = {w \overline{C_{I}} \cap S^{n - 1} : w \in W, I ⊊ S}$ is the set of nonempty faces of a finite spherical simplicial complex (adjoin the empty face): each member is the spherical simplex whose vertices are the points $ρ (w) v_{s} / ∥ v_{s} ∥_{B}$ $(s \notin I)$ in the same sense, the relative interiors $w C_{I} \cap S^{n - 1}$ are pairwise disjoint and cover $S^{n - 1}$ , and the intersection of two members is a common face, possibly empty, by (3); its face poset is isomorphic to the coset face poset by $w W_{I} \mapsto w \overline{C_{I}} \cap S^{n - 1}$ . Consequently the abstract simplicial complex $K$ with vertices the cosets $w W_{S ∖ {s}}$ $(w \in W, s \in S)$ and simplices the empty set and the sets ${w W_{S ∖ {s}} : s \notin I}$ $(w \in W, I ⊊ S)$ is a triangulation of $S^{n - 1}$ : the radial normalization $φ : ∣ K ∣ \to S^{n - 1}$ of the simplexwise affine map into $V$ that sends the barycentric coordinate at the vertex $w W_{S ∖ {s}}$ to the direction of $ρ (w) v_{s}$ is a continuous bijection, $∣ K ∣$ is compact because $K$ is finite (A finite simplicial complex has a compact Hausdorff realization) and $S^{n - 1}$ is Hausdorff, so $φ$ is a homeomorphism (The geometric realization of an abstract simplicial complex, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological). In particular $K$ has exactly $∣ W ∣$ maximal simplices, indexed by the chambers. For $S = \emptyset$ the group is trivial, $V = 0$ , and $S^{- 1} = \emptyset$ is triangulated by the complex with no vertices and sole simplex $\emptyset$ , whose realization is empty.

Facts & Assumptions

Given: A Coxeter system $(W, S)$ of finite type with $S$ finite, $n = ∣ S ∣$ , the space $V = R^{S}$ with the Coxeter form $B$ , the canonical reflection homomorphism $ρ$ with root system $Φ = Φ_{+} ⊔ Φ_{-}$ , the dual action with chamber $C$ , faces $\overline{C_{I}}$ , $C_{I}$ and Tits cone $U$ , and the identification $V ≅ V^{*}$ of The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset.

[F1]

Finite type: $W$ is finite, $Φ = {ρ (w) e_{s} : w \in W, s \in S}$ is the image of the finite set $W \times S$ and hence finite, $B$ is positive definite, $b (v) = B (v, \cdot)$ is a linear isomorphism $V \to V^{*}$ , and every $ρ (w)$ preserves $B$ (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite, clauses (1)-(2), Disconnected diagrams, direct products, and comparison of invariant forms, clause (4), Descent of the reflection representation, unit root norms, and conjugation of reflections, Coxeter diagrams: edges, labels, components and finite type).

[F2]

Roots and signs: $Φ = Φ_{+} ⊔ Φ_{-}$ , $Φ_{-} = - Φ_{+}$ , $e_{s} \in Φ_{+}$ for every $s \in S$ , every root $α$ satisfies $B (α, α) = 1$ and $α \neq 0$ , and for $g \in C^{\circ}$ one has $g (α) > 0$ for $α \in Φ_{+}$ and $g (α) < 0$ for $α \in Φ_{-}$ (Root sign coherence and the action of simple reflections on positive roots, clauses (2)-(3), Descent of the reflection representation, unit root norms, and conjugation of reflections, clause (3)).

[F3]

Chamber system and walls: the chambers are the sets $w C$ and $U = ⋃_{w \in W} w C$ ; with $H_{α} = {v \in V : B (v, α) = 0}$ , $\overline{C_{I}} = {v \in C : B (v, e_{s}) = 0 for s \in I}$ and $S (f) = {s \in S : f (e_{s}) = 0}$ , for all $w \in W$ , $s \in S$ one has $w H_{e_{s}} = H_{ρ (w) e_{s}}$ , and the walls of the chamber system are exactly the root hyperplanes $H_{α}$ , $α \in Φ$ (The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset, The Tits cone, its interior, and the negative-root set of a functional, clauses (1)-(2), Chamber collisions, point stabilizers, and the intersection rule, clause (1)).

[F4]

Collision and strict fundamental domain: if $f, g \in C$ , $w \in W$ and $w \cdot f = g$ , then $f = g$ and $w \in W_{S (f)}$ ; every $W$ -orbit contained in $U$ meets $C$ in exactly one point; the open chambers $w C^{\circ}$ $(w \in W)$ are pairwise disjoint (Chamber collisions, point stabilizers, and the intersection rule, clauses (3) and (6)).

[F5]

Topology: since $S$ is finite, $f \mapsto (f (e_{s}))_{s \in S}$ is a linear bijection $V^{*} \to R^{S}$ , and $d (f, g) : = \max_{s \in S} ∣ f (e_{s}) - g (e_{s}) ∣$ is the metric topology of $V^{*}$ in which all the assertions about open sets, interiors and connected components of $V^{*}$ are read (The Tits cone, its interior, and the negative-root set of a functional, clause (3), Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

[F6]

Finite-negativity criterion: for every $f \in V^{*}$ , $f \in U$ if and only if $Neg (f) = {α \in Φ_{+} : f (α) < 0}$ is finite (The finite-negativity criterion, the reduction step, and convexity of the Tits cone, clause (1)).

[F7]

Supports and standard parabolics: for every $w \in W$ the support $S (w)$ (the letters occurring in a reduced expression) is well defined, $W_{J} = {w \in W : S (w) \subseteq J}$ for $J \subseteq S$ , and $W_{J} \cap S = J$ ; moreover $(W_{J}, J)$ is a Coxeter system whose intrinsic length is the restriction of $ℓ$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, clauses (1)-(2)).

[F8]

The $B$ -dual family: there are elements $v_{s} \in V$ with $B (v_{s}, e_{t}) = δ_{s t}$ , and they form a basis of $V$ ; for $x = \sum_{t} μ_{t} v_{t}$ one has $B (x, e_{t}) = μ_{t}$ (The dual family $(b^{*})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{*} (c) = δ_{b c}$ , The dual family of a finite basis is a basis of the dual space, with the same dimension, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).

[F9]

Simplicial machinery: $∣ K ∣$ is the set of barycentric coordinate functions on the abstract simplicial complex $K$ with the weak topology of the closed simplices (An abstract simplicial complex, The geometric realization of an abstract simplicial complex); a finite complex has compact Hausdorff realization (A finite simplicial complex has a compact Hausdorff realization); a continuous bijection from a compact space to a Hausdorff space is a homeomorphism (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clause (3), Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological); and metric spaces are Hausdorff (Distinct points of a metric space have disjoint balls around them).

[F10]

Convexity and connectedness: a convex subset of $R^{n}$ $(n \geq 1)$ is path-connected and connected (Every convex subset of $R^{n}$ , in particular every ball and $R^{n}$ itself, is path-connected and hence connected, clause 1, Metric continuity characterisations, with countable choice for the sequential converse, clause (a)).

[F11]

Cosets: for $I \subseteq S$ and $w \in W$ the set $w W_{I} = {w u : u \in W_{I}}$ is a left coset of the subgroup $W_{I}$ (Left and right cosets $g H$ and $H g$ of a subgroup, The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[F12]

Continuity toolkit in finite dimensions: for a norm $N$ on a real vector space, $N (\sum_{j} u_{j}) \leq \sum_{j} N (u_{j})$ and $∣ N (u) - N (w) ∣ \leq N (u - w)$ (The finite and reverse triangle inequalities for a norm; and for $n \geq 1$ every norm $N$ on $R^{n}$ satisfies $N (x) \leq C ∥ x ∥_{1}$ and is Lipschitz, hence continuous, for $d_{2}$ , clause 1); and sums, scalar multiples and (where the denominator does not vanish) quotients of continuous real-valued functions are continuous (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, Metric continuity characterisations, with countable choice for the sequential converse, clause (a)).

[F13]

For $0 < θ \leq π / 2$ , $\sin θ > 0$ , $\sin^{2} θ + \cos^{2} θ = 1$ , and $\arccos (\cos θ) = θ$ (Signs, monotonicity intervals, and ranges of sine and cosine, Quarter-turn values and shifts by pi/2 and pi, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Principal inverse sine and inverse cosine). The principal angle of two $B$ -unit vectors is here defined as the arccosine of their inner product.

Proof

technique · direct; the case $S=\emptyset$ is separated first and every later step assumes $S\ne\emptyset$

1.1givenF1algebra

If $S = \emptyset$ then $W = {1}$ , $V = V^{*} = {0}$ , $Φ = \emptyset$ , $C = C^{\circ} = {0} = \overline{C_{\emptyset}} = C_{S}$ and the coset face poset is empty, so (1)-(3) hold in the vacuous form described in the statement and (4) is exactly its stated convention for $S = \emptyset$ ; assume $S \neq \emptyset$ from now on, so that $n \geq 1$ .

1.2F1F2F6algebra

By [F1] the root system $Φ$ is finite, so $Neg (f) \subseteq Φ_{+}$ is finite for every $f \in V^{*}$ ; the criterion [F6] therefore gives $V^{*} \subseteq U$ , and $U \subseteq V^{*}$ holds by definition, so $U = V^{*}$ and every $f \in V^{*}$ lies in some chamber $w C$ .

1.3F8F1algebra

By [F8] the $B$ -dual basis $(v_{s})_{s \in S}$ exists, with $B (v_{s}, e_{t}) = δ_{s t}$ and $B (x, e_{t}) = μ_{t}$ for $x = \sum_{t} μ_{t} v_{t}$ ; hence $C = {\sum_{s} λ_{s} v_{s} : λ_{s} \geq 0}$ , $\overline{C_{I}} = {\sum_{s \notin I} λ_{s} v_{s} : λ_{s} \geq 0}$ and $C_{I} = {\sum_{s \notin I} λ_{s} v_{s} : λ_{s} > 0}$ for every $I \subseteq S$ , so $C_{S} = {0}$ and $C_{I} \neq \emptyset$ for $I ⊊ S$ ; and the cone $C$ has extreme rays $R_{\geq 0} v_{s}$ , because $v_{s} = x + y$ with $x = \sum μ_{t} v_{t}$ , $y = \sum ν_{t} v_{t}$ in $C$ forces $μ_{t} + ν_{t} = 0$ for $t \neq s$ and $μ_{s} + ν_{s} = 1$ with $μ, ν \geq 0$ , hence $x, y \in R_{\geq 0} v_{s}$ ; any vector with at least two positive coefficients splits into two nonproportional vectors of $C$ , so it spans no extreme ray.

1.4F2F3algebra

For $g \in C^{\circ}$ and $α \in Φ$ one has $α \in Φ_{+}$ or $α \in Φ_{-}$ , and $g (α) \neq 0$ by [F2]; for every $w \in W$ the point $ρ (w) g$ lies in $w C^{\circ}$ and $B (ρ (w) g, α) = B (g, ρ (w)^{- 1} α) \neq 0$ because $Φ$ is stable under $ρ$ , so $ρ (w) g \notin H_{α}$ and $w C^{\circ} \cap H_{α} = \emptyset$ .

2.1F5F10F1step 1.3F12algebra

$C^{\circ} = ⋂_{s \in S} {f : f (e_{s}) > 0}$ is open, as a finite intersection of preimages of the open half-line under the coordinate functionals $f \mapsto f (e_{s})$ , which are continuous for the topology of [F5]; it is convex, because those functionals are linear; and in the coordinates $V^{*} ≅ R^{S}$ it is a nonempty convex subset, hence path-connected and connected by [F10]. Each $w C^{\circ} = ρ (w) C^{\circ}$ is the image of $C^{\circ}$ under the linear isomorphism $ρ (w)$ [F1], whose coordinate matrix and inverse give Lipschitz maps for the metric of [F5], so $w C^{\circ}$ too is open and connected.

2.2step 1.3algebra

From step 1.3: $\overline{C_{I}} \cap \overline{C_{J}} = \overline{C_{I \cup J}}$ , and $\overline{C_{I}} \subseteq \overline{C_{J}}$ if and only if $J \subseteq I$ ; hence $\overline{C_{I}} = \overline{C_{J}}$ if and only if $I = J$ . Moreover $C_{I}$ is the relative interior of $\overline{C_{I}}$ in its affine span, which is $span {v_{s} : s \notin I}$ , because by step 1.3 the relative interior consists exactly of the combinations with all coefficients positive.

2.3step 1.3F7algebra

For every $I \subseteq S$ each $u \in W_{I}$ fixes $\overline{C_{I}}$ pointwise and fixes each $v_{s}$ with $s \notin I$ : by [F7] $u$ is a product of elements of $I$ , so it suffices to check the generators; for $s \in I$ and $x \in \overline{C_{I}} \subseteq C$ one has $B (x, e_{s}) = 0$ , hence $r_{s} (x) = x - 2 B (x, e_{s}) e_{s} = x$ by the reflection formula, while for $t \in I$ and $s \notin I$ one has $t \neq s$ and $B (v_{s}, e_{t}) = δ_{s t} = 0$ , hence $r_{t} (v_{s}) = v_{s}$ .

2.4step 1.2step 1.3F4algebra

Every $W$ -orbit in $V$ meets $C$ in exactly one point, by the strict-fundamental-domain clause [F4] together with $U = V^{*}$ (step 1.2); and the map $w \mapsto w C^{\circ}$ is injective: if $w C^{\circ} = v C^{\circ}$ and $g \in C^{\circ}$ (nonempty by step 1.3), then $g = u \cdot h$ for some $h \in C^{\circ}$ , where $u : = w^{- 1} v$ , and the collision rule [F4] applied to $g \in C$ and $h \in C$ gives $h = g$ and $u \in W_{S (g)} = W_{\emptyset} = {1}$ , so $w = v$ .

2.5step 1.3F1F3F13algebra

Let $s \neq t$ , put $θ : = π / m (s, t)$ , $c : = \cos θ$ , and $P : = R e_{s} + R e_{t}$ . The order convention in Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups and finiteness of $W$ give $m (s, t) < \infty$ , so $0 < θ \leq π / 2$ . In the $B$ -orthonormal coordinates $a = e_{s}$ , $b = (e_{t} + c e_{s}) / \sin θ$ of $P$ , one has $e_{t} = - c a + \sin θ b$ , using the Gram entries and [F13]. Orthogonal projection onto $P$ preserves the two values $B (v, e_{s})$ and $B (v, e_{t})$ , because its kernel is $P^{⊥} = H_{e_{s}} \cap H_{e_{t}}$ ; every point of the resulting sector in $P$ is its own projection. Write a point of $P$ as $x a + y b$ . The sector is $x \geq 0$ , $- c x + \sin θ y \geq 0$ , bounded by the rays through $b$ and $\sin θ a + c b$ . These are unit vectors with inner product $c$ , and the nonnegative sector between them has principal angle $\arccos c = θ$ by [F13]. This proves the stated dihedral-angle formula.

3.1step 1.2step 1.3step 1.4step 2.1F2F4F5algebra

$V ∖ ⋃_{α \in Φ} H_{α} = ⨆_{w \in W} w C^{\circ}$ , these sets are exactly the connected components of $V ∖ ⋃_{α} H_{α}$ , and $\overline{w C^{\circ}} = w C$ : the inclusion $\supseteq$ holds by step 1.4; for $\subseteq$ take $f \notin ⋃_{α} H_{α}$ , so $f \in w C$ for some $w$ by step 1.2, and $(w^{- 1} \cdot f) (e_{s}) = f (ρ (w) e_{s}) \neq 0$ for every $s$ because $ρ (w) e_{s} \in Φ$ [F2], whence $w^{- 1} \cdot f \in C^{\circ}$ and $f \in w C^{\circ}$ ; the sets $w C^{\circ}$ are pairwise disjoint by [F4], open and connected by step 2.1 and nonempty by step 1.3, so each is a component, since a connected set meeting two of them would be separated by the partition into $w C^{\circ}$ and its complement, both of which are open in $V$ minus the hyperplanes; finally each $w C$ is closed, being the intersection of the finitely many closed half-spaces ${v : B (v, ρ (w) e_{s}) \geq 0}$ , and every $x \in w C$ is a limit of points $ρ (w) (w^{- 1} x + ε u) \in w C^{\circ}$ with $u = \sum_{s} v_{s}$ and $ε > 0$ , because $B (w^{- 1} x + ε u, e_{s}) = B (w^{- 1} x, e_{s}) + ε > 0$ and $B (u, e_{s}) = 1$ ; hence $\overline{w C^{\circ}} = w C$ and the closed chambers are exactly the sets $w C$ .

3.2step 2.3F7F11algebra

If $w W_{I} = v W_{J}$ with $I, J \subseteq S$ , then $u : = w^{- 1} v$ satisfies $u W_{J} = W_{I}$ , so $1 = u j$ for some $j \in W_{J}$ (because $1 \in W_{I}$ ), giving $u = j^{- 1} \in W_{J}$ and $W_{I} = u W_{J} = W_{J}$ , so $I = W_{I} \cap S = W_{J} \cap S = J$ by [F7]; then $u \in W_{I}$ fixes $\overline{C_{I}}$ pointwise by step 2.3, so $w \overline{C_{I}} = v u^{- 1} \overline{C_{I}} = v \overline{C_{I}} = v \overline{C_{J}}$ , and the assignment $w W_{I} \mapsto w \overline{C_{I}}$ is well defined.

3.3step 1.3step 2.3F3F4F7algebra

For all $w, v \in W$ and $I, J \subseteq S$ the intersection formula $w \overline{C_{I}} \cap v \overline{C_{J}} = w \overline{C_{I \cup J \cup S (v^{- 1} w)}}$ holds: if $x$ lies in the intersection, then $f : = w^{- 1} x \in \overline{C_{I}}$ and $g : = v^{- 1} x \in \overline{C_{J}}$ are points of $C$ with $(v^{- 1} w) \cdot f = g$ , so [F4] gives $f = g$ and $v^{- 1} w \in W_{S (f)}$ ; then $I \subseteq S (f)$ and $J \subseteq S (f)$ by [F3] and $S (v^{- 1} w) \subseteq S (f)$ by [F7], so $f \in \overline{C_{I \cup J \cup S (v^{- 1} w)}}$ and $x \in w \overline{C_{I \cup J \cup S (v^{- 1} w)}}$ ; conversely if $x = w f$ with $f \in \overline{C_{I \cup J \cup S (v^{- 1} w)}}$ , then $S (v^{- 1} w) \subseteq S (f)$ and hence $v^{- 1} w \in W_{S (f)}$ by [F7], so $v^{- 1} w$ fixes $\overline{C_{S (f)}}$ pointwise by step 2.3, whence $v^{- 1} x = v^{- 1} w f = f \in \overline{C_{J}}$ and $x \in w \overline{C_{I}} \cap v \overline{C_{J}}$ .

3.4step 1.2step 1.3step 2.3F4algebra

The faces $w C_{I}$ with $I ⊊ S$ are nonempty by step 1.3, cover $V ∖ {0}$ and have pairwise disjoint relative interiors, so $V ∖ {0} = ⨆ w C_{I}$ over the cosets $w W_{I}$ with $I ⊊ S$ ; moreover $w C_{I} \cap v C_{J} \neq \emptyset$ forces $w W_{I} = v W_{J}$ and hence $w C_{I} = v C_{J}$ : if $x$ lies in $w C_{I} \cap v C_{J}$ , then $f : = w^{- 1} x \in C_{I}$ and $g : = v^{- 1} x \in C_{J}$ are points of $C$ with $(v^{- 1} w) \cdot f = g$ , so [F4] gives $f = g$ and $v^{- 1} w \in W_{S (f)} = W_{I}$ by step 1.3, and then $v = w u^{- 1}$ with $u : = v^{- 1} w \in W_{I}$ and $v C_{I} = w u^{- 1} C_{I} = w C_{I}$ by step 2.3; for the covering, let $x \neq 0$ and choose $w$ with $x \in w C$ by step 1.2, so that $w^{- 1} x \neq 0$ and therefore $S (w^{- 1} x) ⊊ S$ (else $w^{- 1} x = 0$ by step 1.3), giving $w^{- 1} x \in C_{S (w^{- 1} x)}$ by step 1.3 and $x \in w C_{S (w^{- 1} x)}$ .

3.5step 1.3step 2.3F4algebra

For $f \in C$ one has ${Stab}_{W} (f) = W_{S (f)}$ : if $h \cdot f = f$ , then the collision rule [F4] applied to the two points $f \in C$ gives $h \in W_{S (f)}$ , and conversely every $u \in W_{S (f)}$ fixes $\overline{C_{S (f)}} ∋ f$ pointwise by step 2.3; hence for $x \in w C_{I}$ , writing $x = w f$ with $f \in C_{I}$ , one has $S (f) = I$ by step 1.3 and ${Stab}_{W} (x) = w {Stab}_{W} (f) w^{- 1} = w W_{I} w^{- 1}$ .

3.6step 1.3step 2.2F1F3F8algebra

For every $s \in S$ one has $C \cap S^{n - 1} \cap ⋂_{t \neq s} H_{e_{t}} = {v_{s} / ∥ v_{s} ∥_{B}}$ : a point $x$ of that intersection equals $μ_{s} v_{s}$ with $μ_{s} \geq 0$ by step 1.3, and $∥ x ∥_{B} = 1$ forces $μ_{s} = 1 / ∥ v_{s} ∥_{B} \neq 0$ , while each $v_{s} / ∥ v_{s} ∥_{B}$ does lie in the intersection; moreover $ρ (w) \overline{C_{I}} = {\sum_{s \notin I} λ_{s} ρ (w) v_{s} : λ_{s} \geq 0}$ with the vectors $ρ (w) v_{s}$ $(s \notin I)$ linearly independent because $ρ (w)$ is invertible [F1], and the walls of $w C$ are $w H_{e_{s}} = H_{ρ (w) e_{s}}$ by [F3].

4.1step 1.3step 2.2step 2.3step 3.2step 3.3step 3.4F7algebra

For all $w, v \in W$ and $I, J \subseteq S$ : (a) $w W_{I} = v W_{J}$ if and only if $w \overline{C_{I}} = v \overline{C_{J}}$ , because the forward implication is step 3.2 and if $w \overline{C_{I}} = v \overline{C_{J}}$ then the intersection formula (step 3.3) gives $w \overline{C_{I \cup J \cup S (v^{- 1} w)}} = w \overline{C_{I}}$ , so $I = I \cup J \cup S (v^{- 1} w)$ by step 2.2, whence $J \subseteq I$ and $v^{- 1} w \in W_{I}$ by [F7], and symmetrically $I \subseteq J$ , so $I = J$ and $w W_{I} = v W_{I} = v W_{J}$ ; (b) $w W_{I} \subseteq v W_{J}$ is equivalent to $v^{- 1} w \in W_{J}$ and $I \subseteq J$ : containment gives $w \in v W_{J}$ and, after multiplying by $w^{- 1}$ , $W_{I} \subseteq W_{J}$ , whence $I \subseteq J$ by [F7]; conversely these conditions give containment. By step 3.3, $v \overline{C_{J}} \subseteq w \overline{C_{I}}$ is equivalent to $v \overline{C_{J \cup I \cup S (w^{- 1} v)}} = v \overline{C_{J}}$ , hence to $I \subseteq J$ and $S (w^{- 1} v) \subseteq J$ by step 2.2, which is the same pair of conditions by [F7] and closure of $W_{J}$ under inverses; (c) $w C_{I} \cap v C_{J} \neq \emptyset$ implies $w W_{I} = v W_{J}$ by step 3.4, and conversely $w W_{I} = v W_{J}$ gives $I = J$ and $v^{- 1} w \in W_{I}$ by (a), hence $v C_{I} = w C_{I}$ by step 2.3 and this face is nonempty by step 1.3. Consequently $w W_{I} \mapsto w \overline{C_{I}}$ is a bijection from the coset face poset onto the set of proper faces, and by (b) it reverses inclusions, so it is an isomorphism from the coset face poset ordered by reverse inclusion onto the face poset ordered by containment.

4.2step 2.2step 3.2step 3.6F7F9algebra

Define $V e r t : = {w W_{S ∖ {s}} : w \in W, s \in S}$ and $σ (w W_{I}) : = {w W_{S ∖ {s}} : s \notin I} \subseteq V e r t$ for $w \in W$ , $I ⊊ S$ , and put $\hat{u} (w W_{S ∖ {s}}) : = ρ (w) v_{s} / ∥ ρ (w) v_{s} ∥_{B}$ . The direction is well defined: if $w W_{S ∖ {s}} = w^{'} W_{S ∖ {s^{'}}}$ , then step 3.2 gives $S ∖ {s} = S ∖ {s^{'}}$ and $w \overline{C_{S ∖ {s}}} = w^{'} \overline{C_{S ∖ {s}}}$ , so $s = s^{'}$ and, by step 3.6 applied to these one-dimensional cones, $ρ (w) v_{s}$ and $ρ (w^{'}) v_{s}$ are positive multiples of one another, whence $∥ ρ (w) v_{s} ∥_{B} = ∥ v_{s} ∥_{B}$ because $B$ is $ρ$ -invariant [F1] and the unit directions agree. The simplex is well defined: for $s \notin I$ and $u \in W_{I}$ one has $W_{I} \subseteq W_{S ∖ {s}}$ , hence $w u W_{S ∖ {s}} = w W_{S ∖ {s}}$ and $σ (w u W_{I}) = σ (w W_{I})$ . For $I ⊊ S$ , the intersection of its vertex cosets is $⋂_{s \notin I} w W_{S ∖ {s}} = w W_{I}$ , since [F7] identifies the intersection of these subgroups with the support condition $S (u) \subseteq I$ . Thus the vertex set determines the original coset. Finally a subset of $σ (w W_{I})$ keeping precisely the indices $s \in T \subseteq S ∖ I$ is $σ (w W_{S ∖ T})$ , so $K : = {\emptyset} \cup {σ (w W_{I}) : w \in W, I ⊊ S}$ is closed under subsets, contains the empty set and every singleton, and is finite.

5.1step 3.5step 4.1F11algebra

For $x \in w C_{I}$ the setwise stabiliser of the face $w \overline{C_{I}}$ is $w W_{I} w^{- 1}$ : one has $w^{'} w \overline{C_{I}} = w \overline{C_{I}}$ if and only if $w^{'} w W_{I} = w W_{I}$ by step 4.1(a), that is, if and only if $w^{'} \in w W_{I} w^{- 1}$ ; this subgroup contains ${Stab}_{W} (x) = w W_{I} w^{- 1}$ from step 3.5.

5.2step 3.6step 4.2F5F9F12algebra

Define $φ (α) : = (\sum_{v \in supp α} α (v) \hat{u} (v)) / ∥ \sum_{v \in supp α} α (v) \hat{u} (v) ∥_{B}$ for $α \in ∣ K ∣$ . This is well defined and takes values in $S^{n - 1}$ : on a simplex $σ (w W_{I})$ containing $supp α$ the sum is $\sum_{s \notin I} α (w W_{S ∖ {s}}) ρ (w) v_{s} / ∥ v_{s} ∥_{B}$ , a combination with nonnegative coefficients, not all zero, of the linearly independent vectors $ρ (w) v_{s}$ of step 3.6, so the numerator does not vanish; and if $α$ lies in two simplices, both contain the minimal simplex $supp α$ , on which the formula is the same. On each closed simplex $∣ σ (w W_{I}) ∣$ , which is compact and carries the Euclidean simplex topology of [F9], the barycentric coordinates $α \mapsto α (w W_{S ∖ {s}})$ are continuous, so the numerator is continuous as a map into $V$ (finite sums of scalar multiples of the fixed vectors, read in the coordinates of [F5]) and the denominator is a continuous positive real function by [F12], so $φ ∣_{∣ σ (w W_{I}) ∣}$ is continuous by [F12]; every simplex of $K$ is a face of one of the finitely many maximal simplices, whose traces are therefore continuous, and $∣ K ∣$ carries the weak topology of [F9], so $φ$ is continuous.

5.3step 1.3step 3.4step 3.6step 4.2algebra

The map $φ$ is surjective: for $x \in S^{n - 1}$ step 3.4 gives a coset $w W_{I}$ with $x \in w C_{I}$ , so $x = \sum_{s \notin I} λ_{s} ρ (w) v_{s}$ with all $λ_{s} > 0$ by step 1.3; putting $Λ : = \sum_{s \notin I} λ_{s} ∥ v_{s} ∥_{B}$ and $α (w W_{S ∖ {s}}) : = λ_{s} ∥ v_{s} ∥_{B} / Λ$ on the vertices of $σ (w W_{I})$ and $α : = 0$ elsewhere defines a point of $∣ σ (w W_{I}) ∣ \subseteq ∣ K ∣$ whose numerator is $w (\sum_{s \notin I} (λ_{s} / Λ) v_{s})$ , so that $φ (α) = x / ∥ x ∥_{B} = x$ . It is injective: if $φ (α) = x$ and $σ (w W_{I})$ is the minimal simplex supporting $α$ , so that all $α (w W_{S ∖ {s}}) > 0$ for $s \notin I$ , then the defining identity expresses $w^{- 1} x$ as the positive multiple $y / ∥ y ∥_{B}$ of $y : = \sum_{s \notin I} (α (w W_{S ∖ {s}}) / ∥ v_{s} ∥_{B}) v_{s}$ ; since $w^{- 1} x \in C_{I}$ is itself the combination $\sum_{s \notin I} λ_{s} v_{s}$ with $λ_{s} > 0$ (step 1.3) and $(v_{s})$ is a basis, the coefficients satisfy $α (w W_{S ∖ {s}}) = ∥ y ∥_{B} λ_{s} ∥ v_{s} ∥_{B}$ , and the coset $w W_{I}$ together with the numbers $λ_{s}$ is determined by $x$ (steps 3.4 and 1.3), summing the coefficients to $1$ gives $∥ y ∥_{B} = 1 / (\sum_{s \notin I} λ_{s} ∥ v_{s} ∥_{B})$ , so $α$ is determined by $x$ .

6.1step 1.1step 3.4step 3.6step 4.1step 4.2step 5.2step 5.3F9algebra∎

The realization $∣ K ∣$ is compact by [F9] and finiteness of $K$ (step 4.2), the sphere $S^{n - 1}$ is Hausdorff, since distinct points are separated by the intersections with $S^{n - 1}$ of disjoint metric balls in $V$ [F9], and $φ$ is a continuous bijection by steps 5.2 and 5.3; hence $φ$ is a homeomorphism by [F9] and $K$ is a triangulation of $S^{n - 1}$ , with the members of $Σ$ , the relative interiors and the face poset as described in steps 3.6, 3.4 and 4.1. The maximal simplices of $K$ are exactly the $σ (w W_{\emptyset}) = {w W_{S ∖ {s}} : s \in S}$ , one for each $w \in W$ : indeed $σ (w W_{I}) \subseteq σ (v W_{J})$ holds if and only if $v W_{J} \subseteq w W_{I}$ (both sides are equivalent to the pair of conditions $J \subseteq I$ and $v^{- 1} w \in W_{I}$ , by the argument of step 4.1(b) applied to vertex sets), so a simplex is maximal exactly when $I = \emptyset$ , and $σ (w W_{\emptyset}) = σ (v W_{\emptyset})$ forces $w = v$ . Thus $K$ has exactly $∣ W ∣$ maximal simplices, indexed by the chambers $w C$ , and the case $S = \emptyset$ was disposed of in step 1.1.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-10-08Open item page →

The longest element as the opposition of the chamber, and longest elements of finite parabolics

Statement

Let $(W, S)$ be a Coxeter system with $S$ finite, length function $ℓ$ , canonical reflection representation $ρ$ on $V$ , form $B$ , root system $Φ = Φ_{+} ⊔ Φ_{-}$ and reflections $T$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The canonical reflection homomorphism, roots, reflections, and the positive cone, Root sign coherence and the action of simple reflections on positive roots), with inversion sets $N (w) = {α \in Φ_{+} : ρ (w) α \in Φ_{-}}$ (The geometric inversion set $N (w)$ of an element of a Coxeter group), and let $C$ and $C^{\circ}$ be the chamber and its interior in $V^{*}$ (The dual action, chambers, faces, and root hyperplanes).

(1) The opposition and the longest element of a finite Coxeter system. Suppose $W$ is finite. Then:

(i) there is a unique $w_{0} \in W$ with $w_{0} \cdot C = - C$ ; equivalently $w_{0}$ is the unique element of $W$ with $N (w_{0}) = Φ_{+}$ (equivalently with $N (w_{0}^{- 1}) = Φ_{+}$ );

(ii) $ℓ (w_{0}) = ∣ N (w_{0}) ∣ = ∣ Φ_{+} ∣ = ∣ T ∣$ , and $ρ (w_{0}) Φ_{+} = Φ_{-}$ ;

(iii) $ℓ (w_{0} w) = ℓ (w_{0}) - ℓ (w)$ and $ℓ (w w_{0}) = ℓ (w_{0}) - ℓ (w)$ for every $w \in W$ ; consequently $ℓ (w_{0}) = \max {ℓ (w) : w \in W}$ and $w_{0}$ is the unique element of that length (the element of longest length);

(iv) $w_{0}^{2} = 1$ ;

(v) $w_{0} S w_{0}^{- 1} = S$ , and there is a permutation $σ$ of $S$ with $ρ (w_{0}) e_{s} = - e_{σ (s)}$ , equivalently $w_{0} s w_{0} = σ (s)$ , for every $s \in S$ .

(2) Longest elements of finite parabolics. Let $I \subseteq S$ with $W_{I} = ⟨ s : s \in I ⟩$ finite (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups); this holds automatically for every $I \subseteq S$ when $W$ is finite. Then $W_{I}$ has a unique longest element $w_{0} (I)$ : it is the unique $u \in W_{I}$ with $ℓ (u w) = ℓ (u) - ℓ (w)$ for every $w \in W_{I}$ , and it satisfies $w_{0} (I)^{2} = 1$ , $ℓ (w_{0} (I)) = \max {ℓ (w) : w \in W_{I}}$ , $ℓ (w_{0} (I) w) = ℓ (w_{0} (I)) - ℓ (w)$ and $ℓ (w w_{0} (I)) = ℓ (w_{0} (I)) - ℓ (w)$ for every $w \in W_{I}$ , and $w_{0} (I) I w_{0} (I)^{- 1} = I$ . With the parabolic subsystem $V_{I} : = s p a n {e_{s} : s \in I}$ , $V_{I}^{+} : = {\sum_{s \in I} λ_{s} e_{s} : λ_{s} \geq 0}$ and $Φ_{I} : = {ρ (u) e_{s} : u \in W_{I}, s \in I}$ , one has $ℓ (w_{0} (I)) = ∣ Φ_{I} \cap V_{I}^{+} ∣$ .

Facts & Assumptions

Given: A Coxeter system $(W, S)$ with $S$ finite, the space $V = R^{S}$ with Coxeter form $B$ , the canonical reflection homomorphism $ρ$ , root system $Φ = Φ_{+} ⊔ Φ_{-}$ , reflections $T$ , length $ℓ$ , inversion sets $N (w)$ , chamber $C$ and its interior $C^{\circ}$ in $V^{*}$ . For part (1), $W$ is finite, so the identification $V ≅ V^{*}$ of The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset is available; for part (2) only the subsystem is identified with its dual after its finiteness is assumed.

[F1]

Structure and elementary length facts: $B$ is symmetric bilinear with $B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ (or $- 1$ for $m (s, t) = \infty$ ), every $ρ (w)$ preserves $B$ , the reflection formula is $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ for $B (a, a) \neq 0$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, clauses (1)-(2), Descent of the reflection representation, unit root norms, and conjugation of reflections); the length is the least word length, so $ℓ (x) = 0$ if and only if $x = 1$ , reversing a word gives $ℓ (w^{- 1}) = ℓ (w)$ , and $ℓ (x) = 1$ means $x \in S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups); and every root satisfies $B (α, α) = 1$ (Descent of the reflection representation, unit root norms, and conjugation of reflections, clause (3)).

[F2]

Tiling of the finite chamber system: $V ∖ ⋃_{α \in Φ} H_{α}$ has the sets $w C^{\circ}$ $(w \in W)$ as its connected components, the map $w \mapsto w C^{\circ}$ is a bijection from $W$ onto the set of chambers and $\overline{w C^{\circ}} = w C$ (The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere, clause (1)).

[F3]

Root signs: $Φ = Φ_{+} ⊔ Φ_{-}$ , $Φ_{-} = - Φ_{+}$ and $e_{s} \in Φ_{+}$ for every $s$ ; moreover for $α \in Φ$ one has $α \in Φ_{+}$ if and only if $f (α) > 0$ for every $f \in C^{\circ}$ , and $α \in Φ_{-}$ if and only if $f (α) < 0$ for every $f \in C^{\circ}$ (Root sign coherence and the action of simple reflections on positive roots, clauses (2)-(3)).

[F4]

Inversions and the root-reflection dictionary: $∣ N (w) ∣ = ℓ (w)$ for every $w$ , the map $Φ_{+} \to T$ , $α \mapsto t_{α}$ , is a bijection, and for $α, β \in Φ$ one has $t_{α} = t_{β}$ if and only if $α = \pm β$ (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange, clauses (1)-(2)).

[F5]

Conjugation: for $w \in W$ , $s \in S$ and $α \in Φ$ one has $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ and $ρ (t_{α}) = r_{α}$ (Descent of the reflection representation, unit root norms, and conjugation of reflections, clause (4), The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange, clause (1)(ii)).

[F6]

Standard parabolics: for $I \subseteq S$ the pair $(W_{I}, I)$ is a Coxeter system, its intrinsic length function agrees on $W_{I}$ with the restriction of $ℓ$ , and $W_{I} \cap S = I$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification, clauses (1)-(2)).

[F7]

Universal property of the presented group: any assignment of generators satisfying the Coxeter relations extends uniquely to a homomorphism (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

Proof

technique · direct; (1) is proved for a general finite Coxeter system from the tiling, the root signs and the inversion dictionary, and (2) instantiates it to the parabolic subsystem

1.1F2F3algebra

Since $Φ_{-} = - Φ_{+}$ [F3], the arrangement $⋃_{α \in Φ} H_{α}$ is invariant under the antipodal map $v \mapsto - v$ (as $H_{- α} = H_{α}$ ), and that map is a homeomorphism permuting the connected components of the complement; hence $- C^{\circ}$ is a connected component of $V ∖ ⋃_{α} H_{α}$ , and by the tiling [F2] there is a unique $w_{0} \in W$ with $- C^{\circ} = w_{0} C^{\circ}$ ; taking closures with [F2] gives $- C = w_{0} C$ .

1.2F1algebra

By [F1] the length is the least word length, so $ℓ (x) = 0$ if and only if $x = 1$ ; reversing a word gives $ℓ (w^{- 1}) = ℓ (w)$ for every $w$ ; and $ℓ (x) = 1$ means that $x$ is represented by a one-letter word, that is, $x \in S$ .

2.1step 1.1F3algebra

For $α \in Φ$ one has $α \in Φ_{+}$ exactly when $f (α) > 0$ for every $f \in C^{\circ}$ , and $α \in Φ_{-}$ exactly when $f (α) < 0$ for every $f \in C^{\circ}$ [F3]; as $f$ runs over $C^{\circ}$ the points $w_{0} \cdot f$ run over $- C^{\circ}$ , and $(- g) (α) = - g (α)$ , so $α \in Φ_{+}$ is equivalent to $f (ρ (w_{0})^{- 1} α) < 0$ for every $f \in C^{\circ}$ , that is, to $ρ (w_{0})^{- 1} α \in Φ_{-}$ ; hence $ρ (w_{0})^{- 1} Φ_{+} = Φ_{-}$ , equivalently $ρ (w_{0}) Φ_{+} = Φ_{-}$ , which is the equality $N (w_{0}) = Φ_{+}$ , and $ρ (w_{0})^{- 1} Φ_{+} = Φ_{-}$ is $N (w_{0}^{- 1}) = Φ_{+}$ .

3.1step 1.1step 2.1F2F3algebra

If $w^{'} \in W$ satisfies $w^{'} C = - C$ , then $w^{'} C^{\circ} = int (w^{'} C) = int (- C) = - C^{\circ} = w_{0} C^{\circ}$ , so $w^{'} = w_{0}$ by the injectivity of $w \mapsto w C^{\circ}$ in [F2]; together with step 1.1 this proves uniqueness of the opposite chamber. If $v \in W$ satisfies $N (v) = Φ_{+}$ or $N (v^{- 1}) = Φ_{+}$ , then $ρ (v)^{- 1} Φ_{+} = Φ_{-}$ : in the first case $ρ (v) Φ_{+} \subseteq Φ_{-}$ is equality because $Φ$ is finite and the two sign sets have equal cardinality, and negation gives $ρ (v) Φ_{-} = Φ_{+}$ ; in the second case this is the inversion-set definition. For $f \in C^{\circ}$ and $s \in S$ , [F3] now gives $(v \cdot f) (e_{s}) = f (ρ (v)^{- 1} e_{s}) < 0$ , so $v C^{\circ} \subseteq - C^{\circ}$ . Both sets are connected components by [F2] and step 1.1, so they are equal and $v = w_{0}$ by [F2]. Conversely step 2.1 gives both full inversion sets for $w_{0}$ , proving all equivalences in (i).

3.2step 1.2step 2.1F4algebra

By [F4] one has $ℓ (w_{0}) = ∣ N (w_{0}) ∣ = ∣ Φ_{+} ∣ = ∣ T ∣$ , using step 2.1 and the bijection $Φ_{+} \to T$ ; and for every $w \in W$ the set $N (w_{0} w) = {α \in Φ_{+} : ρ (w_{0}) ρ (w) α \in Φ_{-}}$ equals ${α \in Φ_{+} : ρ (w) α \in Φ_{+}} = Φ_{+} ∖ N (w)$ , because $ρ (w_{0}) β \in Φ_{-}$ if and only if $β \in ρ (w_{0})^{- 1} Φ_{-} = Φ_{+}$ by step 2.1; hence $ℓ (w_{0} w) = ∣ N (w_{0} w) ∣ = ∣ Φ_{+} ∣ - ∣ N (w) ∣ = ℓ (w_{0}) - ℓ (w)$ by [F4]. This proves (ii) and the first identity of (iii).

4.1step 1.2step 3.2F1algebra

Taking $w = w_{0}$ in step 3.2 gives $ℓ (w_{0}^{2}) = ℓ (w_{0}) - ℓ (w_{0}) = 0$ , so $w_{0}^{2} = 1$ by step 1.2; this is (iv).

4.2step 1.2step 3.2F1algebra

By step 3.2, $0 \leq ℓ (w_{0} w) = ℓ (w_{0}) - ℓ (w)$ for every $w$ , so $ℓ (w) \leq ℓ (w_{0})$ and $w_{0}$ has maximal length; if $ℓ (v) = ℓ (w_{0})$ then $ℓ (w_{0} v^{- 1}) = ℓ (w_{0}) - ℓ (v^{- 1}) = ℓ (w_{0}) - ℓ (v) = 0$ by steps 3.2 and 1.2, so $w_{0} v^{- 1} = 1$ by step 1.2 and $v = w_{0}$ ; hence $ℓ (w_{0}) = \max {ℓ (w) : w \in W}$ and $w_{0}$ is the unique element of that length.

5.1step 1.2step 3.2step 4.1F1algebra

For every $w \in W$ the identity of step 3.2 applied to $w^{- 1}$ gives $ℓ (w_{0} w^{- 1}) = ℓ (w_{0}) - ℓ (w^{- 1})$ , and inversion invariance together with $w_{0}^{- 1} = w_{0}$ from step 4.1 gives $ℓ (w w_{0}) = ℓ (w_{0}) - ℓ (w)$ by step 1.2; this is the second identity of (iii).

6.1step 1.2step 3.2step 5.1step 4.1F1algebra

For $s \in S$ , step 5.1 with $w = s w_{0}$ gives $ℓ (s w_{0} w_{0}) = ℓ (w_{0}) - ℓ (s w_{0})$ , that is, $ℓ (s) = ℓ (w_{0}) - ℓ (s w_{0})$ by step 4.1, so $ℓ (s w_{0}) = ℓ (w_{0}) - 1$ ; then step 3.2 with $w = s w_{0}$ gives $ℓ (w_{0} s w_{0}) = ℓ (w_{0}) - ℓ (s w_{0}) = 1$ , so $w_{0} s w_{0} = : s^{'} \in S$ by step 1.2. Conjugation by $w_{0}$ therefore maps $S$ into $S$ , and since $w_{0}^{2} = 1$ and $S$ is finite it maps $S$ bijectively onto $S$ , that is, $w_{0} S w_{0} = w_{0} S w_{0}^{- 1} = S$ .

7.1step 2.1step 6.1F3F4F5algebra

For $s \in S$ let $σ (s) : = s^{'}$ , so that $w_{0} s w_{0} = σ (s)$ by step 6.1; applying the homomorphism $ρ$ and the conjugation formula [F5] gives $r_{ρ (w_{0}) e_{s}} = ρ (w_{0}) ρ (s) ρ (w_{0})^{- 1} = ρ (σ (s)) = r_{e_{σ (s)}}$ , that is, $t_{ρ (w_{0}) e_{s}} = t_{e_{σ (s)}}$ in the dictionary [F4]; since $t_{α} = t_{β}$ for $α, β \in Φ$ only when $α = \pm β$ , one has $ρ (w_{0}) e_{s} = \pm e_{σ (s)}$ , and the sign is negative because $ρ (w_{0}) e_{s} \in Φ_{-}$ by step 2.1 while $e_{σ (s)} \in Φ_{+}$ [F3]; hence $ρ (w_{0}) e_{s} = - e_{σ (s)}$ for every $s \in S$ , and $σ$ is a permutation of $S$ . This completes (v).

8.1step 7.1F1F2F3F4F5F6F7algebra

Let $I \subseteq S$ with $W_{I}$ finite. By [F6] the pair $(W_{I}, I)$ is a Coxeter system, its intrinsic length function is the restriction of $ℓ$ , and $W_{I} \cap S = I$ ; moreover $I$ is finite. The canonical reflection representation of $(W_{I}, I)$ (on $V_{I} = R^{I}$ with the restricted Coxeter form $B_{I} = B ∣_{V_{I} \times V_{I}}$ ) is the restriction of $ρ$ to $V_{I}$ : each $s \in I$ has $ρ (s) e_{t} = e_{t} - 2 B (e_{t}, e_{s}) e_{s} \in V_{I}$ for $t \in I$ and $ρ (s) v = v$ for $v \in V_{I}^{⊥}$ , so $ρ$ preserves $V_{I}$ and acts there by the reflection of $B_{I}$ with normal $e_{s}$ , and both maps are homomorphisms on the presented group $(W_{I}, I)$ agreeing on generators, hence equal by the universal property [F7]. Therefore the argument of steps 1.1-7.1, which used only the tiling [F2], the root signs [F3], the inversion dictionary [F4], the conjugation formula [F5] and the length facts [F1] - all available verbatim for the Coxeter system $(W_{I}, I)$ with its intrinsic length - applies to $(W_{I}, I)$ : there is a unique $w_{0} (I) \in W_{I}$ with $N_{I} (w_{0} (I)) = (Φ_{I})_{+}$ , it satisfies $w_{0} (I)^{2} = 1$ , it has maximal length in $W_{I}$ and is the unique element with $ℓ (w_{0} (I) w) = ℓ (w_{0} (I)) - ℓ (w)$ and $ℓ (w w_{0} (I)) = ℓ (w_{0} (I)) - ℓ (w)$ for all $w \in W_{I}$ , and $w_{0} (I) I w_{0} (I)^{- 1} = I$ . To verify the asserted uniqueness for the length-complement characterization, if $u \in W_{I}$ satisfies $ℓ (u w) = ℓ (u) - ℓ (w)$ for every $w \in W_{I}$ , put $w = w_{0} (I)$ and $L : = ℓ (w_{0} (I))$ . Then $0 \leq ℓ (u w_{0} (I)) = ℓ (u) - L$ , while maximality gives $ℓ (u) \leq L$ , so $ℓ (u) = L$ and uniqueness of the maximal-length element gives $u = w_{0} (I)$ .

9.1step 8.1F3F6algebra∎

By the instance (ii) of the argument in step 8.1 one has $ℓ (w_{0} (I)) = ∣ (Φ_{I})_{+} ∣$ , where $(Φ_{I})_{+}$ is the positive root system of the subsystem; by the intrinsic form of [F3] applied to $(W_{I}, I)$ (whose dual chamber is ${g \in V_{I}^{*} : g (e_{s}) \geq 0 for s \in I}$ ) a root $α \in Φ_{I} \subseteq V_{I}$ is positive exactly when it lies in the subsystem's positive cone $V_{I} \cap V_{+}$ , which is $V_{I}^{+} = {\sum_{s \in I} λ_{s} e_{s} : λ_{s} \geq 0}$ ; hence $(Φ_{I})_{+} = Φ_{I} \cap V_{I}^{+}$ and $ℓ (w_{0} (I)) = ∣ Φ_{I} \cap V_{I}^{+} ∣$ , as asserted in (2).

5 · Examples, counterexamples and false statements

None yet.

Finite Reflection Arrangements and Spherical Coxeter Complexes

1 · Prerequisites

2 · Summary

Prerequisites and reading

3 · Logical flowchart

4 · Definitions, theorems and proofs

The finite reflection arrangement, its chambers, the spherical chamber complex, and the coset face poset

Definition

The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere

Statement

Facts & Assumptions

Proof

The longest element as the opposition of the chamber, and longest elements of finite parabolics

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources