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Real Forms and Reflection Geometry

1 · Prerequisites

2 · Summary

Reflections require a form and a nonisotropic normal, not an unstated Euclidean structure. This page begins with finite-dimensional real vector spaces and distinguishes positive-definite, indefinite and degenerate forms before assigning any chamber interpretation.

The real Coxeter form, its radical, reflections, and form-preserving maps fixes a finite set $S$ , a Coxeter matrix $m$ and the real vector space $V = R^{S}$ , and defines the Coxeter form $B$ on the basis $e_{s}$ by $B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ (and $- 1$ when $m (s, t) = \infty$ ), together with its radical, its $B$ -preserving linear maps and the reflection $r_{a} (v) = v - 2 B (v, a) a / B (a, a)$ for $B (a, a) \neq 0$ . Neither positive definiteness nor nondegeneracy is presumed, and $r_{a}$ is left undefined for null $a$ . Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order proves that this form is well defined, that every $r_{a}$ is a $B$ -preserving linear involution with fixed hyperplane $\ker B (-, a)$ , and computes the rank-two plane $R e_{s} + R e_{t}$ : the Gram matrix $(\begin{array}{cc} 1 & - c \\ - c & 1 \end{array})$ is positive definite for finite $m$ (the square-sum identity $(x_{s} - c x_{t})^{2} + \sin^{2} (π / m) x_{t}^{2}$ ) and positive semidefinite with radical $R (e_{s} + e_{t})$ for $m = \infty$ ; the product $r_{s} r_{t}$ has determinant $1$ , exact order $m$ for finite $m$ (trace $2 \cos (2 π / m)$ ) and is unipotent of infinite order for $m = \infty$ .

The canonical reflection homomorphism, roots, reflections, and the positive cone names the canonical reflection homomorphism $ρ : W \to G L (V)$ with $ρ (s) = r_{s}$ through the universal property of the presented Coxeter group, together with the roots $Φ$ , the reflections $T$ and the positive and negative cones; Descent of the reflection representation, unit root norms, and conjugation of reflections verifies the relators, proves existence and uniqueness of $ρ$ , shows that every root has $B$ -norm $1$ and that $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ . Positivity, faithfulness, discreteness and nondegeneracy are deliberately left to later pages.

The dual action, chambers, faces, and root hyperplanes passes to the algebraic dual $V^{*}$ with the contragredient action $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ — used even when $B$ is degenerate — and defines the closed chamber, its interior, the faces $C_{I}$ and the root hyperplanes $H_{α}$ . The dual action, the faces, and the rank-two chamber tiling verifies the action axioms, exhibits every face $C_{I}$ explicitly and proves the rank-two chamber structure: the finite case tiles $P^{*}$ by $2 m$ sectors on which $W_{s, t}$ acts simply transitively, and in the infinite case the chambers have pairwise disjoint interiors with union ${φ : φ (e_{s} + e_{t}) > 0} \cup {0}$ — the closed half-plane ${φ : φ (e_{s} + e_{t}) \geq 0}$ with the nonzero points of its boundary line removed — and wall traces on ${φ : φ (e_{s} + e_{t}) = 1}$ exactly the integers. All six items are choice-free. Their worked examples are collected on real-forms-and-reflection-geometry-examples.

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 real Coxeter form, its radical, reflections, and form-preserving maps

Definition

Let $S$ be a finite set and let $m$ be a Coxeter matrix on $S$ , so that $m (s, s) = 1$ and $m (s, t) = m (t, s) \in {2, 3, \dots} \cup {\infty}$ for $s \neq t$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

(1) The space and its distinguished functions. Let $V : = R^{S}$ be the real vector space of all functions $S \to R$ , with pointwise operations (The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ ); it is a vector space over the ordered field $R$ (The reals form a totally ordered field). Write $u (s)$ for the value of $u$ at $s$ and put $e_{s} (s) : = 1$ , $e_{s} (t) : = 0$ for $t \neq s$ .

(2) The Coxeter form. For finite $m (s, t)$ put $c (s, t) : = \cos (π / m (s, t))$ (Sine and cosine defined by their real power series, Pi as twice the smallest positive zero of cosine) and for $m (s, t) = \infty$ put $c (s, t) : = 1$ ; then $c (s, t) = c (t, s)$ and $c (s, s) = \cos π = - 1$ . The Coxeter form $B$ is the unique symmetric bilinear form on $V$ with $B (e_{s}, e_{t}) = - c (s, t)$ for all $s, t \in S$ ; in particular $B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ , while $B (e_{s}, e_{t}) = - 1$ when $m (s, t) = \infty$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V \to V^{*}$ ). No positive definiteness, definiteness or nondegeneracy of $B$ is presumed: $rad (B) : = {u : B (u, v) = 0 for all v \in V}$ may be nonzero (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space), and the form may be indefinite (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form). A linear map $g : V \to V$ is $B$ -preserving when $B (g u, g w) = B (u, w)$ for all $u, w \in V$ .

(3) Reflections. For $a \in V$ with $B (a, a) \neq 0$ define the reflection with normal $a$ by $r_{a} (v) : = v - \frac{2 B (v, a)}{B (a, a)} a .$ The definition asserts neither that $r_{a}$ is linear or an involution nor that $\ker B (-, a) = {v : B (v, a) = 0}$ is a hyperplane; these are proved in Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order ↗. When $B (a, a) = 0$ the symbol $r_{a}$ is not defined.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order

Statement

Let $S$ , $m$ , $V = R^{S}$ , $B$ and $r_{a}$ be as in The real Coxeter form, its radical, reflections, and form-preserving maps.

(1) Well-definedness of $B$ . The functions $e_{s}$ ( $s \in S$ ) form a basis of $V$ , and there is exactly one symmetric bilinear form $B$ on $V$ with $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ and $B (e_{s}, e_{t}) = - 1$ for $m (s, t) = \infty$ ; it satisfies $B (u, w) = \sum_{s, t \in S} u (s) w (t) B (e_{s}, e_{t})$ for all $u, w \in V$ .

(2) Reflection identities. Let $a \in V$ with $B (a, a) \neq 0$ . Then $r_{a}$ is linear, $r_{a}^{2} = {i d}_{V}$ , $r_{a} (a) = - a$ , $r_{a} (v) = v$ for every $v$ with $B (v, a) = 0$ , and $B (r_{a} u, r_{a} w) = B (u, w)$ for all $u, w \in V$ . Moreover $\ker B (-, a)$ is a linear subspace of dimension $\dim V - 1$ (a hyperplane of $V$ , in the codimension-one sense) and is fixed pointwise by $r_{a}$ .

(3) The rank-two plane. Let $s \neq t$ in $S$ , put $P : = R e_{s} + R e_{t}$ and $c : = c (s, t)$ as in The real Coxeter form, its radical, reflections, and form-preserving maps (so $c = 1$ when $m (s, t) = \infty$ ). Then:

(i) $B ∣_{P}$ has Gram matrix $(\begin{array}{cc} 1 & - c \\ - c & 1 \end{array})$ ; for finite $m$ it is positive definite, with $B (x_{s} e_{s} + x_{t} e_{t}, x_{s} e_{s} + x_{t} e_{t}) = (x_{s} - c x_{t})^{2} + \sin^{2} (π / m) x_{t}^{2}$ ; for $m = \infty$ it is positive semidefinite, i.e. $B (x, x) \geq 0$ for all $x \in P$ , with radical $R (e_{s} + e_{t})$ .

(ii) $r_{s}$ and $r_{t}$ fix every element of $P^{⊥} = {v \in V : B (v, e_{s}) = B (v, e_{t}) = 0}$ pointwise; and if $m < \infty$ then $V = P \oplus P^{⊥}$ .

(iii) The product $A : = r_{s} r_{t}$ has, in the ordered basis $(e_{s}, e_{t})$ of $P$ , the matrix $A = (\begin{array}{cc} 4 c^{2} - 1 & - 2 c \\ 2 c & - 1 \end{array})$ , of determinant $1$ .

(iv) If $m < \infty$ then $tr A = 2 \cos (2 π / m)$ , $A^{m} = {i d}_{V}$ , and $A^{k} \neq {i d}_{V}$ for $0 < k < m$ . If $m = \infty$ then $A = {i d}_{P} + N$ on $P$ for some $N$ with $N \neq 0$ and $N^{2} = 0$ ; hence $(r_{s} r_{t})^{k} ∣_{P} = {i d}_{P} + k N$ for every $k \in Z$ , so $r_{s} r_{t}$ has infinite order on $V$ .

Facts & Assumptions

Given: a finite set $S$ , a Coxeter matrix $m$ on $S$ , the real vector space $V = R^{S}$ , the Coxeter form $B$ and the maps $r_{a}$ defined for $B (a, a) \neq 0$ of The real Coxeter form, its radical, reflections, and form-preserving maps, and, for the rank-two clauses, distinct $s, t \in S$ with $c : = c (s, t)$ .

[F1]

The data fixed by the Statement: $m (s, s) = 1$ and $m (s, t) = m (t, s) \in {2, 3, \dots} \cup {\infty}$ for $s \neq t$ ; $e_{s} \in V$ is the function with $e_{s} (s) = 1$ and $e_{s} (t) = 0$ for $t \neq s$ ; and $c (s, t) = \cos (π / m (s, t))$ for finite $m (s, t)$ , while $c (s, t) = 1$ when $m (s, t) = \infty$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ ).

[F2]

A bilinear form on $V$ is a function $V \times V \to R$ that is linear in each variable separately, and it is symmetric when $B (u, v) = B (v, u)$ for all $u, v \in V$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, The real Coxeter form, its radical, reflections, and form-preserving maps).

[F3]

For a linear map $T$ of a finite-dimensional vector space over $R$ one has $\dim V = \dim (\ker T) + \dim (im T)$ , and $\dim_{R} R = 1$ ; if a linear map $V \to R$ has a nonzero value then its image is all of $R$ and its rank is $1$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ , Rank and nullity of a linear map with finite-dimensional domain, The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[F4]

In ordered bases the matrix of a composite is the product of the matrices of the factors, matrix powers are iterated products, and $I_{2}$ denotes the identity matrix (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases, $[S \circ T]_{B}^{D} = [S]_{C}^{D} [T]_{B}^{C}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F5]

Trigonometric values and identities: the addition formulas $\sin (x + y) = \sin x \cos y + \cos x \sin y$ and $\cos (x + y) = \cos x \cos y - \sin x \sin y$ ; $\sin^{2} x + \cos^{2} x = 1$ ; $\sin x = 0$ if and only if $x \in π Z$ , and $\cos x = 0$ if and only if $x = (k + \frac{1}{2}) π$ for some $k \in Z$ ; $\cos (π / 2) = 0$ , $\cos π = - 1$ , $\cos (x + π) = - \cos x$ , and $\cos 0 = 1$ by the defining series evaluated at $0$ (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi, Sine and cosine defined by their real power series).

[F6]

For linear subspaces: $V = U \oplus W$ means $U + W = V$ and $U \cap W = {0_{V}}$ ; a symmetric bilinear form is positive definite when $B (v, v) > 0$ for every $v \neq 0$ ; the left radical of a bilinear form is ${rad}_{L} (B) = {u : B (u, v) = 0 for every v}$ (Internal direct sum $V = ⨁_{i < n} U_{i}$ : the sum is everything and each summand meets the sum of the others only in $0_{V}$ , $V = ⨁_{i < n} U_{i}$ if and only if every $v \in V$ is $\sum_{i < n} u_{i}$ with $u_{i} \in U_{i}$ in exactly one way; equivalently, if and only if the sum is $V$ and $\sum_{i < n} u_{i} = 0_{V}$ with $u_{i} \in U_{i}$ forces every $u_{i} = 0_{V}$ , Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form, The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space).

[F7]

In the ordered field $R$ one has $2 \neq 0$ , so $2 a = 0_{V}$ implies $a = 0_{V}$ (The reals form a totally ordered field).

Proof

technique · direct finite-dimensional computation, in layers: first the basis and reflection algebra, then the rank-two plane with its trigonometry, and finally the order statements on the whole space

1.1givenF1F2

The functions $e_{s}$ are linearly independent: if $\sum_{s \in S} λ_{s} e_{s} = 0_{V}$ then evaluating at $s$ gives $λ_{s} = 0$ . They span $V$ : every $u \in V$ satisfies $u = \sum_{s \in S} u (s) e_{s}$ , since the right side is a finite sum ( $S$ is finite) whose value at any $t \in S$ is $u (t)$ . So $(e_{s})_{s \in S}$ is a basis of $V$ . Consequently, for all $u, w \in V$ one has $B (u, w) = \sum_{s, t \in S} u (s) w (t) B (e_{s}, e_{t})$ : expanding $u$ and $w$ in the basis and distributing with bilinearity in each variable gives that finite sum. In particular a bilinear form on $V$ is determined by the numbers $B (e_{s}, e_{t})$ .

1.2givenF1F2algebra

The displayed $r_{a}$ for $B (a, a) \neq 0$ is linear in $v$ , because $v \mapsto B (v, a)$ is linear and $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ is a sum of the identity and the composite of that functional with scalar multiplication by $a$ ; also $r_{a} (a) = a - \frac{2 B (a, a)}{B (a, a)} a = a - 2 a = - a$ , and $r_{a} (v) = v$ for every $v \in \ker B (-, a)$ . Moreover $B (r_{a} v, a) = B (v, a) - \frac{2 B (v, a)}{B (a, a)} B (a, a) = - B (v, a)$ , so $r_{a} (r_{a} (v)) = r_{a} (v) - \frac{2 B (r_{a} v, a)}{B (a, a)} a = r_{a} (v) + \frac{2 B (v, a)}{B (a, a)} a = v$ ; hence $r_{a}^{2} = {i d}_{V}$ . Finally, expanding with bilinearity and cancelling the equal cross terms via symmetry $B (u, a) = B (a, u)$ and $B (w, a) = B (a, w)$ , $B (r_{a} u, r_{a} w) = B (u, w) - \frac{2 B (u, a) B (a, w)}{B (a, a)} - \frac{2 B (w, a) B (a, u)}{B (a, a)} + \frac{4 B (u, a) B (w, a)}{B (a, a)} = B (u, w),$ so $r_{a}$ is $B$ -preserving.

1.3givenF1F2F3

Put $φ (v) : = B (v, a)$ . By $B (a, a) \neq 0$ the value $φ (a)$ is nonzero, so the image of $φ$ is all of $R$ and its rank is $1$ ; the kernel $\ker B (-, a) = \ker φ$ is a linear subspace and rank-nullity gives $\dim \ker B (-, a) = \dim V - 1$ .

1.4givenF1F2F4algebra

For distinct $s, t \in S$ , evaluating the definition of $r$ at $e_{s}$ and $e_{t}$ with $B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - c$ gives $r_{s} (e_{s}) = - e_{s}$ , $r_{s} (e_{t}) = e_{t} + 2 c e_{s}$ , $r_{t} (e_{t}) = - e_{t}$ and $r_{t} (e_{s}) = e_{s} + 2 c e_{t}$ . Hence in the ordered basis $(e_{s}, e_{t})$ the matrices are $[r_{s}] = (\begin{array}{cc} - 1 & 2 c \\ 0 & 1 \end{array})$ and $[r_{t}] = (\begin{array}{cc} 1 & 0 \\ 2 c & - 1 \end{array})$ , and the composite $A : = r_{s} r_{t}$ has matrix $[A] = [r_{s}] [r_{t}] = (\begin{array}{cc} 4 c^{2} - 1 & - 2 c \\ 2 c & - 1 \end{array}), \det [A] = (4 c^{2} - 1) (- 1) - (2 c) (- 2 c) = 1.$

1.5givenF1F2F5F6algebra

On $P = R e_{s} + R e_{t}$ write $x = x_{s} e_{s} + x_{t} e_{t}$ . Bilinearity and the values $B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ , $B (e_{s}, e_{t}) = - c$ give $B (x, x) = x_{s}^{2} - 2 c x_{s} x_{t} + x_{t}^{2} = (x_{s} - c x_{t})^{2} + (1 - c^{2}) x_{t}^{2}$ . For finite $m$ one has $1 - c^{2} = \sin^{2} (π / m)$ and $\sin (π / m) \neq 0$ because $0 < π / m \leq π / 2$ and the sine zero set is $π Z$ ; hence $B (x, x) > 0$ whenever $x \neq 0$ , so $B ∣_{P}$ is positive definite. For $m = \infty$ one has $c = 1$ and $B (x, x) = (x_{s} - x_{t})^{2} \geq 0$ , which vanishes exactly when $x_{s} = x_{t}$ , i.e. on $R (e_{s} + e_{t})$ ; and $R (e_{s} + e_{t})$ is precisely the radical of $B ∣_{P}$ , since $B (x, e_{s}) = x_{s} - c x_{t}$ and $B (x, e_{t}) = x_{t} - c x_{s}$ vanish for all such $x$ , while $x_{s} \neq x_{t}$ makes $B (x, e_{s}) \neq 0$ .

1.6givenF1F4F5algebra

Assume $m < \infty$ and put $θ : = π / m$ , so $c = \cos θ$ . The trace of the matrix in 1.4 is $tr [A] = 4 c^{2} - 2 = 2 (2 c^{2} - 1) = 2 \cos 2 θ$ by the double-angle instance of the addition formulas, and the $2 \times 2$ Cayley-Hamilton identity, verified directly from the displayed entries, reads $A^{2} = 2 \cos 2 θ A - I_{2}$ . For $m \geq 3$ one has $2 θ \in (0, π)$ and $\sin 2 θ \neq 0$ , and for $m = 2$ one has $c = \cos (π / 2) = 0$ , so $[A] = - I_{2}$ .

2.1givenF5step 1.6algebra

Assume $m \geq 3$ ; then $\sin 2 θ \neq 0$ and induction on $k \geq 1$ using $A^{2} = 2 \cos 2 θ A - I_{2}$ and the product-to-sum identity $2 \sin x \cos y = \sin (x + y) + \sin (x - y)$ gives $A^{k} = \frac{\sin (2 k θ) A - \sin (2 (k - 1) θ) I_{2}}{\sin 2 θ}$ for every $k \geq 1$ : the case $k = 1$ is $\sin 0 = 0$ , and the induction step replaces $A^{k + 1} = A \cdot A^{k}$ using the displayed identity. Hence $A^{m} = I_{2}$ because $\sin (2 m θ) = \sin (2 π) = 0$ and $\sin (2 (m - 1) θ) = - \sin 2 θ$ . If $1 \leq k < m$ and $A^{k} = I_{2}$ , then $\sin (2 k θ) A = (\sin 2 θ + \sin (2 (k - 1) θ)) I_{2}$ ; were $\sin (2 k θ) \neq 0$ , the matrix $A$ would be scalar, contradicting its nonzero off-diagonal entry $- 2 c$ (for $m \geq 3$ the value $c$ is nonzero, since $c = 0$ would force $π / m = (j + \frac{1}{2}) π$ , that is $2 / m = 2 j + 1 > 0$ for an integer $j$ , so $j \geq 0$ and $2 / m \geq 1$ , impossible because $m \geq 3$ ). So $\sin (2 k θ) = 0$ , and then $\sin (2 (k - 1) θ) = \sin (2 k θ - 2 θ) = - \cos (2 k θ) \sin 2 θ$ forces $\cos (2 k θ) = 1$ ; with $\sin (2 k θ) = 0$ it gives $2 k θ \in 2 π Z$ (indeed $\sin x = 0$ writes $x = ℓ π$ , and $\cos (ℓ π) = (- 1)^{ℓ}$ follows from $\cos 0 = 1$ and $\cos (x + π) = - \cos x$ , so $\cos (2 k θ) = 1$ forces $ℓ$ even), hence $m ∣ k$ , contradicting $1 \leq k < m$ . Hence $A^{k} \neq I_{2}$ for $0 < k < m$ .

2.2givenF4F7step 1.4algebra

Assume $m = \infty$ , so $c = 1$ and $[A] = (\begin{array}{cc} 3 & - 2 \\ 2 & - 1 \end{array}) = I_{2} + N$ with $N = (\begin{array}{cc} 2 & - 2 \\ 2 & - 2 \end{array})$ . Then $N \neq 0$ and $N^{2} = 0$ by direct multiplication, so the binomial theorem gives $A^{k} = I_{2} + k N$ for every $k \geq 0$ , while $A^{- 1} = I_{2} - N$ because $(I_{2} + N) (I_{2} - N) = I_{2} - N^{2} = I_{2}$ , so $(I_{2} - N)^{j} = I_{2} + (- j) N$ for $j \geq 0$ and $A^{k} = I_{2} + k N$ for every $k \in Z$ . For $k \neq 0$ the matrix $k N$ has entry $2 k \neq 0$ in its first column, so $A^{k} \neq I_{2}$ ; the product $A = r_{s} r_{t}$ therefore has infinite order on $P$ .

2.3givenF1F2F5step 1.1

The Coxeter form $B$ of the Statement is well defined: the prescription $B (u, w) : = \sum_{s, t \in S} u (s) w (t) (- c (s, t))$ is a finite sum of scalar multiples of products of coordinate values, hence a function $V \times V \to R$ linear in each variable; it is symmetric because $c (s, t) = c (t, s)$ , which follows from $m (s, t) = m (t, s)$ and the symmetry of cosine; and it has $B (e_{s}, e_{t}) = - c (s, t)$ for all $s, t$ . By 1.1 it is the unique symmetric bilinear form with these values, and $c (s, s) = \cos (π / 1) = \cos π = - 1$ gives $B (e_{s}, e_{s}) = 1$ .

2.4givenF1F2F6step 1.2step 1.4step 1.5

The maps $r_{s}$ and $r_{t}$ fix $P^{⊥} = {v \in V : B (v, e_{s}) = B (v, e_{t}) = 0}$ pointwise, since $r_{s} (v) = v - \frac{2 B (v, e_{s})}{B (e_{s}, e_{s})} e_{s} = v$ and likewise for $t$ . If $m < \infty$ , then $V = P \oplus P^{⊥}$ : the Gram matrix $(\begin{array}{cc} 1 & - c \\ - c & 1 \end{array})$ of $B ∣_{P}$ has determinant $1 - c^{2} = \sin^{2} (π / m) \neq 0$ , so for given $v$ the $2 \times 2$ system $B (p, e_{s}) = B (v, e_{s})$ , $B (p, e_{t}) = B (v, e_{t})$ has a unique solution $p \in P$ ; then $z : = v - p$ satisfies $B (z, e_{s}) = B (z, e_{t}) = 0$ , hence $z \in P^{⊥}$ and $v = p + z$ , so $P + P^{⊥} = V$ . If $p \in P \cap P^{⊥}$ then $B (p, p) = 0$ with $p \in P$ , and positive definiteness of $B ∣_{P}$ forces $p = 0$ ; thus $P \cap P^{⊥} = {0}$ and $V = P \oplus P^{⊥}$ .

3.1givenstep 1.6step 2.1step 2.2step 2.4∎

Let $A = r_{s} r_{t}$ . If $m < \infty$ , then $A^{m} ∣_{P} = I_{2}$ and $A^{k} ∣_{P} \neq I_{2}$ for $0 < k < m$ by steps 1.6 and 2.1 and the case $m = 2$ of step 1.6, while $A ∣_{P}$ has the matrix displayed in step 1.4; on $P^{⊥}$ the map $A$ fixes every vector by step 2.4, so $A^{m} = {i d}_{V}$ and $A^{k} \neq {i d}_{V}$ for $0 < k < m$ because its restriction to the direct summand $P$ differs from $I_{2}$ . If $m = \infty$ , then $[A] = I_{2} + N$ with $N \neq 0$ , $N^{2} = 0$ and $A^{k} ∣_{P} = I_{2} + k N \neq I_{2}$ for every $k \neq 0$ by step 2.2, so $A^{k} \neq {i d}_{V}$ for every $k \neq 0$ and $r_{s} r_{t}$ has infinite order on $V$ .

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

The canonical reflection homomorphism, roots, reflections, and the positive cone

Definition

Let $S$ , $m$ , $V$ , $B$ be as in The real Coxeter form, its radical, reflections, and form-preserving maps, let $W$ be the presented Coxeter group of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups with its universal property, and put $r_{s} : = r_{e_{s}}$ for $s \in S$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

(1) Write $G L (V)$ for the group of invertible linear maps $V \to V$ , the unit group of the monoid of linear endomorphisms under composition (Invertible linear maps, linear isomorphisms, and inverse linear maps, The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication, Identity maps and composites of linear maps are linear, The invertible elements of a monoid form a group under the restricted operation). The canonical reflection homomorphism is the group homomorphism $ρ : W \to G L (V)$ (Monoid homomorphism and group homomorphism) that satisfies $ρ (s) = r_{s}$ for every $s \in S$ ; its existence and uniqueness are established by Descent of the reflection representation, unit root norms, and conjugation of reflections ↗, which is this definition's recorded justifier. For $w \in W$ and $v \in V$ write $ρ (w) v$ for the image.

(2) The root system of the pair is $Φ : = {ρ (w) e_{s} : w \in W, s \in S} \subset V$ ; its elements are the roots. The set of reflections of $W$ is $T : = {w s w^{- 1} : w \in W, s \in S} \subset W$ ; that $ρ (w s w^{- 1})$ is the reflection $r_{ρ (w) e_{s}}$ and that every root is $B$ -non-isotropic are proved in Descent of the reflection representation, unit root norms, and conjugation of reflections ↗.

(3) The positive cone is $V_{+} : = {\sum_{s \in S} λ_{s} e_{s} : λ_{s} \in R, λ_{s} \geq 0}$ and the negative cone is $- V_{+}$ (Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ ).

No positivity of $ρ (w) e_{s}$ for fixed $w$ , no faithfulness or injectivity of $ρ$ , no discreteness and no nondegeneracy of $B$ is asserted by this definition.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

Descent of the reflection representation, unit root norms, and conjugation of reflections

Statement

Let $S$ , $m$ , $V$ , $B$ , $W$ , $r_{s}$ , $ρ$ , $Φ$ , $T$ be as in The canonical reflection homomorphism, roots, reflections, and the positive cone.

(1) Relators and descent. For every $s \in S$ one has $r_{s}^{2} = {i d}_{V}$ , and for all $s \neq t$ with $m (s, t) < \infty$ one has $(r_{s} r_{t})^{m (s, t)} = {i d}_{V}$ . Consequently the assignment $s \mapsto r_{s}$ sends every relator of the presentation to the identity, and by the universal property of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups there is a unique group homomorphism $ρ : W \to G L (V)$ with $ρ (s) = r_{s}$ for all $s \in S$ .

(2) $ρ$ preserves the form. For every $w \in W$ and all $u, w^{'} \in V$ , $B (ρ (w) u, ρ (w) w^{'}) = B (u, w^{'})$ .

(3) Roots have unit norm. For every $w \in W$ and $s \in S$ , $B (ρ (w) e_{s}, ρ (w) e_{s}) = 1$ ; hence every root $α \in Φ$ satisfies $B (α, α) = 1 \neq 0$ and $r_{α}$ is defined.

(4) Conjugation of reflections. Let $g \in G L (V)$ be $B$ -preserving and let $a \in V$ with $B (a, a) \neq 0$ . Then $g r_{a} g^{- 1} = r_{g a}$ . In particular, for every $w \in W$ and $s \in S$ , $ρ (w s w^{- 1}) = ρ (w) r_{s} ρ (w)^{- 1} = r_{ρ (w) e_{s}},$ so every $t = w s w^{- 1} \in T$ acts on $V$ as the reflection in the root $ρ (w) e_{s} \in Φ$ .

Facts & Assumptions

Given: a finite set $S$ , a Coxeter matrix $m$ , the space $V = R^{S}$ , the Coxeter form $B$ , the reflections $r_{a}$ and the presented Coxeter group $W$ with its universal property, all as in the Statement of The canonical reflection homomorphism, roots, reflections, and the positive cone; here $r_{s} : = r_{e_{s}}$ for $s \in S$ .

[F1]

For $a \in V$ with $B (a, a) \neq 0$ the map $r_{a}$ is linear, satisfies $r_{a}^{2} = {i d}_{V}$ and $B (r_{a} u, r_{a} w) = B (u, w)$ for all $u, w \in V$ ; and for distinct $s, t \in S$ with $m (s, t) < \infty$ the product $A = r_{s} r_{t}$ satisfies $A^{m (s, t)} = {i d}_{V}$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, clauses (2) and (3)(iv)).

[F2]

The Coxeter form satisfies $B (e_{s}, e_{s}) = 1$ for every $s \in S$ , the symbol $r_{s}$ abbreviates $r_{e_{s}}$ , and a linear map $g$ is $B$ -preserving when $B (g u, g w) = B (u, w)$ for all $u, w \in V$ (The real Coxeter form, its radical, reflections, and form-preserving maps).

[F3]

A subset $H$ of a group $W$ that is closed under the group operation and under inverses and contains a generating set $S$ equals $W$ , because $W = ⟨ S ⟩$ is the smallest subgroup containing $S$ ; a group homomorphism satisfies $φ (g h) = φ (g) φ (h)$ and $φ (g^{- 1}) = φ (g)^{- 1}$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, Group and abelian group, Monoid homomorphism and group homomorphism).

[F4]

For a vector space $V$ the set of linear maps $V \to V$ is a monoid under composition with identity ${i d}_{V}$ , its unit group is the group $G L (V)$ of invertible linear maps, and composition is associative (The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication, Identity maps and composites of linear maps are linear, The invertible elements of a monoid form a group under the restricted operation, Invertible linear maps, linear isomorphisms, and inverse linear maps, Linear map between vector spaces over the same field).

Proof

technique · verify the relators, descend through the presentation's universal property, and propagate the form preservation along the subgroup generated by the images of the generators

1.1givenF1F2F4

For every $s \in S$ the map $r_{s} = r_{e_{s}}$ is linear, $B$ -preserving and satisfies $r_{s}^{2} = {i d}_{V}$ : these are the identities of [F1] for $a = e_{s}$ , whose hypothesis $B (e_{s}, e_{s}) \neq 0$ holds by [F2]. Since $r_{s}^{2} = {i d}_{V}$ , the map $r_{s}$ is invertible with $r_{s}^{- 1} = r_{s}$ , so $r_{s} \in G L (V)$ .

1.2givenF1F2

For all distinct $s, t \in S$ with $m (s, t) < \infty$ one has $(r_{s} r_{t})^{m (s, t)} = {i d}_{V}$ , directly from the exact order statement of [F1] applied to the plane spanned by $e_{s}$ and $e_{t}$ : the product $r_{s} r_{t}$ acts there with exact order $m (s, t)$ , so its $m (s, t)$ -th power is the identity on $V$ .

1.3givenF1F2algebra

Conjugation formula. Let $g \in G L (V)$ be $B$ -preserving and let $a \in V$ with $B (a, a) \neq 0$ . Then $B (g a, g a) = B (a, a) \neq 0$ , and for every $v \in V$ the $B$ -preservation of $g$ gives $B (g^{- 1} v, a) = B (g (g^{- 1} v), g a) = B (v, g a)$ . Substituting into the definition of $r_{g a}$ and writing $v = g (g^{- 1} v)$ , $r_{g a} (v) = g (g^{- 1} v) - \frac{2 B (g^{- 1} v, a)}{B (a, a)} g a = g (g^{- 1} v - \frac{2 B (g^{- 1} v, a)}{B (a, a)} a) = g r_{a} g^{- 1} (v),$ so $g r_{a} g^{- 1} = r_{g a}$ .

2.1givenF4step 1.1step 1.2

Descent. The assignment $s \mapsto r_{s}$ sends the relator $s^{2}$ to $r_{s}^{2} = {i d}_{V}$ by 1.1 and the relator $(s t)^{m (s, t)}$ to $(r_{s} r_{t})^{m (s, t)} = {i d}_{V}$ by 1.2; these are exactly the relators of the presented Coxeter group $W$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, and the images $r_{s}$ lie in the group $G L (V)$ . So the universal property of that presentation gives a unique group homomorphism $ρ : W \to G L (V)$ with $ρ (s) = r_{s}$ for every $s \in S$ .

3.1givenF2F3step 1.1step 2.1

$ρ$ preserves the form, and roots have unit norm. Let $H : = {w \in W : B (ρ (w) u, ρ (w) w^{'}) = B (u, w^{'}) for all u, w^{'} \in V}$ . Every $s \in S$ lies in $H$ by 1.1, since $ρ (s) = r_{s}$ . If $g, h \in H$ then $B (ρ (g h) u, ρ (g h) w^{'}) = B (ρ (g) ρ (h) u, ρ (g) ρ (h) w^{'}) = B (ρ (h) u, ρ (h) w^{'}) = B (u, w^{'})$ , using the homomorphism property of 2.1 and the $B$ -preservation of $ρ (g)$ and then of $ρ (h)$ , so $g h \in H$ ; and if $g \in H$ , then $B (ρ (g^{- 1}) u, ρ (g^{- 1}) w^{'}) = B (ρ (g) ρ (g^{- 1}) u, ρ (g) ρ (g^{- 1}) w^{'}) = B (u, w^{'})$ , so $g^{- 1} \in H$ . Thus $H$ is a subgroup containing $S$ , and since every element of $W$ is the value of a finite word in $S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so that the images of the elements of $S$ generate $W$ , [F3] gives $H = W$ . In particular $B (ρ (w) e_{s}, ρ (w) e_{s}) = B (e_{s}, e_{s}) = 1$ for all $w \in W$ and $s \in S$ by [F2], so every root $α \in Φ$ has $B (α, α) = 1 \neq 0$ and the reflection $r_{α}$ is defined.

4.1givenstep 1.3step 2.1step 3.1∎

Conjugation of reflections by $ρ$ . Let $w \in W$ and $s \in S$ . By 3.1 the map $ρ (w)$ is $B$ -preserving, and $r_{s} = ρ (s)$ by 2.1; applying the conjugation formula 1.3 with $g = ρ (w)$ and $a = e_{s}$ therefore gives $ρ (w) r_{s} ρ (w)^{- 1} = r_{ρ (w) e_{s}}$ . Since $ρ$ is a homomorphism, $ρ (w)^{- 1} = ρ (w^{- 1})$ and $ρ (w) ρ (s) ρ (w)^{- 1} = ρ (w s w^{- 1})$ , so $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}} .$ Hence for every $t = w s w^{- 1} \in T$ the element $ρ (t)$ is the reflection in the root $ρ (w) e_{s} \in Φ$ , as asserted.

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

The dual action, chambers, faces, and root hyperplanes

Definition

Let $S$ , $m$ , $V$ , $B$ , $W$ , $ρ$ , $Φ$ be as in The canonical reflection homomorphism, roots, reflections, and the positive cone, and let $V^{*}$ be the algebraic dual of $V$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ).

(1) The dual action. For $w \in W$ and $f \in V^{*}$ define $w \cdot f \in V^{*}$ by $(w \cdot f) (v) : = f (ρ (w)^{- 1} v) .$ This is the dual (contragredient) action of $ρ$ ; that it really is a left action by linear maps is proved in The dual action, the faces, and the rank-two chamber tiling ↗. The dual action is used even when $B$ is degenerate: no identification of $V$ with $V^{*}$ through $B$ is made or assumed.

(2) Chambers, faces, root hyperplanes. The closed chamber is $C : = {f \in V^{*} : f (e_{s}) \geq 0 for all s \in S}$ , its interior is $C^{\circ} : = {f \in V^{*} : f (e_{s}) > 0 for all s \in S}$ , and for $I \subseteq S$ the face $C_{I}$ is $C_{I} : = {f \in V^{*} : f (e_{s}) = 0 for s \in I and f (e_{s}) > 0 for s \notin I},$ the set of functionals of $C$ vanishing exactly on $I$ . For a root $α \in Φ$ the root hyperplane is $H_{α} : = {f \in V^{*} : f (α) = 0} = \ker ({e v}_{α})$ , the kernel of the evaluation functional $f \mapsto f (α)$ (Kernel and image of a linear map).

Neither the nonemptiness of the $C_{I}$ nor any orbit or tiling property is asserted here; existence of the faces and the exact rank-two chamber structure are proved in The dual action, the faces, and the rank-two chamber tiling ↗.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08Open item page →

The dual action, the faces, and the rank-two chamber tiling

Statement

With the notation of The dual action, chambers, faces, and root hyperplanes:

(1) The dual action is an action by linear maps. For all $w_{1}, w_{2} \in W$ and $f \in V^{*}$ one has $i d \cdot f = f$ and $w_{1} \cdot (w_{2} \cdot f) = (w_{1} w_{2}) \cdot f$ , and for each $w$ the map $f \mapsto w \cdot f$ is linear.

(2) The faces are nonempty. Let $f_{s} \in V^{*}$ be the dual basis functionals of the basis $(e_{s})$ of $V$ , i.e. $f_{s} (e_{t}) = δ_{s t}$ (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). For $I \subseteq S$ the functional $f_{I} : = \sum_{s \notin I} f_{s}$ satisfies $f_{I} (e_{s}) = 1$ for $s \notin I$ and $f_{I} (e_{s}) = 0$ for $s \in I$ ; hence $f_{I} \in C_{I}$ and every face is nonempty. In particular $C^{\circ} ∋ \sum_{s \in S} f_{s}$ is nonempty and $C_{S} = {0}$ .

(3) Rank two. Let $s \neq t$ in $S$ and put $P : = R e_{s} + R e_{t}$ . Let $P^{*} : = H o m (P, R)$ be the algebraic dual space of the $2$ -dimensional space $P$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ), with the dual basis $f_{s}, f_{t}$ of the basis $(e_{s}, e_{t})$ of $P$ ; functionals on $P$ are the restrictions of functionals on $V$ . Let $W_{s, t} : = ⟨ ρ (s), ρ (t) ⟩ \subseteq G L (V)$ be the matrix subgroup generated by the two reflections. It preserves $P$ . For $g \in W_{s, t}$ and $φ \in P^{*}$ define $(g \cdot φ) (v) : = φ (g^{- 1} v)$ for $v \in P$ ; this is a left action of $W_{s, t}$ on $P^{*}$ by linear maps, the contragredient of its restriction to $P$ . The abstract subgroup $⟨ s, t ⟩ \leq W$ acts through its image $W_{s, t}$ ; no action of all of $W$ on $P^{*}$ is asserted. Writing $c : = c (s, t)$ (so $c = 1$ for $m (s, t) = \infty$ ) and coordinates $(y_{s}, y_{t})$ on $P^{*}$ , so that $φ = y_{s} f_{s} + y_{t} f_{t}$ and $φ (e_{s}) = y_{s}$ , $φ (e_{t}) = y_{t}$ , the dual generators act by $ρ (s)^{*} (y_{s}, y_{t}) = (- y_{s}, 2 c y_{s} + y_{t}), ρ (t)^{*} (y_{s}, y_{t}) = (y_{s} + 2 c y_{t}, - y_{t});$ each is an involution fixing the line ${φ : φ (e_{s}) = 0} = {y_{s} = 0}$ pointwise, respectively ${φ : φ (e_{t}) = 0} = {y_{t} = 0}$ , and the functional $δ (φ) : = φ (e_{s} + e_{t})$ is preserved by $W_{s, t}$ when $m (s, t) = \infty$ . Put $Φ_{P} : = W_{s, t} {e_{s}, e_{t}} \subset P$ , the rank-two root system. For $β \in Φ_{P}$ the rank-two root hyperplane is $H_{β} \cap P^{*} : = {φ \in P^{*} : φ (β) = 0}$ .

(i) If $m : = m (s, t) < \infty$ : $W_{s, t}$ is a dihedral group of order $2 m$ , and the $2 m$ chambers $w C_{P}$ ( $w \in W_{s, t}$ ), where $C_{P} : = {f \in P^{*} : f (e_{s}) \geq 0, f (e_{t}) \geq 0}$ , are exactly the $2 m$ closed sectors cut out in $P^{*}$ by the $m$ root hyperplanes $H_{β} \cap P^{*}$ $(β \in Φ_{P})$ , $Φ_{P} : = W_{s, t} {e_{s}, e_{t}}$ . They have pairwise disjoint interiors, their union is all of $P^{*}$ , $W_{s, t}$ acts simply transitively on them, and any two distinct chambers are separated by at least one root hyperplane, i.e. there is $β \in Φ_{P}$ with the two interiors in opposite open half-planes of $H_{β}$ .

(ii) If $m (s, t) = \infty$ : the chambers $w C_{P}$ ( $w \in W_{s, t}$ ) have pairwise disjoint interiors and their union is ${φ \in P^{*} : φ (e_{s} + e_{t}) > 0} \cup {0}$ , that is, the closed half-plane ${φ : φ (e_{s} + e_{t}) \geq 0}$ with the nonzero points of its boundary line removed; any two distinct chambers are separated by at least one root hyperplane $H_{β} \cap P^{*}$ $(β \in Φ_{P})$ , and the traces $H_{β} \cap {φ : φ (e_{s} + e_{t}) = 1}$ are exactly the integers in the coordinate $y_{s}$ on that affine line.

Facts & Assumptions

Given: a finite set $S$ , a Coxeter matrix $m$ , the space $V = R^{S}$ , the Coxeter form $B$ , the reflections $r_{a}$ , the presented group $W$ , the canonical homomorphism $ρ : W \to G L (V)$ with roots $Φ$ , and the dual action, chambers, faces and root hyperplanes on $V^{*}$ of The dual action, chambers, faces, and root hyperplanes; for distinct $s, t \in S$ the plane $P = R e_{s} + R e_{t}$ and the number $c = c (s, t)$ .

[F1]

$B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - c$ for all $s, t$ ; for $B (a, a) \neq 0$ the map $r_{a}$ is a linear involution preserving $B$ with $r_{a} (a) = - a$ and $r_{a}$ the identity on $\ker B (-, a)$ ; for $a = e_{s}$ this gives $r_{s} (e_{s}) = - e_{s}$ and $r_{s} (e_{t}) = e_{t} + 2 c e_{s}$ , and symmetrically $r_{t} (e_{t}) = - e_{t}$ , $r_{t} (e_{s}) = e_{s} + 2 c e_{t}$ ; and for finite $m (s, t)$ the restriction $B ∣_{P}$ is positive definite (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, clauses (2) and (3)(i)–(iv), and Proof step 2.1). In particular, $r_{s} r_{t}$ has exact order $m$ on $P$ for finite $m$ , while for $m = \infty$ its restriction is $I_{2} + N$ with $N = (\begin{array}{cc} 2 & - 2 \\ 2 & - 2 \end{array})$ and $(r_{s} r_{t})^{k} ∣_{P} = I_{2} + k N$ .

[F2]

$ρ : W \to G L (V)$ is the unique homomorphism with $ρ (s) = r_{s}$ for $s \in S$ , it satisfies $ρ (g h) = ρ (g) ρ (h)$ and $ρ (g^{- 1}) = ρ (g)^{- 1}$ , every $ρ (w)$ preserves $B$ , every root has $B$ -norm $1$ , and $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ for all $w \in W$ , $s \in S$ , so all reflections of $W$ act as the reflections in the roots $ρ (w) e_{s} \in Φ$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections, clauses (1)–(4)).

[F3]

The dual action is $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ ; the closed chamber is $C = {f \in V^{*} : f (e_{s}) \geq 0 for all s \in S}$ , its interior is $C^{\circ} = {f : f (e_{s}) > 0 for all s}$ , the face of type $I$ is $C_{I} = {f : f (e_{s}) = 0 for s \in I, f (e_{s}) > 0 for s \notin I}$ , and $H_{α} = {f \in V^{*} : f (α) = 0}$ for $α \in Φ$ (The dual action, chambers, faces, and root hyperplanes).

[F4]

For a finite-dimensional space the dual space, the dual family of a basis and its basis property are as recorded in the algebraic-dual and dual-basis items; in particular $f_{s} (e_{t}) = δ_{s t}$ defines unique elements $f_{s}$ of the dual of $P$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ , 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).

[F5]

$V = R^{S}$ consists of the functions $S \to R$ with pointwise operations, the functions $e_{s}$ form a basis of $V$ , and $u = \sum_{s \in S} u (s) e_{s}$ for $u \in V$ (The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = {0, 1, \dots, n - 1}$ , Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, clause (1)).

[F6]

Trigonometric facts: the addition formulas; $\sin^{2} x + \cos^{2} x = 1$ ; $\sin x = 0$ if and only if $x \in π Z$ , and $\cos x = 0$ if and only if $x = (k + \frac{1}{2}) π$ for some $k \in Z$ ; $\cos (x + π) = - \cos x$ , $\cos π = - 1$ ; $\cos 2 x = \cos^{2} x - \sin^{2} x$ and $\sin 2 x = 2 \sin x \cos x$ follow from the addition formulas (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi).

[F7]

$π > 0$ and $R$ is a totally ordered field, so $0 < π / m \leq π / 2$ for $m \geq 2$ and the order arithmetic below is the order of $R$ (Pi as twice the smallest positive zero of cosine, The reals form a totally ordered field).

[F8]

Every real $τ$ has an integer $j$ with $j \leq τ < j + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[F9]

A nonzero linear map from a finite-dimensional real vector space to $R$ is onto, because any nonzero value can be scaled to any prescribed real value. Its kernel therefore has codimension one by Rank-nullity: $\dim_{F} V = nullity T + rank T$ .

Proof

technique · first the action and face identities, then the explicit two-dimensional model, then the finite and infinite chamber computations

1.1givenF2F3F4

The dual action is an action by linear maps. For $w \in W$ and $f \in V^{*}$ the formula $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ defines an element of $V^{*}$ because a composite of linear maps is linear, and $f \mapsto w \cdot f$ is linear because evaluation at $v$ is linear in $f$ . Moreover $i d \cdot f = f$ , since $ρ (i d) = i d$ , and $w_{1} \cdot (w_{2} \cdot f) = f \circ ρ (w_{2})^{- 1} \circ ρ (w_{1})^{- 1} = f \circ ρ (w_{1} w_{2})^{- 1} = (w_{1} w_{2}) \cdot f,$ using $ρ (w_{1} w_{2}) = ρ (w_{1}) ρ (w_{2})$ and hence $ρ (w_{1} w_{2})^{- 1} = ρ (w_{2})^{- 1} ρ (w_{1})^{- 1}$ .

1.2givenF1F2F3F4algebra

Generators on the dual plane. The formulas in [F1] show $r_{s} (P) = P = r_{t} (P)$ , since each map preserves $P$ and is its own inverse. Hence every $g \in W_{s, t} = ⟨ r_{s}, r_{t} ⟩$ and its inverse preserve $P$ . The restriction $g \mapsto g ∣_{P}$ is a homomorphism into $G L (P)$ , so $(g \cdot φ) (v) = φ (g^{- 1} v)$ is well-defined on $P^{*}$ and is a left action: $(g h) \cdot φ = φ \circ h^{- 1} \circ g^{- 1} = g \cdot (h \cdot φ)$ , and the identity acts trivially. Here $s \cdot φ$ and $t \cdot φ$ denote the actions of $r_{s}$ and $r_{t}$ respectively. The dual basis vectors satisfy $(s \cdot f_{s}) (e_{s}) = f_{s} (ρ (s)^{- 1} e_{s}) = f_{s} (- e_{s}) = - 1$ and $(s \cdot f_{s}) (e_{t}) = f_{s} (ρ (s) e_{t}) = f_{s} (e_{t} + 2 c e_{s}) = 2 c$ , while $(s \cdot f_{t}) (e_{s}) = f_{t} (- e_{s}) = 0$ and $(s \cdot f_{t}) (e_{t}) = f_{t} (e_{t} + 2 c e_{s}) = 1$ ; hence in coordinates $s \cdot (y_{s} f_{s} + y_{t} f_{t}) = (- y_{s}) f_{s} + (2 c y_{s} + y_{t}) f_{t}$ , that is $ρ (s)^{*} (y_{s}, y_{t}) = (- y_{s}, 2 c y_{s} + y_{t}) .$ The same computation with $s$ replaced by $t$ gives $ρ (t)^{*} (y_{s}, y_{t}) = (y_{s} + 2 c y_{t}, - y_{t})$ . Both displayed maps are involutions: applying the first twice returns $(y_{s}, y_{t})$ , and applying the second twice returns $(y_{s}, y_{t})$ . The first fixes the line ${φ : φ (e_{s}) = 0} = {y_{s} = 0}$ pointwise and the second fixes ${φ : φ (e_{t}) = 0} = {y_{t} = 0}$ pointwise, since these are exactly the vectors whose coordinate in question vanishes. For the functional $δ (φ) = φ (e_{s} + e_{t}) = y_{s} + y_{t}$ one has $δ (s \cdot φ) = (2 c - 1) y_{s} + y_{t}$ and $δ (t \cdot φ) = y_{s} + (2 c - 1) y_{t}$ , so each generator preserves $δ$ exactly when $c = 1$ , i.e. when $m (s, t) = \infty$ ; in that case $W_{s, t}$ preserves $δ$ because it is generated by $r_{s}$ and $r_{t}$ .

1.3givenF2F4F5F9

Faces. Let $I \subseteq S$ and put $f_{I} : = \sum_{s \notin I} f_{s} \in V^{*}$ . By the dual-basis property $f_{I} (e_{s}) = 1$ for $s \notin I$ and $f_{I} (e_{s}) = 0$ for $s \in I$ ; hence $f_{I} \in C_{I}$ and every face is nonempty. In particular $I = \emptyset$ gives $\sum_{s \in S} f_{s} \in C^{\circ}$ , so the interior is nonempty, and $I = S$ gives $C_{S} = {f : f (e_{s}) = 0 for all s \in S} = {0}$ , since a functional vanishing on the basis $(e_{s})$ of $V$ is zero. For a root $α$ , $B (α, α) = 1$ makes $α \neq 0$ , so some coordinate $α (s) \neq 0$ and $f_{s} (α) = α (s) \neq 0$ . Thus ${e v}_{α} : V^{*} \to R$ is a nonzero linear map and $H_{α}$ is a hyperplane by [F9].

1.4givenF1F2F5F6F7algebra

The finite rank-two model. Assume $m < \infty$ and put $θ : = π / m$ and $c = \cos θ$ . Then $0 < θ \leq π / 2$ , $\sin θ \neq 0$ and $1 - c^{2} = \sin^{2} θ$ . On $P$ put $u : = e_{s}$ and $v : = (e_{t} + c e_{s}) / \sin θ$ . Then $B (u, u) = B (e_{s}, e_{s}) = 1$ , $B (u, v) = (B (e_{s}, e_{t}) + c B (e_{s}, e_{s})) / \sin θ = (- c + c) / \sin θ = 0$ and $B (v, v) = (B (e_{t}, e_{t}) + 2 c B (e_{s}, e_{t}) + c^{2} B (e_{s}, e_{s})) / \sin^{2} θ = (1 - 2 c^{2} + c^{2}) / \sin^{2} θ = (1 - c^{2}) / \sin^{2} θ = 1$ : thus $(u, v)$ is a $B ∣_{P}$ -orthonormal basis of $P$ and $e_{s} = u$ , $e_{t} = - c u + \sin θ v$ . Since $r_{s}$ fixes $\ker B (-, e_{s}) \cap P = R v$ pointwise and sends $u = e_{s}$ to $- u$ , in the coordinates $(x_{u}, x_{v})$ of the basis $(u, v)$ it is $r_{s} (x_{u}, x_{v}) = (- x_{u}, x_{v})$ ; since $\ker B (-, e_{t}) \cap P$ is spanned by $(\sin θ, c)$ and $r_{t} (e_{t}) = - e_{t}$ , the map $r_{t}$ is the reflection in the line $R (\sin θ, c)$ . Consequently the composite $A : = r_{s} r_{t}$ satisfies $A (u) = r_{s} (e_{s} + 2 c e_{t}) = r_{s} ((1 - 2 c^{2}) u + 2 c \sin θ v) = (2 c^{2} - 1) u + 2 c \sin θ v = (\cos 2 θ) u + (\sin 2 θ) v$ and $A (v) = - \sin 2 θ u + \cos 2 θ v$ , so in the orthonormal coordinates $A (x_{u}, x_{v}) = (x_{u} \cos 2 θ - x_{v} \sin 2 θ, x_{u} \sin 2 θ + x_{v} \cos 2 θ),$ the rotation through $2 θ$ , whose order is exactly $m$ .

2.1givenF1F2F6step 1.4

The rank-two roots. Assume $m < \infty$ and keep the orthonormal coordinates $(u, v)$ of 1.4, in which $A$ is the rotation through $2 θ$ . Then $A^{k} e_{s} = (\cos 2 k θ) u + (\sin 2 k θ) v$ for every $k \in Z$ , and $r_{s} A^{k} e_{s} = (- \cos 2 k θ) u + (\sin 2 k θ) v = (\cos (m - 2 k) θ) u + (\sin (m - 2 k) θ) v$ because $π = m θ$ and $\cos (π - α) = - \cos α$ , $\sin (π - α) = \sin α$ . Moreover $e_{t} = - \cos θ u + \sin θ v = (\cos (m - 1) θ) u + (\sin (m - 1) θ) v$ , and the same two formulas applied to $e_{t}$ give $A^{k} e_{t}$ and $r_{s} A^{k} e_{t}$ at angles $(m - 1 + 2 k) θ$ and $(1 - 2 k) θ$ . As $k$ varies over $Z$ the angles $2 k θ$ are exactly the even multiples of $θ$ and the angles $(1 - 2 k) θ$ are exactly the odd multiples, so every element of $S : = {(\cos j θ) u + (\sin j θ) v : j = 0, 1, \dots, 2 m - 1}$ is one of the four displayed families and $S \subseteq Φ_{P}$ ; conversely $S$ contains $e_{s}$ and $e_{t}$ and is stable under $A$ (which adds $2 θ$ to an angle) and under $r_{s}$ (which sends an angle $φ$ to $π - φ = (m θ - φ)$ ), hence stable under $W_{s, t} = ⟨ ρ (s), ρ (t) ⟩ = ⟨ ρ (s), A ⟩$ , so $Φ_{P} \subseteq S$ . Therefore $Φ_{P} = W_{s, t} {e_{s}, e_{t}} = {(\cos j θ) u + (\sin j θ) v : j = 0, 1, \dots, 2 m - 1},$ a set of exactly $2 m$ unit vectors, and $Φ_{P}$ is stable under $β \mapsto - β$ .

2.2givenF1F2F8step 1.2algebra

The infinite case. Assume $m = \infty$ , so $c = 1$ and by 1.2 the generators act by $s \cdot (y_{s}, y_{t}) = (- y_{s}, 2 y_{s} + y_{t})$ and $t \cdot (y_{s}, y_{t}) = (y_{s} + 2 y_{t}, - y_{t})$ and preserve $δ = y_{s} + y_{t}$ . On the affine line ${δ = 1}$ parametrized by $τ = y_{s}$ these maps read $τ \mapsto - τ$ and $τ \mapsto 2 - τ$ , the two reflections of the line about $0$ and $1$ . Every word in the two involutions rewrites to $A^{k}$ or $A^{k} r_{s}$ with $k \in Z$ , where $A = r_{s} r_{t}$ ; their actions on this affine line are respectively $τ \mapsto τ - 2 k$ and $τ \mapsto - τ - 2 k$ . These are pairwise distinct, so restriction to $P$ is faithful on $W_{s, t}$ . The interval images are $A^{k} [0, 1] = [- 2 k, 1 - 2 k]$ and $A^{k} r_{s} [0, 1] = [- 1 - 2 k, - 2 k]$ , so they are exactly $[j, j + 1]$ for $j \in Z$ , with distinct elements giving distinct intervals. These cover the affine line by [F8] and have pairwise disjoint interiors. Since $C_{P}$ is the cone on $[0, 1]$ in the half-plane ${δ \geq 0}$ and the action is linear, the chamber $w C_{P}$ is the cone on the interval corresponding to $w$ ; these cones have pairwise disjoint interiors and their union is ${φ : δ (φ) > 0} \cup {0}$ , because every point with $δ > 0$ is a positive multiple of a unique point of ${δ = 1}$ lying in a unit interval, while every point of a cone on a unit interval has $δ \geq 0$ and only $0$ has $δ = 0$ . By [F1] and the normal forms above, the roots are precisely the four families $A^{k} e_{s} = (1 + 2 k, 2 k)$ , $A^{k} e_{t} = (- 2 k, 1 - 2 k)$ , $A^{k} r_{s} e_{s} = (- 1 - 2 k, - 2 k)$ and $A^{k} r_{s} e_{t} = (2 + 2 k, 1 + 2 k)$ for $k \in Z$ . Equivalently $Φ_{P} = {(j + 1, j), (j, j + 1) : j \in Z}$ , since the first and fourth families cover the even and odd values of $j$ in the first pair, and the second and third do so in the second pair. On ${δ = 1}$ the equation $φ (a e_{s} + b e_{t}) = 0$ is $(a - b) y_{s} + b = 0$ , so the two root families give wall traces $- j$ and $j + 1$ , exactly the integers. If two chamber intervals are $[j, j + 1]$ and $[l, l + 1]$ with $j < l$ , the wall whose trace is $j + 1$ places their interiors in opposite open half-planes, because its defining linear form on ${δ > 0}$ has the sign of $y_{s} / δ - (j + 1)$ up to a fixed nonzero factor.

3.1givenF1F2F3F6step 1.4step 2.1

The rank-two walls. Assume $m < \infty$ . The map $♭ : P \to P^{*}$ , $x \mapsto B (x, \cdot)$ , is an isomorphism: in the orthonormal basis $(u, v)$ of 1.4 its inverse sends $φ$ to $φ (u) u + φ (v) v$ . Moreover it intertwines the two actions, since $(g \cdot ♭ (x)) (z) = B (x, g^{- 1} z) = B (g x, z) = ♭ (g x) (z)$ for $g \in W_{s, t}$ . For $β \in P$ one has $♭ (β^{⊥}) = H_{β} \cap P^{*}$ , where $β^{⊥} = {x \in P : B (x, β) = 0}$ : indeed $♭ (x) (β) = B (x, β)$ vanishes exactly when $x \in β^{⊥}$ . By 2.1 each $β \in Φ_{P}$ is a unit vector at an angle $j θ$ , so $β^{⊥}$ is the line at angle $j θ + π / 2$ , and the lines $H_{β} \cap P^{*}$ depend only on the pair ${β, - β}$ : there are exactly $m$ of them, at angles $π / 2 + j θ$ , $j = 0, \dots, m - 1$ , equally spaced by $θ$ . The chamber $C_{P} = {φ \in P^{*} : φ (e_{s}) \geq 0, φ (e_{t}) \geq 0}$ corresponds under $♭$ to ${x \in P : B (x, e_{s}) \geq 0, B (x, e_{t}) \geq 0}$ , whose boundary lines are $e_{s}^{⊥}$ (the line at angle $π / 2$ ) and $e_{t}^{⊥}$ (the line at angle $(m - 1) θ + π / 2$ , i.e. $π / 2 - θ$ ): so $C_{P} = ♭$ of the closed sector between these two consecutive walls, of angle $θ$ , and no root hyperplane line meets $C_{P}^{\circ}$ because the $m$ walls are equally spaced by $θ$ and contain both boundary lines.

4.1givenF1F2F3step 1.2step 1.4step 2.1step 3.1∎

The finite chamber tiling. Assume $m < \infty$ . Put $a : = r_{s}$ , $b : = r_{t}$ , and $A : = a b$ . Since $a^{2} = b^{2} = 1$ and $a A a = A^{- 1}$ , every word in $a, b$ rewrites to $A^{k}$ or $A^{k} a$ . The finite relation $A^{m} = 1$ therefore gives at most $2 m$ elements, represented by $0 \leq k < m$ . Their restrictions to $P$ are all distinct: the rotations $A^{k} ∣_{P}$ are distinct by their exact order $m$ , the maps $A^{k} a ∣_{P}$ are distinct after cancellation of $a ∣_{P}$ , and a rotation cannot equal a reflection because their determinants on $P$ are $1$ and $- 1$ . Thus restriction to $P$ is faithful on this matrix subgroup, $∣ W_{s, t} ∣ = 2 m$ , and it is the dihedral group of order $2 m$ . Its contragredient action on $P^{*}$ has the same distinct elements. The action maps the set of $m$ walls to itself, because $g \cdot (H_{β} \cap P^{*}) = H_{g β} \cap P^{*}$ for $g \in W_{s, t}$ and $Φ_{P}$ is $W_{s, t}$ -stable, so it permutes the $2 m$ closed sectors; it maps the sector $C_{P}$ to the sectors obtained by successive rotations through $2 θ$ and by the reflection $ρ (s)$ , which realizes all $2 m$ of them: under $♭^{- 1}$ , $C_{P}$ has angular interval $[π / 2 - θ, π / 2]$ and $r_{s} C_{P}$ has interval $[π / 2, π / 2 + θ]$ ; their $A^{k}$ images add $2 k θ$ , giving all consecutive sectors modulo $2 π$ . Thus the orbit of $C_{P}$ is exactly the set of closed sectors cut out by the $m$ root hyperplanes. Since $∣ W_{s, t} ∣ = 2 m$ equals the number of sectors, the action on the orbit is simply transitive, and the sectors have pairwise disjoint interiors and union $P^{*}$ as the $2 m$ sectors of $m$ distinct lines through the origin. Finally, a sign choice for each of the $m$ wall forms defines an intersection of open half-planes. If nonempty it is convex, by linearity of those forms, and so lies in a single sector: the segment joining any two of its points meets no wall. Thus distinct sector interiors have different signs for some wall form, which supplies a root hyperplane separating their entire interiors.

Remarks

Davis, Example D.2.1(i), printed p. 442, describes the infinite-dihedral Tits cone as the closed half-plane. For the literal union of the chamber images, step 2.2 shows that every nonzero point of its boundary is absent; the closed half-plane is the closure of that union.

5 · Examples, counterexamples and false statements

None yet.

Real Forms and Reflection Geometry

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

The real Coxeter form, its radical, reflections, and form-preserving maps

Definition

Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order

Statement

Facts & Assumptions

Proof

The canonical reflection homomorphism, roots, reflections, and the positive cone

Definition

Descent of the reflection representation, unit root norms, and conjugation of reflections

Statement

Facts & Assumptions

Proof

The dual action, chambers, faces, and root hyperplanes

Definition

The dual action, the faces, and the rank-two chamber tiling

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, counterexamples and false statements

Sources