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Canonical Roots, Signs, and Faithful Reflections — Examples

1 · Prerequisites

2 · Summary

This companion is a dependency leaf: its examples use the theory of canonical-roots-signs-and-faithful-reflections, that page's prerequisite closure, and the free-product universal property from free-products-and-amalgamation, and no other page or item depends on them.

Roots, inversions and chamber images in $I_{2} (5)$ , $A_{2}$ and infinite dihedral type computes the full rank-two data for $m = 3, 5, \infty$ : the positive roots, the inversion sets $N (u_{k})$ of the alternating words together with the complement formula $N (u_{k}) = Φ_{+} ∖ N (u_{k - 5}^{'})$ and the explicit value of $N (u_{7})$ in $I_{2} (5)$ , and the chamber geometry — six sectors $C^{\circ}, s C^{\circ}, s t C^{\circ}, s t s C^{\circ}, t s C^{\circ}, t C^{\circ}$ with $s t s C^{\circ} = - C^{\circ}$ in $A_{2}$ , the ten sectors cut out by the five root lines with the wall-separation ranges in $I_{2} (5)$ , and, for infinite dihedral type, the roots $e_{s} + k u$ , $e_{t} + k u$ along the null vector $u = e_{s} + e_{t}$ with the chambers realised as cones over the intervals $(j, j + 1)$ of the affine line $f (e_{s}) + f (e_{t}) = 1$ .

An indefinite Coxeter form with a faithful canonical reflection representation shows that an indefinite or degenerate Coxeter form does not obstruct faithfulness: for the all-infinite rank-three matrix the form has signature $(2, 1)$ , and for the degenerate rank-two form its radical is $R (e_{s} + e_{t})$ ; the root-length criterion applies verbatim to both, while what fails in the degenerate case is only the identification of $V$ with $V^{*}$ through $B$ . A vector with mixed signs is not a root, while every root has a sign marks the exact scope of the sign theorem: the vector $e_{s} - e_{t}$ has mixed signs and $B$ -norm $2 + 2 c \neq 1$ and so is not a root, although every element of the $W$ -orbit of the simple roots indeed has a sign; the comparison cases $m = 3$ and $m = \infty$ are evaluated explicitly.

The results tested here are proved on the theory page: the rank-two chamber and length facts of The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ , the sign partition of Root sign coherence and the action of simple reflections on positive roots, the faithfulness of The root-length criterion and faithfulness of the canonical reflection representation, the inversion recursion of The geometric inversion set $N (w)$ of an element of a Coxeter group and the inversion formula of The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange. The examples are evidence within their computed scope and do not replace those proofs.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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

Roots, inversions and chamber images in $I_{2} (5)$ , $A_{2}$ and infinite dihedral type

Example

Let $S = {s, t}$ with $m : = m (s, t) \in {3, 5, \infty}$ , $V = R^{S}$ , and let $B$ be the Coxeter form with $c = \cos (π / m)$ for $m < \infty$ and $c = 1$ for $m = \infty$ , so that $B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - c$ (The real Coxeter form, its radical, reflections, and form-preserving maps). Let $ρ, Φ, Φ_{\pm}, N, C^{\circ}, W, ℓ$ be as on this page (The canonical reflection homomorphism, roots, reflections, and the positive cone, Root sign coherence and the action of simple reflections on positive roots, The geometric inversion set $N (w)$ of an element of a Coxeter group, The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange), and let $u_{k}$ denote the alternating word $s t s \dots$ of $k$ letters. The following data are computed and verified.

(i) $A_{2} = I_{2} (3)$ . $c = \frac{1}{2}$ and $Φ_{+} = {e_{s}, e_{t}, e_{s} + e_{t}}$ . For the six elements, $N (1) = \emptyset, N (s) = {e_{s}}, N (t) = {e_{t}}, N (s t) = {e_{t}, e_{s} + e_{t}}, N (t s) = {e_{s}, e_{s} + e_{t}}, N (s t s) = Φ_{+},$ each of cardinality $ℓ (w)$ . In the dual plane $V^{*}$ the three walls $H_{e_{s}}, H_{e_{t}}, H_{e_{s} + e_{t}}$ cut out six sectors, and the six chambers $w C^{\circ}$ occur in the cyclic order $C^{\circ}, s C^{\circ}, s t C^{\circ}, s t s C^{\circ}, t s C^{\circ}, t C^{\circ}$ , with $s t s C^{\circ} = - C^{\circ}$ ; explicitly $H_{e_{s}}$ separates $C^{\circ}$ from $u_{1} C^{\circ}, \dots, u_{3} C^{\circ}$ and $H_{e_{t}}$ from $u_{3} C^{\circ}, \dots, u_{5} C^{\circ}$ .

(ii) $I_{2} (5)$ . $2 c = ϕ = \frac{1 + \sqrt{5}}{2}$ , $4 c^{2} - 1 = ϕ$ , and $Φ_{+} = {e_{s}, e_{t}, e_{t} + ϕ e_{s}, e_{s} + ϕ e_{t}, ϕ (e_{s} + e_{t})},$ five roots, each of $B$ -norm one. With $ℓ (u_{k}) = \min (k, 10 - k)$ for $0 \leq k < 10$ , for $1 \leq k \leq 5$ $N (u_{1}) = {e_{s}}, N (u_{2}) = {e_{t}, e_{s} + ϕ e_{t}}, N (u_{3}) = {e_{s}, e_{t} + ϕ e_{s}, ϕ (e_{s} + e_{t})},$ $N (u_{4}) = {e_{t}, e_{s} + ϕ e_{t}, ϕ (e_{s} + e_{t}), ϕ e_{s} + e_{t}}, N (u_{5}) = Φ_{+},$ and for $5 \leq k < 10$ one has $N (u_{k}) = Φ_{+} ∖ N (u_{k - 5}^{'})$ , where $u_{j}^{'}$ is the alternating word of length $j$ beginning with $t$ ; for instance $N (u_{7}) = {e_{t}, e_{s} + ϕ e_{t}, ϕ (e_{s} + e_{t})}$ , of cardinality $3 = ℓ (u_{7})$ . The ten chambers $u_{k} C^{\circ}$ are the sectors cut out by the five root lines; the chambers on the negative side of $H_{e_{s}}$ are exactly $u_{1} C^{\circ}, \dots, u_{5} C^{\circ}$ , and those on the negative side of $H_{e_{t}}$ are exactly $u_{5} C^{\circ}, \dots, u_{9} C^{\circ}$ .

(iii) $I_{2} (\infty)$ . $c = 1$ . Put $u : = e_{s} + e_{t}$ ; then $B (u, \cdot) = 0$ on $R e_{s} + R e_{t}$ , and $Φ_{+} = {e_{s} + k u : k \geq 0} \cup {e_{t} + k u : k \geq 0}$ is infinite. For every $k \geq 0$ , $N ((s t)^{k}) = {j u + e_{t} : 0 \leq j \leq 2 k - 1}, N ((s t)^{k} s) = {j u + e_{s} : 0 \leq j \leq 2 k},$ of cardinalities $2 k = ℓ ((s t)^{k})$ and $2 k + 1 = ℓ ((s t)^{k} s)$ . The chambers $w C^{\circ}$ are the cones over the intervals $(j, j + 1)$ of the affine line ${f : f (e_{s}) + f (e_{t}) = 1}$ , and the walls $H_{e_{s}}, H_{e_{t}}$ meet that line in the points $0$ and $1$ .

Facts & Assumptions

Given: a two-element set $S = {s, t}$ with $m : = m (s, t) \in {3, 5, \infty}$ , the space $V = R^{S}$ with basis vectors $e_{s}, e_{t}$ , the Coxeter form $B$ with constant $c$ ( $c = \cos (π / m)$ for $m < \infty$ , $c = 1$ for $m = \infty$ ), the presented Coxeter group $W$ with length function $ℓ$ , the canonical reflection homomorphism $ρ : W \to G L (V)$ with root system $Φ$ , the partition $Φ = Φ_{+} ⊔ Φ_{-}$ , the inversion sets $N (w)$ of The geometric inversion set $N (w)$ of an element of a Coxeter group, and the alternating words $u_{k} = s t s \dots$ of $k$ letters beginning with $s$ .

[F1]

$B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - c$ ; the reflection $r_{a}$ with normal $a$ , $B (a, a) \neq 0$ , is $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ , so that $r_{s} (e_{s}) = - e_{s}$ , $r_{s} (e_{t}) = e_{t} + 2 c e_{s}$ and $r_{t} (e_{t}) = - e_{t}$ , $r_{t} (e_{s}) = e_{s} + 2 c e_{t}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

[F2]

$ρ$ is a homomorphism with $ρ (s) = r_{s}$ , $ρ (t) = r_{t}$ , so $ρ (u v) = ρ (u) ρ (v)$ and $ρ (1) = {i d}_{V}$ ; the root system is $Φ = {ρ (w) e_{r} : w \in W, r \in {s, t}}$ , every root has $B$ -norm one, and $Φ$ is invariant under every $ρ (w)$ ; moreover $Φ_{+} = Φ \cap V_{+}$ and $Φ_{-} = Φ \cap (- V_{+}) = - Φ_{+}$ with $Φ = Φ_{+} ⊔ Φ_{-}$ , $V_{+} = {λ e_{s} + μ e_{t} : λ, μ \geq 0}$ and $V_{+} \cap (- V_{+}) = {0}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections, Root sign coherence and the action of simple reflections on positive roots (1), (2)).

[F3]

$N (w) = {α \in Φ_{+} : ρ (w) α \in Φ_{-}}$ for $w \in W$ ; $N (1) = \emptyset$ ; and for $u \in W$ , $r \in {s, t}$ with $ℓ (u r) > ℓ (u)$ one has $N (u r) = {e_{r}} ⊔ ρ (r) N (u)$ , while for $ℓ (u r) < ℓ (u)$ one has $N (u r) = ρ (r) (N (u) ∖ {e_{r}})$ , where $ρ (r) N (u) = {ρ (r) β : β \in N (u)}$ (The geometric inversion set $N (w)$ of an element of a Coxeter group (1), (2)).

[F4]

$∣ N (w) ∣ = ℓ (w)$ for every $w \in W$ , and the map $α \mapsto t_{α}$ is a bijection from $Φ_{+}$ onto the reflection set $T$ (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1), (2)).

[F5]

In $W$ one has $s^{2} = t^{2} = 1$ and, when $m < \infty$ , $(s t)^{m} = 1$ ; if $m < \infty$ the elements $u_{0}, \dots, u_{2 m - 1}$ exhaust $W$ and $ℓ (u_{k}) = \min (k, 2 m - k)$ ; if $m = \infty$ the elements $u_{k}$ are pairwise distinct with $ℓ (u_{k}) = k$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ (1)).

[F6]

The dual action is $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ , and $f \in w C^{\circ}$ if and only if $f (ρ (w) e_{s}) > 0$ and $f (ρ (w) e_{t}) > 0$ ; for every $w \in W$ and $r \in {s, t}$ one has $w C^{\circ} \subseteq r B_{r}$ if and only if $ℓ (r w) < ℓ (w)$ . If $m < \infty$ , the $2 m$ chambers $w C^{\circ}$ $(w \in W)$ are exactly the $2 m$ sectors cut out by the $m$ root hyperplanes $H_{β}$ $(β \in Φ)$ , with pairwise disjoint interiors; the corresponding closed sectors cover $V^{*}$ , while the open sectors cover its complement of the root lines. If $m = \infty$ , the chambers have pairwise disjoint interiors, the corresponding closed chambers have union ${f \in V^{*} : f (e_{s} + e_{t}) > 0} \cup {0}$ , distinct chambers are separated by a root hyperplane, and the traces $H_{β} \cap {f : f (e_{s}) + f (e_{t}) = 1}$ are exactly the integers in the coordinate $y_{s} = f (e_{s})$ on that affine line (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling (3), The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ (1), (4)).

[F7]

Addition formulas: $\sin (x + y) = \sin x \cos y + \cos x \sin y$ and $\cos (x + y) = \cos x \cos y - \sin x \sin y$ ; $\cos (- x) = \cos x$ and $\sin^{2} x + \cos^{2} x = 1$ ; $\cos (x + π) = - \cos x$ , $\cos (π / 2) = 0$ and $\cos π = - 1$ ; cosine is strictly decreasing on $[0, π]$ ; $π > 0$ and $π / 2$ is the smallest positive zero of cosine; every $a \geq 0$ has a unique nonnegative square root $\sqrt{a}$ , and $\sqrt{5} > 1$ ; a product of two real numbers is zero only if one factor is zero (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Quarter-turn values and shifts by pi/2 and pi, Signs, monotonicity intervals, and ranges of sine and cosine, Pi as twice the smallest positive zero of cosine, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Verification

technique · direct

1.1F1F7algebra

The case $A_{2}$ : the constant. Put $x : = π / 3$ , $q : = \cos x$ and $r : = \sin x$ . The addition formulas give $\cos 2 x = \cos^{2} x - \sin^{2} x = 2 q^{2} - 1$ and $\sin 2 x = 2 q r$ , hence $\cos 3 x = \cos (2 x + x) = (2 q^{2} - 1) q - (2 q r) r = 2 q^{3} - q - 2 q (1 - q^{2}) = 4 q^{3} - 3 q$ ; since $3 x = π$ and $\cos π = - 1$ , we get $4 q^{3} - 3 q + 1 = 0$ , that is $(q + 1) (2 q - 1)^{2} = 0$ . Now $0 < π / 3 < π / 2 < π$ and cosine is strictly decreasing on $[0, π]$ with $\cos 0 = 1$ and $\cos (π / 2) = 0$ , so $0 < q < 1$ ; hence $q \neq - 1$ and $q = 1 / 2$ . Thus in this case $c = 1 / 2$ , and the reflection formulas of [F1] give $ρ (s) e_{t} = e_{t} + e_{s}$ and $ρ (t) e_{s} = e_{s} + e_{t}$ .

1.2F5F6algebra

The case $I_{2} (5)$ : the chambers. By [F6] the ten chambers $u_{k} C^{\circ}$ $(0 \leq k < 10)$ are exactly the ten sectors cut out by the five root lines. By [F5], $(s t)^{5} = 1$ together with $s^{2} = t^{2} = 1$ gives $s u_{k} = u_{1 - k}$ and $t u_{k} = u_{- 1 - k}$ with indices modulo $10$ ; combined with $ℓ (u_{k}) = \min (k, 10 - k)$ this gives $ℓ (s u_{k}) = ℓ (u_{k}) - 1$ exactly for $1 \leq k \leq 5$ , and $ℓ (t u_{k}) = ℓ (u_{k}) - 1$ exactly for $5 \leq k \leq 9$ . By the shortening equivalence of [F6], the chambers on the negative side of $H_{e_{s}}$ are exactly $u_{1} C^{\circ}, \dots, u_{5} C^{\circ}$ and those on the negative side of $H_{e_{t}}$ exactly $u_{5} C^{\circ}, \dots, u_{9} C^{\circ}$ .

1.3F1F2F5algebra

The case $I_{2} (\infty)$ : the invariant functional direction. Here $c = 1$ , so $B (u, u) = B (e_{s}, e_{s}) + 2 B (e_{s}, e_{t}) + B (e_{t}, e_{t}) = 1 - 2 + 1 = 0$ , $B (u, e_{s}) = B (e_{s}, e_{s}) + B (e_{t}, e_{s}) = 1 - 1 = 0$ and $B (u, e_{t}) = B (e_{s}, e_{t}) + B (e_{t}, e_{t}) = - 1 + 1 = 0$ , where $u : = e_{s} + e_{t}$ ; by bilinearity $B (u, \cdot) = 0$ on $R e_{s} + R e_{t}$ . Moreover $ρ (s) u = ρ (s) e_{s} + ρ (s) e_{t} = - e_{s} + (e_{t} + 2 e_{s}) = u$ and $ρ (t) u = (e_{s} + 2 e_{t}) - e_{t} = u$ , so $ρ (w) u = u$ for every $w \in W$ , because $W$ is generated by $s, t$ and $ρ$ is a homomorphism.

2.1F2F6step 1.1

The case $A_{2}$ : the positive roots. By 1.1, the three vectors $e_{s} = ρ (1) e_{s}$ , $e_{t} = ρ (1) e_{t}$ and $e_{s} + e_{t} = ρ (t) e_{s}$ are roots; they lie in $V_{+} ∖ {0}$ , hence in $Φ_{+}$ , and they are pairwise distinct as functions on ${s, t}$ , with values $(1, 0)$ , $(0, 1)$ , $(1, 1)$ . By [F6] the six chambers are cut out by the three root hyperplanes $H_{β}$ , $β \in Φ$ ; the map $α \mapsto H_{α}$ from $Φ_{+}$ onto these lines is injective, because two roots on one line are proportional and both have $B$ -norm one by [F2], forcing the proportionality factor to be $\pm 1$ , and it is surjective because every $H_{β}$ equals $H_{- β}$ with exactly one of $\pm β$ in $Φ_{+}$ . Hence $∣ Φ_{+} ∣ = 3$ and $Φ_{+} = {e_{s}, e_{t}, e_{s} + e_{t}}$ .

2.2F5F6step 1.1algebra

The case $A_{2}$ : the chambers. Write $y_{s} : = f (e_{s})$ and $y_{t} : = f (e_{t})$ for $f \in V^{*}$ , so that $f$ is described by the pair $(y_{s}, y_{t})$ . By [F6], $f \in w C^{\circ}$ if and only if $f (ρ (w) e_{s}) > 0$ and $f (ρ (w) e_{t}) > 0$ . Using the products $ρ (s) e_{s} = - e_{s}$ , $ρ (s) e_{t} = e_{s} + e_{t}$ , $ρ (t) e_{t} = - e_{t}$ , $ρ (t) e_{s} = e_{s} + e_{t}$ from [F1] and 1.1, together with $ρ (s t) e_{s} = e_{t}$ , $ρ (s t) e_{t} = - e_{s} - e_{t}$ , $ρ (t s) e_{s} = - e_{s} - e_{t}$ , $ρ (t s) e_{t} = e_{s}$ , $ρ (s t s) e_{s} = - e_{t}$ , $ρ (s t s) e_{t} = - e_{s}$ (computed from $ρ (u v) = ρ (u) ρ (v)$ ), this gives $C^{\circ} = {y_{s} > 0, y_{t} > 0}$ , $s C^{\circ} = {y_{s} < 0, y_{s} + y_{t} > 0}$ , $s t C^{\circ} = {y_{t} > 0, y_{s} + y_{t} < 0}$ , $s t s C^{\circ} = {y_{s} < 0, y_{t} < 0} = - C^{\circ}$ , $t s C^{\circ} = {y_{s} > 0, y_{s} + y_{t} < 0}$ and $t C^{\circ} = {y_{t} < 0, y_{s} + y_{t} > 0}$ . On each of these six sets the signs of $(y_{s}, y_{t}, y_{s} + y_{t})$ are the six distinct patterns $(+, +, +)$ , $(-, +, +)$ , $(-, +, -)$ , $(-, -, -)$ , $(+, -, -)$ , $(+, -, +)$ ; these are exactly the six combinations not excluded by one of the three equations $y_{s} = 0$ , $y_{t} = 0$ , $y_{s} + y_{t} = 0$ , so the six chambers are the six sectors, in the stated cyclic order. By [F5], $s^{2} = t^{2} = 1$ and $(s t)^{3} = 1$ give $s (s t)^{j} = (t s)^{j} s$ and $t (s t)^{j} = (t s)^{j} t$ for all $j \geq 0$ and $(t s)^{j} = (s t)^{- j}$ , whence $s u_{k} = u_{1 - k}$ and $t u_{k} = u_{- 1 - k}$ with indices read modulo $6$ ; combined with $ℓ (u_{k}) = \min (k, 6 - k)$ this gives $ℓ (s u_{k}) = ℓ (u_{k}) - 1$ exactly for $k \in {1, 2, 3}$ and $ℓ (t u_{k}) = ℓ (u_{k}) - 1$ exactly for $k \in {3, 4, 5}$ . By the shortening equivalence of [F6], $H_{e_{s}}$ separates $C^{\circ}$ from $u_{1} C^{\circ}, u_{2} C^{\circ}, u_{3} C^{\circ}$ and $H_{e_{t}}$ from $u_{3} C^{\circ}, u_{4} C^{\circ}, u_{5} C^{\circ}$ .

2.3F7step 1.1algebra

The case $I_{2} (5)$ : the constant. Put $c : = \cos (π / 5)$ (the constant of the form at $m = 5$ ) and $q : = \cos (2 π / 5)$ . By 1.1, $\cos (π / 3) = 1 / 2$ ; since $0 < π / 5 < π / 3 < π$ and cosine is strictly decreasing on $[0, π]$ , we have $c > 1 / 2 > 0$ . The double-angle formula gives $q = 2 c^{2} - 1$ , and since $4 π / 5 = π - π / 5$ with $\cos (π - x) = \cos ((- x) + π) = - \cos (- x) = - \cos x$ , we get $\cos (4 π / 5) = - c$ ; on the other hand $\cos (4 π / 5) = 2 q^{2} - 1 = 2 (2 c^{2} - 1)^{2} - 1$ . Hence $2 (2 c^{2} - 1)^{2} - 1 + c = 0$ , that is $8 c^{4} - 8 c^{2} + c + 1 = 0$ , and because $8 c^{4} - 8 c^{2} + c + 1 = (2 c - 1) (c + 1) (4 c^{2} - 2 c - 1)$ we obtain $4 c^{2} - 2 c - 1 = 0$ , as $c > 1 / 2$ and $c \neq - 1$ . Finally $4 X^{2} - 2 X - 1 = 4 (X - \frac{1 + \sqrt{5}}{4}) (X - \frac{1 - \sqrt{5}}{4})$ by expansion, so $c$ is one of the two numbers $\frac{1 \pm \sqrt{5}}{4}$ ; since $\sqrt{5} > 1$ gives $\frac{1 - \sqrt{5}}{4} < 0 < c$ , we conclude $c = \frac{1 + \sqrt{5}}{4} = ϕ / 2$ with $ϕ : = \frac{1 + \sqrt{5}}{2}$ , and then $2 c = ϕ$ and $4 c^{2} - 1 = 2 c + 1 - 1 = 2 c = ϕ$ .

2.4F1F2step 1.3algebra

The case $I_{2} (\infty)$ : the positive roots. Put $Ψ : = {e_{s} + k u : k \geq 0} \cup {e_{t} + k u : k \geq 0}$ . First, $Ψ \subseteq Φ_{+}$ : clearly $e_{s}, e_{t} \in Ψ \cap Φ_{+}$ , and for $k \geq 1$ the identities $e_{s} + k u = ρ (s) (e_{t} + (k - 1) u)$ and $e_{t} + k u = ρ (t) (e_{s} + (k - 1) u)$ hold by 1.3, so induction on $k$ shows that every element of $Ψ$ is a root; every element of $Ψ$ has nonnegative coefficients in $e_{s}, e_{t}$ , so $Ψ \subseteq Φ \cap V_{+} ∖ {0} = Φ_{+}$ . Conversely, $Ψ \cup (- Ψ)$ is stable under $ρ (s)$ and $ρ (t)$ : by 1.3 one has $ρ (s) (e_{s} + k u) = - e_{s} + k u$ , which is $e_{t} + (k - 1) u$ for $k \geq 1$ and $- e_{s}$ for $k = 0$ ; $ρ (s) (e_{t} + k u) = (e_{t} + 2 e_{s}) + k u = e_{s} + (k + 1) u$ ; $ρ (t) (e_{t} + k u) = - e_{t} + k u$ , which is $e_{s} + (k - 1) u$ for $k \geq 1$ and $- e_{t}$ for $k = 0$ ; and $ρ (t) (e_{s} + k u) = (e_{s} + 2 e_{t}) + k u = e_{t} + (k + 1) u$ . Since $W$ is generated by $s, t$ and $ρ$ is a homomorphism, $Ψ \cup (- Ψ)$ is stable under every $ρ (w)$ ; it contains $e_{s}, e_{t}$ , so it contains the orbit $Φ$ . Hence $Φ \subseteq Ψ \cup (- Ψ)$ , and an element of $Φ_{+}$ lies in $Ψ$ : it lies in $Ψ \cup (- Ψ)$ , and no element of $- Ψ$ has all coefficients nonnegative, while the elements of $V_{+}$ do. Therefore $Φ_{+} = Ψ$ ; finally the two families $e_{s} + k u$ and $e_{t} + k u$ are disjoint and each is injective in $k$ , because $u \neq 0$ while evaluating $e_{s} + k u = e_{s} + k^{'} u$ at $s$ gives $k = k^{'}$ , and $e_{s} + k u = e_{t} + k^{'} u$ would give $1 + k = k^{'}$ at $s$ and $k = 1 + k^{'}$ at $t$ , hence $k = k + 2$ , a contradiction; so $Φ_{+}$ is infinite.

2.5F3F4F5step 1.3algebra

The case $I_{2} (\infty)$ : the inversion sets. By [F5] the $u_{k}$ are pairwise distinct with $ℓ (u_{k}) = k$ , and $u_{2 k} = (s t)^{k}$ , $u_{2 k + 1} = (s t)^{k} s$ . We prove by induction on $k \geq 0$ the two formulas $N ((s t)^{k}) = {j u + e_{t} : 0 \leq j \leq 2 k - 1}$ and $N ((s t)^{k} s) = {j u + e_{s} : 0 \leq j \leq 2 k}$ . For $k = 0$ the first is $N (1) = \emptyset$ and the second is $N (s) = {e_{s}}$ . Assume the second formula for some $k \geq 0$ . Since $ℓ ((s t)^{k + 1}) = 2 k + 2 > 2 k + 1 = ℓ ((s t)^{k} s)$ , the recursion of [F3] gives $N ((s t)^{k + 1}) = {e_{t}} ⊔ ρ (t) N ((s t)^{k} s)$ ; by 1.3, $ρ (t) e_{s} = e_{s} + 2 e_{t} = u + e_{t}$ , so $ρ (t) {j u + e_{s} : 0 \leq j \leq 2 k} = {j u + u + e_{t} : 0 \leq j \leq 2 k} = {(j + 1) u + e_{t} : 0 \leq j \leq 2 k}$ and therefore $N ((s t)^{k + 1}) = {j u + e_{t} : 0 \leq j \leq 2 k + 1}$ , which is the first formula with $k + 1$ . Since $ℓ ((s t)^{k + 1} s) = 2 k + 3 > 2 k + 2 = ℓ ((s t)^{k + 1})$ , again, but with $ρ (s) e_{t} = e_{t} + 2 e_{s} = u + e_{s}$ from 1.3, $N ((s t)^{k + 1} s) = {e_{s}} ⊔ ρ (s) {j u + e_{t} : 0 \leq j \leq 2 k + 1} = {j u + e_{s} : 0 \leq j \leq 2 k + 2}$ , the second formula with $k + 1$ . So both formulas hold for all $k \geq 0$ . Their elements are pairwise distinct because $u \neq 0$ , so the two sets have cardinalities $2 k$ and $2 k + 1$ , equal to $ℓ ((s t)^{k}) = 2 k$ and $ℓ ((s t)^{k} s) = 2 k + 1$ by [F5], in agreement with [F4].

3.1F3F5step 1.1step 2.1algebra

The case $A_{2}$ : the inversion sets. By [F3] and 1.1, and since $ℓ (u_{0}) = 0 < ℓ (u_{1}) = 1 < ℓ (u_{2}) = 2 < ℓ (u_{3}) = 3$ by [F5], one has $N (s) = {e_{s}} ⊔ ρ (s) N (1) = {e_{s}}$ , $N (t) = {e_{t}}$ , $N (s t) = {e_{t}} ⊔ ρ (t) N (s) = {e_{t}, e_{s} + e_{t}}$ , $N (t s) = {e_{s}} ⊔ ρ (s) N (t) = {e_{s}, e_{s} + e_{t}}$ and $N (s t s) = {e_{s}} ⊔ ρ (s) N (s t) = {e_{s}, e_{s} + e_{t}, ρ (s) (e_{s} + e_{t})} = {e_{s}, e_{s} + e_{t}, e_{t}} = Φ_{+}$ , where we used $ρ (s) e_{t} = e_{s} + e_{t}$ and $ρ (s) (e_{s} + e_{t}) = ρ (s) e_{s} + ρ (s) e_{t} = - e_{s} + (e_{s} + e_{t}) = e_{t}$ ; also $N (1) = \emptyset$ . The cardinalities $0, 1, 1, 2, 2, 3$ equal $ℓ (1), ℓ (s), ℓ (t), ℓ (s t), ℓ (t s), ℓ (s t s)$ by [F5], as asserted.

3.2F1F2F6step 2.3algebra

The case $I_{2} (5)$ : the positive roots. By 2.3 and [F1], $ρ (s) e_{t} = e_{t} + ϕ e_{s}$ and $ρ (t) e_{s} = e_{s} + ϕ e_{t}$ , and using $ρ (u v) = ρ (u) ρ (v)$ and $ϕ^{2} = ϕ + 1$ one computes $ρ (s t) e_{s} = ρ (s) (e_{s} + ϕ e_{t}) = - e_{s} + ϕ (e_{t} + ϕ e_{s}) = (ϕ^{2} - 1) e_{s} + ϕ e_{t} = ϕ (e_{s} + e_{t})$ . Hence the five vectors $e_{s} = ρ (1) e_{s}$ , $e_{t} = ρ (1) e_{t}$ , $e_{t} + ϕ e_{s} = ρ (s) e_{t}$ , $e_{s} + ϕ e_{t} = ρ (t) e_{s}$ and $ϕ (e_{s} + e_{t}) = ρ (s t) e_{s}$ are roots; each lies in $V_{+} ∖ {0}$ , so all five lie in $Φ_{+}$ , and they are pairwise distinct as functions on ${s, t}$ , with values $(1, 0)$ , $(0, 1)$ , $(ϕ, 1)$ , $(1, ϕ)$ , $(ϕ, ϕ)$ . By [F6] the ten chambers are cut out by the five root hyperplanes $H_{β}$ , $β \in Φ$ , and as in 2.1 the map $α \mapsto H_{α}$ is a bijection from $Φ_{+}$ onto these lines, so $∣ Φ_{+} ∣ = 5$ and the five listed vectors exhaust $Φ_{+}$ ; each of them has $B$ -norm one because it is a root, by [F2].

4.1F3F4F5step 3.2algebra

The case $I_{2} (5)$ : the inversion sets $N (u_{k})$ , $1 \leq k \leq 5$ . By [F5], $ℓ (u_{k}) = k$ for $1 \leq k \leq 5$ , so each step below is a shortening-free step of the recursion of [F3] and the union is disjoint. Starting from $N (1) = \emptyset$ and using the reflection values $ρ (t) e_{s} = e_{s} + ϕ e_{t}$ , $ρ (s) e_{t} = e_{t} + ϕ e_{s}$ , $ρ (s) (e_{s} + ϕ e_{t}) = ϕ (e_{s} + e_{t})$ , $ρ (t) (e_{t} + ϕ e_{s}) = ϕ (e_{s} + e_{t})$ , $ρ (t) ϕ (e_{s} + e_{t}) = ϕ e_{s} + e_{t}$ and $ρ (s) ϕ (e_{s} + e_{t}) = e_{s} + ϕ e_{t}$ (all from [F1], $ρ (u v) = ρ (u) ρ (v)$ and $ϕ^{2} = ϕ + 1$ ) one computes $N (u_{1}) = {e_{s}}$ , $N (u_{2}) = {e_{t}} ⊔ ρ (t) N (u_{1}) = {e_{t}, e_{s} + ϕ e_{t}}$ , $N (u_{3}) = {e_{s}} ⊔ ρ (s) N (u_{2}) = {e_{s}, e_{t} + ϕ e_{s}, ϕ (e_{s} + e_{t})}$ , $N (u_{4}) = {e_{t}} ⊔ ρ (t) N (u_{3}) = {e_{t}, e_{s} + ϕ e_{t}, ϕ (e_{s} + e_{t}), e_{t} + ϕ e_{s}}$ and $N (u_{5}) = {e_{s}} ⊔ ρ (s) N (u_{4}) = {e_{s}, e_{t} + ϕ e_{s}, ϕ (e_{s} + e_{t}), e_{s} + ϕ e_{t}, e_{t}} = Φ_{+}$ , the last equality by 3.2. This gives the displayed sets $N (u_{1}), \dots, N (u_{5})$ , of cardinalities $k = ∣ N (u_{k}) ∣ = ℓ (u_{k})$ as required by [F4].

5.1F3F5step 4.1algebra

The case $I_{2} (5)$ : the inversion sets $N (u_{k})$ , $5 \leq k < 10$ . Since $N (u_{5}) = Φ_{+}$ by 4.1, the reflection $ρ (u_{5})$ maps $Φ_{+}$ bijectively onto $Φ_{-}$ (as $N (u_{5}) = Φ_{+}$ means $ρ (u_{5}) Φ_{+} \subseteq Φ_{-}$ , and $ρ (u_{5})$ induces a bijection of $Φ$ ); hence for $α \in Φ_{+}$ : $α \in N (u_{5} w)$ if and only if $ρ (w) α \in Φ_{+}$ , because $ρ (u_{5})$ maps $Φ_{+}$ onto $Φ_{-}$ and $Φ_{-}$ onto $Φ_{+}$ . Comparing with the definition of $N (w)$ this gives $N (u_{5} w) = Φ_{+} ∖ N (w)$ for every $w \in W$ . For $5 \leq k < 10$ one has $u_{k} = u_{5} u_{k - 5}^{'}$ , where $u_{j}^{'}$ is the alternating word of length $j$ beginning with $t$ ; hence $N (u_{k}) = Φ_{+} ∖ N (u_{k - 5}^{'})$ . In particular $N (u_{6}) = Φ_{+} ∖ {e_{t}}$ and, using the recursion of [F3] once more for $u_{2}^{'} = t s$ (read with the roles of $s$ and $t$ exchanged, $ℓ (u_{2}^{'}) = 2 > ℓ (t) = 1$ ), $N (u_{2}^{'}) = {e_{s}} ⊔ ρ (s) N (u_{1}^{'}) = {e_{s}, e_{t} + ϕ e_{s}}$ , so that $N (u_{7}) = Φ_{+} ∖ N (u_{2}^{'}) = {e_{t}, e_{s} + ϕ e_{t}, ϕ (e_{s} + e_{t})}$ , of cardinality $3 = ℓ (u_{7})$ .

6.1F1F5F6step 1.3algebra∎

The case $I_{2} (\infty)$ : the chambers. Let $δ (f) : = f (e_{s}) + f (e_{t})$ for $f \in V^{*}$ , so that $δ (f) = f (u)$ ; since $ρ (w) u = u$ for all $w$ by 1.3, the value $δ (f)$ is preserved by the dual action: $δ (w \cdot f) = (w \cdot f) (u) = f (ρ (w)^{- 1} u) = f (u) = δ (f)$ . On $L : = {f : δ (f) = 1}$ with coordinate $y_{s} = f (e_{s})$ , $C^{\circ}$ has trace $(0, 1)$ . By [F1] and the dual action in [F6], $s$ acts by $y_{s} \mapsto - y_{s}$ and $t$ by $y_{s} \mapsto 2 - y_{s}$ ; hence $s t$ acts by translation $y_{s} \mapsto y_{s} - 2$ . It follows that $(s t)^{k} C^{\circ} \cap L = (- 2 k, 1 - 2 k)$ and $(s t)^{k} s C^{\circ} \cap L = (- 2 k - 1, - 2 k)$ for every integer $k$ . These are all intervals $(j, j + 1)$ , and all group elements are alternating words by [F5] (cancelling adjacent equal letters); therefore these are exactly the open chamber traces, and no integer point belongs to one. Every chamber is a nonempty cone contained in ${δ > 0}$ , since this holds for $C^{\circ}$ and $δ$ is invariant. Each chamber $K$ equals the cone ${λ x : λ > 0, x \in K \cap L}$ over its trace, because for $f \in K$ its normalisation $f / δ (f)$ lies in $K \cap L$ and $f = δ (f) (f / δ (f))$ with $δ (f) > 0$ ; hence the chambers are exactly the cones over the intervals $(j, j + 1)$ . Finally $H_{e_{s}} \cap L$ is the point with $y_{s} = 0$ , and $H_{e_{t}} \cap L$ is the point with $y_{s} = 1$ , because $y_{t} = 1 - y_{s}$ on $L$ . Together with 1.1-1.3, 2.1-2.5, 3.1-3.2, 4.1 and 5.1 this verifies all the claims of (i), (ii) and (iii).

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An indefinite Coxeter form with a faithful canonical reflection representation

Example

Let $S = {s, t, r}$ and let $m$ be the Coxeter matrix with $m (u, u) = 1$ and $m (u, v) = \infty$ for all distinct $u, v \in S$ , so that the presentation has only the involutions $s^{2} = t^{2} = r^{2} = 1$ and $W$ is the free product of three copies of $Z / 2$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The free product of an arbitrary family of groups). Let $V = R^{S}$ , $B$ the Coxeter form and $ρ : W \to G L (V)$ the canonical reflection homomorphism (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Then:

(i) in the basis $(e_{s}, e_{t}, e_{r})$ one has $[B] = (\begin{array}{ccc} 1 & - 1 & - 1 \\ - 1 & 1 & - 1 \\ - 1 & - 1 & 1 \end{array}) = 2 I - J, J = (\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{array}),$ with eigenvalues $2, 2, - 1$ ; hence $B$ is indefinite and nondegenerate with inertia $(2, 1, 0)$ and scalar signature $2 - 1 = 1$ in the convention of 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 positive/negative index pair is $(2, 1)$ , also called the signature in the pair convention used by the companion page; $V_{+}$ is a proper cone;

(ii) nevertheless $ρ$ is faithful (The root-length criterion and faithfulness of the canonical reflection representation (3)), and its behaviour is governed by the sign criterion: $ρ (s) e_{t} = e_{t} + 2 e_{s} \in Φ_{+}$ , $ρ (s t) e_{t} = - e_{t} - 2 e_{s} \in Φ_{-}$ (consistent with $ℓ (s t t) = ℓ (s) = 1 < 2 = ℓ (s t)$ ), $ρ (s t) e_{s} = 3 e_{s} + 2 e_{t} \in Φ_{+}$ (consistent with $ℓ (s t s) = 3 > 2 = ℓ (s t)$ ), and $ρ (s t r) e_{s} = 15 e_{s} + 6 e_{t} + 2 e_{r} \neq e_{s}$ , so $ρ (s t r) \neq 1$ ;

(iii) the degenerate rank-two case behaves the same way: for $S^{'} = {s, t}$ with $m (s, t) = \infty$ one has $[B] = (\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array})$ , positive semidefinite of rank one with radical $R (e_{s} + e_{t})$ , and $ρ$ is still faithful, so neither indefiniteness nor degeneracy of $B$ obstructs faithfulness; what fails in the degenerate case is only the identification of $V$ with $V^{*}$ through $B$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (3)(ii)).

Facts & Assumptions

Given: the three-element set $S = {s, t, r}$ with $m (u, u) = 1$ and $m (u, v) = \infty$ for distinct $u, v$ , the space $V = R^{S}$ with basis $(e_{s}, e_{t}, e_{r})$ , the Coxeter form $B$ , the canonical reflection homomorphism $ρ$ with root system $Φ = Φ_{+} ⊔ Φ_{-}$ , the positive cone $V_{+}$ , and the rank-two subspace $P = R e_{s} + R e_{t}$ .

[F1]

Here $c (u, v) = 1$ for all distinct $u, v$ , so $B (e_{u}, e_{u}) = 1$ and $B (e_{u}, e_{v}) = - 1$ for $u \neq v$ ; the reflection is $r_{a} (v) = v - 2 B (v, a) a$ for $B (a, a) = 1$ , giving $ρ (s) e_{t} = e_{t} + 2 e_{s}$ , $ρ (s) e_{r} = e_{r} + 2 e_{s}$ , $ρ (t) e_{s} = e_{s} + 2 e_{t}$ , $ρ (t) e_{r} = e_{r} + 2 e_{t}$ , $ρ (s) e_{s} = - e_{s}$ and $ρ (r) e_{s} = e_{s} + 2 e_{r}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

[F2]

$W$ is presented by $(S, m)$ , with universal property and length $ℓ$ ; the alternating words $s t s \dots$ of every length are reduced because $m (s, t) = \infty$ , so $ℓ (s) = 1$ , $ℓ (s t) = 2$ and $ℓ (s t s) = 3$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness).

[F3]

Root sign and length criterion: $Φ = Φ_{+} ⊔ Φ_{-}$ , $Φ_{-} = - Φ_{+}$ , and for all $w \in W$ , $s \in S$ one has $ℓ (w s) > ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{+}$ and $ℓ (w s) < ℓ (w) ⟺ ρ (w) e_{s} \in Φ_{-}$ ; moreover $ρ$ is injective (Root sign coherence and the action of simple reflections on positive roots, The root-length criterion and faithfulness of the canonical reflection representation).

[F4]

An eigenvector is nonzero and satisfies $A v = λ v$ ; a basis is an independent spanning family. A real symmetric form with a diagonal matrix having two positive entries, one negative entry and no zeros has inertia $(2, 1, 0)$ and scalar signature $1$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F5]

A free product of groups is characterized by its universal property: homomorphisms from the factors into any group extend uniquely to a homomorphism from the free product (The free product of an arbitrary family of groups).

Verification

technique · direct

1.1F1F4algebra

The Gram matrix and inertia. By [F1], $[B] = 2 I - J$ , where $J x = (x_{s} + x_{t} + x_{r}) (1, 1, 1)$ . Put $p = (1, - 1, 0)$ , $q = (1, 1, - 2)$ and $a = (1, 1, 1)$ in the coordinates $(e_{s}, e_{t}, e_{r})$ . The coordinate matrix with columns $p, q, a$ has determinant $6 \neq 0$ , so they form a basis. Since $J p = J q = 0$ and $J a = 3 a$ , the matrix $2 I - J$ has eigenvalues $2, 2, - 1$ in this basis. Direct substitution into $B (x, y) = 2 \sum_{i} x_{i} y_{i} - (\sum_{i} x_{i}) (\sum_{i} y_{i})$ gives $B (p, p) = 4$ , $B (q, q) = 12$ , $B (a, a) = - 3$ , and all three cross terms zero. Thus $B$ has diagonal matrix $diag (4, 12, - 3)$ in this basis, so it is nondegenerate and indefinite with inertia $(2, 1, 0)$ , scalar signature $1$ , and positive/negative index pair $(2, 1)$ . Finally $V_{+}$ is closed under addition and nonnegative scaling, contains no line because $V_{+} \cap (- V_{+}) = {0}$ , and is not all of $V$ because $- e_{s} \notin V_{+}$ . This is (i).

1.2F1F2F3algebra

The signs of the computed roots. By [F2] one has $ℓ (s) = 1$ , $ℓ (s t) = 2$ and $ℓ (s t s) = 3$ , so $ℓ (s t) > ℓ (s)$ and $ℓ (s t s) > ℓ (s t)$ ; the length criterion [F3] therefore gives $ρ (s) e_{t} \in Φ_{+}$ and $ρ (s t) e_{s} \in Φ_{+}$ . By [F1], $ρ (s) e_{t} = e_{t} + 2 e_{s}$ , $ρ (s t) e_{t} = ρ (s) ρ (t) e_{t} = - ρ (s) e_{t} = - e_{t} - 2 e_{s} \in Φ_{-}$ , and $ρ (s t) e_{s} = ρ (s) (e_{s} + 2 e_{t}) = - e_{s} + 2 (e_{t} + 2 e_{s}) = 3 e_{s} + 2 e_{t}$ . The signs are consistent with the criterion also in the first two cases because $ℓ (s t t) = ℓ (s) = 1 < 2 = ℓ (s t)$ , so $ρ (s t) e_{t} \in Φ_{-}$ says exactly that right multiplication by $t$ shortens $s t$ .

1.3F1F4algebra

A nontrivial image. By [F1], $ρ (r) e_{s} = e_{s} + 2 e_{r}$ , so $ρ (t r) e_{s} = ρ (t) (e_{s} + 2 e_{r}) = (e_{s} + 2 e_{t}) + 2 (e_{r} + 2 e_{t}) = e_{s} + 6 e_{t} + 2 e_{r}$ and $ρ (s t r) e_{s} = ρ (s) (e_{s} + 6 e_{t} + 2 e_{r}) = - e_{s} + 6 (e_{t} + 2 e_{s}) + 2 (e_{r} + 2 e_{s}) = 15 e_{s} + 6 e_{t} + 2 e_{r}$ . This differs from $e_{s}$ because its value at $t$ is $6$ while $e_{s} (t) = 0$ , so $ρ (s t r) \neq {i d}_{V}$ .

1.4F1algebra

The degenerate rank-two case. Restrict to $P = R e_{s} + R e_{t}$ . By [F1] the Gram matrix of $B ∣_{P}$ in $(e_{s}, e_{t})$ is $(\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array})$ , and $B (x_{s} e_{s} + x_{t} e_{t}, x_{s} e_{s} + x_{t} e_{t}) = (x_{s} - x_{t})^{2} \geq 0$ , so $B ∣_{P}$ is positive semidefinite; its radical is ${x : B (x, e_{s}) = B (x, e_{t}) = 0} = {x : x_{s} = x_{t}} = R (e_{s} + e_{t})$ , a line, so $B ∣_{P}$ has rank one and in this two-generator case the map $v \mapsto B ∣_{P} (v, -)$ from $P$ to $P^{*}$ has nonzero kernel and is therefore not an identification of $P$ with $P^{*}$ ; this does not assert that the original nondegenerate rank-three form has a kernel.

1.5F2F5algebra

The free-product assertion. For each $v \in {s, t, r}$ , the relation $v^{2} = 1$ defines a homomorphism $ι_{v} : Z / 2 \to W$ sending the nonidentity element to $v$ . A homomorphism from $Z / 2$ into any group $H$ is uniquely determined by an element $h_{v}$ with $h_{v}^{2} = 1$ . Since our Coxeter presentation has no finite off-diagonal relators, its universal property in [F2] gives exactly one homomorphism $W \to H$ sending $v$ to $h_{v}$ for each $v$ . Therefore $(W, ι_{s}, ι_{t}, ι_{r})$ satisfies the free-product universal property [F5].

2.1F3step 1.1step 1.2step 1.3step 1.4∎

Faithfulness. The matrix $m$ is a Coxeter matrix on the finite set $S$ , so the homomorphism $ρ : W \to G L (V)$ is injective by [F3]; this applies to $S$ and to the degenerate two-generator subcase $S^{'} = {s, t}$ as well, so in both cases $ρ$ is faithful even though $B$ is indefinite, respectively degenerate. With (i) from 1.1, (ii) from 1.2 and 1.3 together with [F3], and (iii) from 1.4, all clauses are verified: neither indefiniteness nor degeneracy of $B$ obstructs faithfulness.

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A vector with mixed signs is not a root, while every root has a sign

Example

Let $S = {s, t}$ with $m (s, t) \geq 3$ , $V = R^{S}$ , $B$ the Coxeter form and $Φ$ the root system of the canonical reflection representation (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Then:

(i) the vector $v : = e_{s} - e_{t}$ has coefficient $- 1$ at $e_{t}$ , so $v \notin V_{+} \cup (- V_{+})$ : it has mixed signs;

(ii) consequently $v \notin Φ$ , although it is nonzero and $B$ -non-isotropic, because by Root sign coherence and the action of simple reflections on positive roots (2) every root lies in $V_{+} ∖ {0}$ or in $- V_{+} ∖ {0}$ ; explicitly $B (v, v) = 2 + 2 c (s, t) > 0$ while every root has $B$ -norm $1$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (3)), so $v$ fails both the sign test and the norm test;

(iii) for $m (s, t) = 3$ one has $B (v, v) = 3$ and the comparison case $e_{s} + e_{t}$ has $B (e_{s} + e_{t}, e_{s} + e_{t}) = 1$ and lies in $Φ_{+}$ ; for $m (s, t) = \infty$ one has $B (v, v) = 4$ and $v$ still fails to be a root.

Facts & Assumptions

Given: a two-element set $S = {s, t}$ with $m : = m (s, t) \geq 3$ , the space $V = R^{S}$ with basis $(e_{s}, e_{t})$ , the Coxeter form $B$ , the canonical reflection homomorphism $ρ$ , the root system $Φ = Φ_{+} ⊔ Φ_{-}$ and the positive cone $V_{+}$ .

[F1]

$B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - c$ with $c : = c (s, t)$ equal to $\cos (π / m)$ when $m < \infty$ and to $1$ when $m = \infty$ ; the reflection with normal $e_{t}$ is $r_{t} (u) = u - 2 B (u, e_{t}) e_{t}$ , so $ρ (t) e_{s} = r_{t} e_{s} = e_{s} + 2 c e_{t}$ ; and $V_{+} = {λ e_{s} + μ e_{t} : λ \geq 0, μ \geq 0}$ with $- V_{+} = {u : - u \in V_{+}}$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone).

[F2]

Every root lies in $V_{+} ∖ {0}$ or in $- V_{+} ∖ {0}$ , and not in both; every root $α$ satisfies $B (α, α) = 1$ ; $Φ_{+} = Φ \cap V_{+}$ (Root sign coherence and the action of simple reflections on positive roots, Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F3]

For every $x \in W$ and every generator $s$ one has $x C^{\circ} \subseteq B_{s}$ or $x C^{\circ} \subseteq s B_{s}$ , and $x C^{\circ} \subseteq s B_{s}$ if and only if $ℓ (s x) < ℓ (x)$ ; moreover $ρ (y) e_{s} \in Φ_{+} ⟺ y^{- 1} C^{\circ} \subseteq B_{s}$ for the root $ρ (y) e_{s}$ (The rank-two half-space alternative and the chamber-length induction $(P_{n})$ , $(Q_{n})$ , Root sign coherence and the action of simple reflections on positive roots).

[F4]

The simple generators are distinct in $W$ , so $s \neq t$ , and $ℓ (t) = 1$ ; moreover $ℓ (x) = 0$ implies $x = 1$ (Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action, Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

[F5]

The addition formulas $\sin (x + y) = \sin x \cos y + \cos x \sin y$ , $\cos (x + y) = \cos x \cos y - \sin x \sin y$ hold for all real $x, y$ ; $\sin (- x) = - \sin x$ , $\cos (- x) = \cos x$ and $\sin^{2} x + \cos^{2} x = 1$ ; $\cos (π / 2) = 0$ and $\cos π = - 1$ ; cosine is strictly decreasing on $[0, π]$ ; and $π > 0$ (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Quarter-turn values and shifts by pi/2 and pi, Signs, monotonicity intervals, and ranges of sine and cosine, Pi as twice the smallest positive zero of cosine).

[F6]

Every $x \in V = R^{{s, t}}$ equals $x (s) e_{s} + x (t) e_{t}$ ; evaluating at $s$ and $t$ shows that these coordinates are unique (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (1)).

Verification

technique · direct

1.1F1F5algebra

Set-up. By [F1] the vector $v = e_{s} - e_{t}$ has coordinates $v (s) = 1$ and $v (t) = - 1$ in the basis $(e_{s}, e_{t})$ , and $B (v, v) = B (e_{s}, e_{s}) - 2 B (e_{s}, e_{t}) + B (e_{t}, e_{t}) = 2 + 2 c$ . The constant $c$ is positive: $c = 1 > 0$ for $m = \infty$ , while for finite $m \geq 3$ one has $0 < π / m \leq π / 3 < π / 2$ and cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ by [F5], so $c = \cos (π / m) > \cos (π / 2) = 0$ . Hence $B (v, v) > 2 > 0$ , so $v \neq 0$ and $v$ is $B$ -non-isotropic.

1.2F1F6algebra

The vector has mixed signs. By [F6] the coordinates of a vector in the basis $(e_{s}, e_{t})$ are unique, so $u \in V_{+}$ exactly when both coordinates of $u$ are $\geq 0$ and $u \in - V_{+}$ exactly when both coordinates are $\leq 0$ . Since $v$ has the coordinates $1$ and $- 1$ , it lies in neither cone: $v \notin V_{+} \cup (- V_{+})$ . This is (i).

2.1F1F5step 1.1algebra

The value $c = \cos (π / 3)$ . For $x$ with $\cos x = : q$ and $\sin x = : r$ , the addition formulas and $\sin^{2} x + \cos^{2} x = 1$ of [F5] give $\cos (2 x) = q^{2} - r^{2} = 2 q^{2} - 1$ and $\sin (2 x) = 2 q r$ , hence $\cos (3 x) = \cos (2 x) \cos x - \sin (2 x) \sin x = (2 q^{2} - 1) q - 2 q r^{2} = 2 q^{3} - q - 2 q (1 - q^{2}) = 4 q^{3} - 3 q$ . Applied to $x = π / 3$ and combined with $\cos π = - 1$ this gives $4 c^{3} - 3 c + 1 = 0$ , that is $(c + 1) (2 c - 1)^{2} = 0$ . Since $c > 0$ by 1.1, the factor $c + 1$ does not vanish, so $(2 c - 1)^{2} = 0$ and $c = 1 / 2$ . Consequently $2 c = 1$ , $B (v, v) = 2 + 1 = 3$ , and $ρ (t) e_{s} = e_{s} + 2 c e_{t} = e_{s} + e_{t}$ ; moreover $B (e_{s} + e_{t}, e_{s} + e_{t}) = 1 + 1 - 2 c = 2 - 1 = 1$ .

2.2F2step 1.1step 1.2

$v$ is not a root. By [F2] every root lies in $V_{+} ∖ {0}$ or in $- V_{+} ∖ {0}$ , and every root has $B$ -norm $1$ . By 1.2 the vector $v$ lies in neither cone, so it is not a root; independently, by 1.1 its $B$ -norm is $2 + 2 c > 2 \neq 1$ , so it also fails the norm test for roots. This is (ii).

3.1F3F4step 2.1

The comparison vector $e_{s} + e_{t}$ is positive. By [F3] applied to $x = t$ either $t C^{\circ} \subseteq B_{s}$ or $t C^{\circ} \subseteq s B_{s}$ . In the second case $ℓ (s t) < ℓ (t) = 1$ by [F3], so $ℓ (s t) = 0$ and hence $s t = 1$ by [F4], which gives $s = t^{- 1} = t$ and contradicts $s \neq t$ ; therefore $t C^{\circ} \subseteq B_{s}$ , and the equivalence of [F3] with $y = t$ gives $ρ (t) e_{s} \in Φ_{+}$ . By 2.1, $ρ (t) e_{s} = e_{s} + e_{t}$ when $m = 3$ , so $e_{s} + e_{t} \in Φ_{+}$ with $B (e_{s} + e_{t}, e_{s} + e_{t}) = 1$ .

4.1step 2.1step 2.2step 3.1F1∎

The cases $m = 3$ and $m = \infty$ . If $m = 3$ then $c = 1 / 2$ by 2.1, so $B (v, v) = 3$ , and $e_{s} + e_{t} = ρ (t) e_{s}$ is a positive root of $B$ -norm $1$ by 3.1. If $m = \infty$ then $c = 1$ by [F1], so $B (v, v) = 4$ , and $v \notin Φ$ by 2.2. With (i) from 1.2 and (ii) from 2.2, all three clauses are verified: the sign theorem applies to the $W$ -orbit of the simple roots, not to arbitrary vectors of $V$ .

Canonical Roots, Signs, and Faithful Reflections — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Roots, inversions and chamber images in I2(5)I2​(5), A2A2​ and infinite dihedral type

Example

Facts & Assumptions

Verification

An indefinite Coxeter form with a faithful canonical reflection representation

Example

Facts & Assumptions

Verification

A vector with mixed signs is not a root, while every root has a sign

Example

Facts & Assumptions

Verification

Sources

Roots, inversions and chamber images in $I_{2} (5)$ , $A_{2}$ and infinite dihedral type