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✓ 5 results · all verified · 2 also independently AI-judged

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Finite Coxeter Diagrams and Complete Classification — Examples

1 · Prerequisites

2 · Summary

This companion is a dependency leaf: its examples use the theory of finite-coxeter-diagrams-and-complete-classification, that page's prerequisite closure and earlier examples in this companion; no item outside this companion depends on them.

Dihedral diagrams $I_{2} (m)$ : Gram determinants, the infinite case, and the low-rank coincidences computes the matrix of the rank-two diagram $I_{2} (m)$ with $m \in {3, 4, \dots} \cup {\infty}$ , obtains $\det B = \sin^{2} (π / m) > 0$ in the finite case together with the exact order $2 m$ of the group, and in the infinite case exhibits the kernel vector $e_{s} + e_{t}$ ; it also records the low-rank coincidences $I_{2} (3) = A_{2}$ , $I_{2} (4) = B_{2} = C_{2}$ , $I_{2} (6) = G_{2}$ , $I_{2} (5) = H_{2}$ and $I_{2} (2) = A_{1} \times A_{1}$ . Gram determinants and principal minors of the non-crystallographic types $H_{3}$ and $H_{4}$ evaluates the leading principal minors of $2 C$ for $H_{3}$ and $H_{4}$ , including the second $2 \times 2$ principal minor $(5 - \sqrt{5}) / 2$ of $H_{3}$ , and shows that the neighbouring overlong paths and the star with labels $3, 3, 5$ have negative determinant or an explicit non-positive vector. $B_{n}$ and $C_{n}$ define the same Coxeter diagram and the same Coxeter group proves that the two names denote the same labelled path with one final label- $4$ edge, computes $\det (2 C) (B_{n}) = 2$ with leading minors $2, 3, \dots, n$ , and identifies the standard parabolic $A_{n - 1}$ . A cycle and an overlong arm: explicit non-positive witnesses exhibits the cycle vector $e_{s_{1}} + \dots + e_{s_{r}}$ with $B (u, u) = 0$ , the weighted vector of the overlong star $(1, 2, 5)$ , and the $\sqrt{2}$ -weighted witnesses for two large labels. Path determinants $d_{k} = d_{k - 1} - \cos^{2} (π / m) d_{k - 2}$ and the three-arm inequality proves the path recursion $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ and the two-subpath determinant, and proves the three-arm criterion: for a star with arms of $p, q, r$ vertices positive definiteness is equivalent to $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} > 1$ , whose surviving triples $(1, 1, r)$ , $(1, 2, 2)$ , $(1, 2, 3)$ , $(1, 2, 4)$ and boundary triples $(2, 2, 2)$ , $(1, 3, 3)$ , $(1, 2, 5)$ are then computed.

The computations test the determinant table and the exclusions of the theory page; they are evidence within their stated scope and do not replace the proofs there.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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

A cycle and an overlong arm: explicit non-positive witnesses

Example

Throughout, $S$ is finite, $m$ is a Coxeter matrix with diagram $Γ$ and Coxeter form $B$ on $V = R^{S}$ (Coxeter diagrams: edges, labels, components and finite type, The real Coxeter form, its radical, reflections, and form-preserving maps), and one is asked whether $B$ can be positive definite.

(i) Cycles. Let $Γ$ be the cycle on $r \geq 3$ vertices $s_{1}, \dots, s_{r}$ with all labels $3$ , so that $B (e_{s_{i}}, e_{s_{i + 1}}) = - \frac{1}{2}$ for consecutive pairs (indices modulo $r$ ) and all other off-diagonal entries are $0$ . Then $u = e_{s_{1}} + \dots + e_{s_{r}}$ satisfies $B (u, u) = r - 2 \cdot r \cdot \frac{1}{2} = 0$ , so the cycle is not positive definite; more generally, if the labels on the cycle are $\geq 3$ then $B (u, u) \leq 0$ .

(ii) An overlong arm. Let $Γ = E_{9}$ be the star with central vertex $c$ and arms of $1, 2, 5$ vertices, all edges labelled $3$ , with arm vertices $a$ (length $1$ ), $b_{1} - b_{2}$ (length $2$ , $b_{1}$ adjacent to $c$ ) and $d_{1} - \dots - d_{5}$ (length $5$ , $d_{5}$ adjacent to $c$ ). Then $u = e_{c} + \frac{1}{2} e_{a} + \frac{1}{3} (2 e_{b_{1}} + e_{b_{2}}) + \frac{1}{6} (e_{d_{1}} + 2 e_{d_{2}} + 3 e_{d_{3}} + 4 e_{d_{4}} + 5 e_{d_{5}})$ satisfies $B (u, u) = 0$ ; hence the star $(1, 2, 5)$ , the one-vertex extension of $E_{8}$ , is not positive definite, in agreement with the arm inequality of Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (6), which the triple $(1, 2, 5)$ fails with equality.

(iii) Two large labels. The three-vertex path with both edges labelled $4$ has the witness $u = e_{s_{1}} + \sqrt{2} e_{s_{2}} + e_{s_{3}}$ with $B (u, u) = 0$ , and the four-vertex path with labels $4, 3, 4$ has the witness $u = e_{s_{1}} + \sqrt{2} e_{s_{2}} + \sqrt{2} e_{s_{3}} + e_{s_{4}}$ with $B (u, u) = 0$ ; these are the first members of the family excluded in Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (4)(ii).

Facts & Assumptions

Given: A finite set $S$ with Coxeter matrix $m$ and diagram $Γ$ , the space $V = R^{S}$ with the Coxeter form $B$ , and the specific diagrams of (i), (ii) and (iii).

[F1]

Distinct vertices $s \neq t$ of $Γ$ are joined exactly when $m (s, t) \geq 3$ and carry the label $m (s, t)$ ; the subdiagram $Γ_{T}$ is the induced labelled graph on $T$ , so deleting vertices deletes exactly the incident edges; the neighbours of $s$ are $N (s) = {t \neq s : m (s, t) \geq 3}$ (Coxeter diagrams: edges, labels, components and finite type).

[F2]

$B$ is the unique symmetric bilinear form on $V$ with $B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ and $B (e_{s}, e_{t}) = - 1$ for $m (s, t) = \infty$ (The real Coxeter form, its radical, reflections, and form-preserving maps).

[F3]

If $0 \neq u \in V_{T}$ for some $T \subseteq S$ and $B (u, u) \leq 0$ , then $B$ is not positive definite; and if all coordinates of $u$ are $\geq 0$ while a labelled graph $Γ_{0}$ on $T$ has all labels at most the labels of $Γ_{T}$ , then $B_{T} (u, u) \leq B_{0} (u, u)$ (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (1)).

[F4]

With $c (s, t) : = - B (e_{s}, e_{t})$ for $s \neq t$ : $c (s, t) \in [0, 1]$ , $c (s, t) = 0$ exactly when $m (s, t) = 2$ , and $c (s, t) \geq \frac{1}{2}$ whenever $m (s, t) \geq 3$ (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms).

[F5]

For a vertex $v$ of degree $3$ whose edges all have label $3$ and whose three arms have $p, q, r \geq 1$ vertices, $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} > 1$ is necessary for positive definiteness (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (6)).

[F6]

$\cos (2 x) = 2 \cos^{2} x - 1$ for every real $x$ (Double-angle and quadratic power-reduction identities).

[F7]

$\cos (x + π) = - \cos x$ and $\cos (- x) = \cos x$ for every real $x$ (Quarter-turn values and shifts by pi/2 and pi, Parity and the Pythagorean identity for sine and cosine).

[F8]

Cosine is strictly decreasing on $[0, π]$ , $π > 0$ and $\cos π = - 1$ (Signs, monotonicity intervals, and ranges of sine and cosine, Pi as twice the smallest positive zero of cosine, Quarter-turn values and shifts by pi/2 and pi).

[F9]

$\cos (π / 4) = \sqrt{2} / 2$ and this factor is positive (The finite Viete cosine product and its positive nested-radical factors).

[F10]

$\sqrt{2} \geq 0$ and ${\sqrt{2}}^{2} = 2$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Verification

1.1F6F7F8F9algebra

(The two numerical values.) $\cos (π / 4) = \sqrt{2} / 2$ is [F9]. For $\cos (π / 3)$ , put $c = \cos (π / 3)$ : the shift formula gives $\cos (2 π / 3) = \cos (π - π / 3) = - \cos (- π / 3) = - c$ by [F7], while the double-angle formula gives $\cos (2 π / 3) = 2 c^{2} - 1$ by [F6]; hence $2 c^{2} - 1 = - c$ , i.e. $(2 c - 1) (c + 1) = 0$ . Since $0 < π / 3 < π$ and cosine is strictly decreasing on $[0, π]$ with $\cos π = - 1$ , one has $c > - 1$ [F8], so $c = \cos (π / 3) = 1 / 2$ .

2.1F2step 1.1algebra

(Weighted arms.) Let a path arm on vertices $s_{1}, \dots, s_{p}$ have all its edges labelled $3$ and let $s_{p}$ be the vertex adjacent to the centre $c$ ; put $u_{p} = \sum_{k = 1}^{p} k e_{s_{k}}$ . Every internal edge ${s_{k}, s_{k + 1}}$ contributes $k (k + 1) B (e_{s_{k}}, e_{s_{k + 1}}) = - k (k + 1) / 2$ twice, so $B (u_{p}, u_{p}) = \sum_{k = 1}^{p} k^{2} - \sum_{k = 1}^{p - 1} k (k + 1) = p^{2} - \sum_{k = 1}^{p - 1} k = p (p + 1) / 2$ , by [1.1] and [F2]; and $B (e_{c}, u_{p}) = - p / 2$ since among the pairs $(c, s_{k})$ only ${c, s_{p}}$ is an edge, with coefficient $p$ .

2.2F1F2F3F4step 1.1algebra

(The cycle witness (i).) For the cycle of (i) put $u = \sum_{i = 1}^{r} e_{s_{i}} \neq 0$ , a non-negative vector. Its diagonal contribution is $r$ ; each of the $r$ consecutive pairs contributes $2 B (e_{s_{i}}, e_{s_{i + 1}}) = - 2 c (s_{i}, s_{i + 1}) \leq - 1$ because $c \geq \frac{1}{2}$ on edges [F4], and every other pair is a non-edge, contributing $0$ [F2, F4]; hence $B (u, u) \leq r - r = 0$ , so $B$ is not positive definite by [F3]. If the labels on the cycle are any values $\geq 3$ , the same computation gives $B (u, u) \leq 0$ because each consecutive $c$ is still $\geq \frac{1}{2}$ while every other pair contributes $\leq 0$ (an edge of label $\geq 3$ gives $\leq - 1$ , a non-edge gives $0$ ), and the conclusion is unchanged.

2.3F2F3step 1.1F10algebra

(Two large labels (iii).) For the path with labels $4, 4$ and $u = e_{s_{1}} + \sqrt{2} e_{s_{2}} + e_{s_{3}}$ : the diagonal is $1 + 2 + 1 = 4$ [F2], and the two edges contribute $2 \cdot (- \cos (π / 4)) \cdot \sqrt{2} = - 2 \sqrt{2} \cos (π / 4) = - 2$ each by [F10] and [1.1]; hence $B (u, u) = 4 - 4 = 0$ . For the path with labels $4, 3, 4$ and $u = e_{s_{1}} + \sqrt{2} e_{s_{2}} + \sqrt{2} e_{s_{3}} + e_{s_{4}}$ : the diagonal is $1 + 2 + 2 + 1 = 6$ , the two label- $4$ edges contribute $- 2$ each, and the middle label- $3$ edge contributes $2 \cdot (- \frac{1}{2}) \cdot \sqrt{2} \cdot \sqrt{2} = - 2$ by [1.1]; hence $B (u, u) = 6 - 6 = 0$ . In both cases $u \neq 0$ has non-negative coordinates, so $B$ is not positive definite by [F3].

3.1F1F2F3step 2.1algebra

(The star witness (ii).) In the star of (ii) the three arms meet only at $c$ [F1], so the three arm vectors $u_{1} = \frac{1}{2} e_{a}$ , $u_{2} = \frac{1}{3} (2 e_{b_{1}} + e_{b_{2}})$ and $u_{3} = \frac{1}{6} (e_{d_{1}} + 2 e_{d_{2}} + 3 e_{d_{3}} + 4 e_{d_{4}} + 5 e_{d_{5}})$ , in which the centre-adjacent vertices $a, b_{1}, d_{5}$ carry the weights $1, 2, 5$ , are pairwise $B$ -orthogonal; step 2.1 with $p = 1, 2, 5$ gives $B (u_{1}, u_{1}) = \frac{1}{4}$ , $B (u_{2}, u_{2}) = \frac{1}{9} \cdot 3 = \frac{1}{3}$ , $B (u_{3}, u_{3}) = \frac{1}{36} \cdot 15 = \frac{5}{12}$ , and $B (e_{c}, u_{1}) = - \frac{1}{2} \cdot \frac{1}{2} = - \frac{1}{4}$ , $B (e_{c}, u_{2}) = - \frac{1}{3} \cdot \frac{2}{2} = - \frac{1}{3}$ , $B (e_{c}, u_{3}) = - \frac{1}{6} \cdot \frac{5}{2} = - \frac{5}{12}$ . Therefore $u = e_{c} + u_{1} + u_{2} + u_{3}$ satisfies $B (u, u) = 1 + \frac{1}{4} + \frac{1}{3} + \frac{5}{12} + 2 (- \frac{1}{4} - \frac{1}{3} - \frac{5}{12}) = 1 - (\frac{1}{4} + \frac{1}{3} + \frac{5}{12}) = 0$ , and $u \neq 0$ with all coordinates $\geq 0$ , so $B$ is not positive definite by [F3].

4.1F5step 3.1step 2.3algebra∎

(Agreement with the general exclusions.) The triple $(1, 2, 5)$ of arm lengths in (ii) gives $\frac{1}{1 + 1} + \frac{1}{2 + 1} + \frac{1}{5 + 1} = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1$ , so it fails the necessary inequality [F5] with equality, and the star of (ii) is the first diagram at which the arm inequality becomes non-strict; the equality in the computation of [3.1] is the same equality. The paths of (iii) are the two shortest diagrams with two edges of label $\geq 4$ , so they are the first members of the family excluded in Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (4)(ii), whose witness vector is $\sqrt{2}$ -weighted on the interior of the chain, exactly as in [2.3].

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Dihedral diagrams $I_{2} (m)$ : Gram determinants, the infinite case, and the low-rank coincidences

Example

Let $S = {s, t}$ with $m (s, t) = r \in {3, 4, \dots} \cup {\infty}$ and $m (s, s) = m (t, t) = 1$ ; let $W$ be the presented group (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $B$ the Coxeter form on $V = R^{S}$ (The real Coxeter form, its radical, reflections, and form-preserving maps) and $Γ$ its diagram (Coxeter diagrams: edges, labels, components and finite type).

(i) The diagram and the matrix. $Γ$ is the single edge $s t$ labelled $r$ , and the matrix of $B$ in the basis $(e_{s}, e_{t})$ is $(\begin{array}{cc} 1 & - c \\ - c & 1 \end{array})$ with $c = \cos (π / r)$ for finite $r$ and $c = 1$ for $r = \infty$ .

(ii) Finite case. For finite $r$ one has $\det B = \sin^{2} (π / r) > 0$ and the $1 \times 1$ principal minors are $1$ , so $B$ is positive definite and $W$ is finite; it is the dihedral group $I_{2} (r)$ of order $2 r$ , and the canonical product $ρ (s) ρ (t)$ has exact order $r$ on $V$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (3)(iv), Finiteness criterion: W is finite exactly when the Coxeter form is positive definite).

(iii) Infinite case. For $r = \infty$ one has $\det B = 0$ and the kernel vector $u = e_{s} + e_{t}$ satisfies $B (u, u) = 0$ ; hence $B$ is not positive definite and $W$ is infinite, namely the infinite dihedral group.

(iv) Low-rank coincidences and products. As Coxeter systems $I_{2} (3) = A_{2}$ , $I_{2} (4) = B_{2} = C_{2}$ , $I_{2} (6) = G_{2}$ , $I_{2} (2) = A_{1} \times A_{1}$ (no edge for $m = 2$ ), $I_{2} (5) = H_{2}$ , and $A_{1} = B_{1}$ is the one-vertex diagram; these are the only overlaps among the rank-two families, and in the classification Classification of finite Coxeter systems, including the H and dihedral families they appear under both names. The one-vertex diagram $A_{1}$ has $B = (1)$ positive definite and $W ≅ Z / 2$ ; the disconnected two-vertex diagram $A_{1} \times A_{1}$ is the direct product of two such groups, of order $4$ (The external direct product $G \times H$ with componentwise multiplication, $G \times H$ is a group with identity $(e_{G}, e_{H})$ , coordinatewise inverses, and homomorphic coordinate projections).

Facts & Assumptions

Given: $S = {s, t}$ , a Coxeter matrix with $m (s, t) = r$ and $m (s, s) = m (t, t) = 1$ , the presented group $W$ with canonical homomorphism $ρ$ , the space $V = R^{S}$ with Coxeter form $B$ , and the diagram $Γ$ ; write $c : = \cos (π / r)$ for finite $r$ .

[F1]

In a diagram with two distinct vertices $s, t$ , an edge is drawn exactly when $m (s, t) \geq 3$ and it carries the label $m (s, t)$ ; when $m (s, t) = 2$ no edge is drawn, and the subdiagram on a single vertex is the one-vertex diagram (Coxeter diagrams: edges, labels, components and finite type).

[F2]

$B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ , $B (e_{s}, e_{t}) = - \cos (π / r)$ for finite $r$ and $B (e_{s}, e_{t}) = - 1$ for $r = \infty$ ; more generally $m (s, t)$ is the order of $s t$ in $W$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

[F3]

The reflection $r_{a}$ is linear and an involution, fixes $\ker B (-, a)$ pointwise, and $B (r_{a} u, r_{a} w) = B (u, w)$ ; on the plane $P = R e_{s} + R e_{t}$ the product $r_{s} r_{t}$ has exact order $m (s, t)$ for finite $m (s, t)$ , fixing $P^{⊥}$ pointwise, while for $m (s, t) = \infty$ it is $i d + N$ on $P$ with $N \neq 0$ , $N^{2} = 0$ , so it has infinite order (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2),(3)(ii),(3)(iv)).

[F4]

$ρ : W \to G L (V)$ is a homomorphism with $ρ (s) = r_{e_{s}}$ ; it is injective (The root-length criterion and faithfulness of the canonical reflection representation (3), Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F5]

$W$ is finite if and only if $B$ is positive definite; a symmetric matrix is positive definite if and only if all its leading principal minors are positive; the form is positive definite when its quadratic form is $> 0$ on every nonzero vector, so a nonzero $u$ with $B (u, u) \leq 0$ excludes positive definiteness (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite, Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F6]

$\cos (π / 2) = 0$ ; $\sin^{2} x + \cos^{2} x = 1$ and $\sin x > 0$ for $0 < x < π$ ; $\sin^{2} (π / r) > 0$ for every finite Coxeter label $r \geq 3$ (Quarter-turn values and shifts by pi/2 and pi, Parity and the Pythagorean identity for sine and cosine, Sine and cosine defined by their real power series, Pi as twice the smallest positive zero of cosine, Signs, monotonicity intervals, and ranges of sine and cosine); sine positivity also follows from the positive rank-two quadratic coefficient in [F3].

[F7]

An external direct product $G \times H$ is a group and is finite exactly when both factors are, with $∣ G \times H ∣ = ∣ G ∣ ∣ H ∣$ ; for a disconnected diagram the group is the direct product of the standard parabolics of its components (The external direct product $G \times H$ with componentwise multiplication, $G \times H$ is a group with identity $(e_{G}, e_{H})$ , coordinatewise inverses, and homomorphic coordinate projections, Disconnected diagrams, direct products, and comparison of invariant forms (1)).

Verification

1.1F2F3F5F6algebra

(The matrix, the finite determinant and the infinite kernel vector.) Since $m (s, t) = r \geq 3$ , or $r = \infty$ , the diagram $Γ$ is the single edge $s t$ labelled $r$ [F1], and in the basis $(e_{s}, e_{t})$ the matrix of $B$ is $(\begin{array}{cc} 1 & - c \\ - c & 1 \end{array})$ with $c = \cos (π / r)$ for finite $r$ and $c = 1$ for $r = \infty$ [F2]; this is (i). Its determinant is $1 - c^{2}$ . For finite $r$ the Pythagorean identity gives $1 - c^{2} = \sin^{2} (π / r)$ , which is positive because $0 < π / r \leq π / 3 < π$ [F6]; the $1 \times 1$ principal minors are the diagonal entries $1 > 0$ . For $r = \infty$ we have $c = 1$ and $\det B = 0$ , while $u = e_{s} + e_{t} \neq 0$ satisfies $B (u, u) = 1 + 1 - 2 \cdot 1 \cdot 1 = 0$ [F2]; by [F5] the form is not positive definite, giving (iii) for the form.

2.1F3F4F5step 1.1algebra

(Finite case: positive definiteness, order $2 r$ .) Let $r < \infty$ . The leading principal minors of $B$ are $1$ and $1 - c^{2} = \sin^{2} (π / r) > 0$ by 1.1 [step 1.1], so $B$ is positive definite by [F5] and $W$ is finite by the criterion [F5]. By [F3] the product $ρ (s) ρ (t) = r_{e_{s}} r_{e_{t}}$ acts on $P = R e_{s} + R e_{t}$ as a rotation of exact order $r$ and fixes $P^{⊥}$ pointwise, so $ρ (s) ρ (t)$ has exact order $r$ on $V$ . Put $A : = ρ (s) ρ (t) = ρ (s t)$ , of exact order $r$ ; every element of $W$ is of the form $A_{0}^{k}$ or $s A_{0}^{k}$ with $A_{0} : = s t \in W$ and $k \in Z$ , because every alternating word in the two involutions is a power of $s t$ possibly preceded by $s$ . Hence $ρ (W) = {A^{k}, ρ (s) A^{k} : 0 \leq k < r}$ : the $A^{k}$ are distinct because $A$ has exact order $r$ , the $ρ (s) A^{k}$ are distinct for the same reason, and $A^{k} \neq ρ (s) A^{l}$ because $\det A^{k} = 1$ while $\det (ρ (s) A^{l}) = - 1$ ; so $∣ ρ (W) ∣ = 2 r$ and injectivity of $ρ$ [F4] gives $∣ W ∣ = 2 r$ , $W$ being the dihedral group $I_{2} (r)$ . For $r = \infty$ the same factorisation with $ρ (s) ρ (t)$ of infinite order [F3] exhibits infinitely many distinct elements and $W$ is the infinite dihedral group, completing (ii) and (iii).

3.1F1F5F7step 2.1algebra∎

(The low-rank coincidences and the products (iv).) Reading the labelled graphs: $I_{2} (3)$ is a single edge labelled $3$ , which is the one-edge path $A_{2}$ ; $I_{2} (4)$ is a single edge labelled $4$ , which is $B_{2}$ , also written $C_{2}$ ; $I_{2} (6)$ is a single edge labelled $6$ , conventionally $G_{2}$ ; $m (s, t) = 2$ draws no edge [F1], so the two-vertex diagram with $m = 2$ is the disconnected diagram $A_{1} \times A_{1}$ , which is the direct product of two one-vertex groups by the component statement [F7]; and $I_{2} (5) = H_{2}$ by the rank-two naming convention. The one-vertex diagram $A_{1}$ has $B = (1)$ , which is positive definite [F5], and presentation $⟨ s ∣ s^{2} = 1 ⟩$ , so $W ≅ Z / 2$ ; the disconnected two-vertex diagram has group $(Z / 2) \times (Z / 2)$ , of order $2 \cdot 2 = 4$ [F7]. These coincidences and the group orders are (iv), and the group $I_{2} (r)$ of 2.1 [step 2.1] is the dihedral group appearing under both names in the classification.

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Gram determinants and principal minors of the non-crystallographic types $H_{3}$ and $H_{4}$

Example

Let $Γ = H_{3}$ be the path $s_{1} - s_{2} - s_{3}$ with labels $m (s_{1}, s_{2}) = 3$ , $m (s_{2}, s_{3}) = 5$ , and let $Γ = H_{4}$ be the path $s_{1} - s_{2} - s_{3} - s_{4}$ with labels $3, 3, 5$ (Coxeter diagrams: edges, labels, components and finite type); let $B$ be the Coxeter form and $C$ the cosine matrix (The real Coxeter form, its radical, reflections, and form-preserving maps). Using $\cos (π / 5) = (1 + \sqrt{5}) / 4$ (derived in Verification 1.1):

(i) $H_{3}$ . The leading principal minors of $2 C$ are $2$ , $3$ and $\det (2 C) = 3 - \sqrt{5} > 0$ ; the $2 \times 2$ principal minor on the label- $5$ edge equals $4 \sin^{2} (π / 5) = (5 - \sqrt{5}) / 2 > 0$ ; the minor on ${s_{1}, s_{3}}$ equals $4$ . Hence $B$ is positive definite and the Coxeter group of type $H_{3}$ is finite.

(ii) $H_{4}$ . The leading principal minors of $2 C$ are $2, 3, 4$ (the first three vertices form type $A_{3}$ ) and $\det (2 C) = \frac{7 - 3 \sqrt{5}}{2} > 0$ . Hence $B$ is positive definite and the Coxeter group of type $H_{4}$ is finite.

(iii) Excluded neighbours. The overlong paths with labels $3, 5, 3$ and $3, 3, 5, 3$ have negative determinant, and the star with a degree- $3$ vertex whose three incident edges have labels $3, 3, 5$ has the negative witness value $1 - (\frac{1}{4} + \frac{1}{4} + \cos^{2} (π / 5)) < 0$ ; none of these diagrams is of finite type (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite, Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (3),(4)).

Facts & Assumptions

Given: The paths $H_{3}$ , $H_{4}$ and the three excluded diagrams above, with the Coxeter form $B$ and its cosine matrix $C = (B (e_{s}, e_{t}))$ ; write $d_{k}$ for the determinant of the leading $k \times k$ principal submatrix of $C$ .

[F1]

$B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ and $B (e_{s}, e_{t}) = - 1$ for $m (s, t) = \infty$ , without any positivity assumption; non-adjacent distinct vertices have $m (s, t) = 2$ and hence matrix entry $0$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Coxeter diagrams: edges, labels, components and finite type).

[F2]

$T_{5} (\cos θ) = \cos (5 θ)$ for every real $θ$ , and the Chebyshev polynomials of the first kind satisfy $T_{0} = 1$ , $T_{1} = t$ , $T_{n + 2} = 2 t T_{n + 1} - T_{n}$ ; $\cos (x + π) = - \cos x$ , $\cos π = - 1$ ; $\cos (2 x) = 2 \cos^{2} x - 1$ ; $1 - \cos^{2} x = \sin^{2} x$ and $\sin x > 0$ for $0 < x < π$ ; cosine is strictly decreasing on $[0, π]$ ; and $2 < \sqrt{5} < 3$ , $3 \sqrt{5} < 7$ ( $T_{n} (\cos θ) = \cos (n θ)$ and $U_{n} (\cos θ) \sin θ = \sin ((n + 1) θ)$ for every $n \in N$ , Chebyshev polynomials of the first and second kinds by their three-term recurrences, Quarter-turn values and shifts by pi/2 and pi, Double-angle and quadratic power-reduction identities, Signs, monotonicity intervals, and ranges of sine and cosine, Pi as twice the smallest positive zero of cosine, Parity and the Pythagorean identity for sine and cosine, Squaring is monotone on the nonnegatives, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F3]

A symmetric real matrix is positive definite if and only if its leading principal minors are positive; the form $B$ is positive definite when $B (u, u) > 0$ for every $u \neq 0$ , and $W$ is finite if and only if $B$ is positive definite (Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form, Finiteness criterion: W is finite exactly when the Coxeter form is positive definite).

[F4]

Scaling an $n \times n$ matrix by $2$ multiplies its determinant by $2^{n}$ , and $d_{k}$ is the determinant of the leading $k \times k$ principal submatrix (Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring, The determinant is the unique normalized alternating multilinear function on the columns).

[F5]

$H_{3}$ and $H_{4}$ are among the standard diagrams of the classification, and every proper principal submatrix of a listed diagram is a block diagonal matrix whose blocks are listed diagrams of smaller rank (Classification of finite Coxeter systems, including the H and dihedral families (3)).

[F6]

Laplace expansion along any row or column expresses the determinant as the sum of entries times their cofactors; the cofactor sign is $(- 1)^{i + j}$ and the determinant of the empty deleted matrix is $1$ (Laplace expansion computes the determinant along every row and every column over a commutative ring, Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring).

Verification

1.1F2algebra

(The three trigonometric values and the small determinants.) $\cos (π / 3) = 1 / 2$ follows from the double-angle formula $\cos (2 x) = 2 \cos^{2} x - 1$ at $x = π / 3$ together with $\cos (2 π / 3) = \cos (π - π / 3) = - \cos (π / 3)$ , which gives $2 c^{2} - 1 = - c$ , i.e. $(2 c - 1) (c + 1) = 0$ , and $c = \cos (π / 3) > - 1$ because $π / 3 \in (0, π)$ and cosine is strictly decreasing on $[0, π]$ with $\cos π = - 1$ [F2]; thus $c = 1 / 2$ . Also $\cos (π / 5) = (1 + \sqrt{5}) / 4$ : with $c_{5} = \cos (π / 5)$ , iterating the recurrence of [F2] gives $T_{5} (t) = 16 t^{5} - 20 t^{3} + 5 t$ , so $T_{5} (c_{5}) = \cos π = - 1$ , i.e. $16 c_{5}^{5} - 20 c_{5}^{3} + 5 c_{5} + 1 = (c_{5} + 1) (4 c_{5}^{2} - 2 c_{5} - 1)^{2} = 0$ by expansion, while $0 < π / 5 < π / 2$ gives $0 < c_{5} < 1$ [F2], so $c_{5} \neq - 1$ , $4 c_{5}^{2} - 2 c_{5} - 1 = 0$ and $(c_{5} - \frac{1}{4})^{2} = \frac{5}{16}$ ; by uniqueness of nonnegative square roots $c_{5} = \frac{1}{4} \pm \frac{\sqrt{5}}{4}$ , and the positive value is $(1 + \sqrt{5}) / 4$ because the other is negative as $2 < \sqrt{5}$ [F2]. Thus $\sin^{2} (π / 5) = 1 - \cos^{2} (π / 5) = (5 - \sqrt{5}) / 8 > 0$ [F2]; also $\sqrt{5} < 3$ and $3 \sqrt{5} < 7$ [F2], by squaring the positive quantities ( $5 < 9$ and $45 < 49$ ).

1.2F1F6algebra

(The recurrence without positivity.) For any of the paths in the Example, let $C_{k}$ be its leading $k \times k$ cosine matrix and set $d_{0} : = 1$ , $d_{1} = 1$ . For $k \geq 2$ put $a : = \cos (π / m_{k - 1})$ . By [F1] the last row has only the potentially nonzero entries $- a$ in column $k - 1$ and $1$ in column $k$ . In the deleted matrix for the first entry, the last column has only the bottom entry $- a$ , whose cofactor is $d_{k - 2}$ ; its determinant is therefore $- a d_{k - 2}$ by [F6]. The last-row cofactor sign at $(k, k - 1)$ is $- 1$ , while the diagonal cofactor is $d_{k - 1}$ . Thus Laplace expansion yields $d_{k} = d_{k - 1} - a^{2} d_{k - 2}$ [F6]. This identity uses no definiteness assumption and applies equally to the excluded paths.

2.1F1F2F3F4step 1.1step 1.2algebra

(The $H_{3}$ minors and positive definiteness.) With $d_{0} = 1$ , $d_{1} = 1$ and $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ by 1.2 [step 1.2]: $d_{2} = 1 - \cos^{2} (π / 3) = \frac{3}{4}$ , $d_{3} = \frac{3}{4} - \cos^{2} (π / 5) \cdot 1 = \frac{3}{4} - \frac{3 + \sqrt{5}}{8} = \frac{3 - \sqrt{5}}{8} > 0$ by 1.1 [step 1.1]. Multiplying by $2^{k}$ [F4] gives for $2 C$ the leading minors $2 d_{1} = 2$ , $4 d_{2} = 3$ , $8 d_{3} = 3 - \sqrt{5} > 0$ , and $\det (2 C) = 3 - \sqrt{5}$ ; the principal submatrix of $C$ on the label- $5$ edge ${s_{2}, s_{3}}$ is $(\begin{array}{cc} 1 & - \cos (π / 5) \\ - \cos (π / 5) & 1 \end{array})$ with determinant $\sin^{2} (π / 5)$ , so the determinant of the corresponding submatrix of $2 C$ is $4 \sin^{2} (π / 5) = (5 - \sqrt{5}) / 2 > 0$ [F1, F2]. The submatrix of $2 C$ on ${s_{1}, s_{3}}$ is $diag (2, 2)$ with determinant $4$ . All leading principal minors of $2 C$ are positive, so $2 C$ and hence $C$ is positive definite by [F3], and $H_{3}$ has finite Coxeter group.

2.2F1F2F3F4step 1.1step 1.2algebra

(The $H_{4}$ minors and positive definiteness.) Using the recurrence of 1.2 [step 1.2] for the path with labels $3, 3, 5$ : $d_{1} = 1$ , $d_{2} = \frac{3}{4}$ , $d_{3} = \frac{3}{4} - \frac{1}{4} \cdot 1 = \frac{1}{2}$ , $d_{4} = \frac{1}{2} - \cos^{2} (π / 5) \cdot \frac{3}{4} = \frac{1}{2} - \frac{3 + \sqrt{5}}{8} \cdot \frac{3}{4} = \frac{16 - 9 - 3 \sqrt{5}}{32} = \frac{7 - 3 \sqrt{5}}{32} > 0$ by 1.1 [step 1.1]. The first three vertices carry the all- $3$ path $A_{3}$ with the computed doubled leading minors $2, 3, 4$ [F4], and $\det (2 C) = 16 d_{4} = (7 - 3 \sqrt{5}) / 2 > 0$ because $3 \sqrt{5} < 7$ [F2]; all leading principal minors of $2 C$ are positive, so $C$ is positive definite by [F3] and $H_{4}$ has finite Coxeter group.

2.3F1F2F3step 1.1step 1.2algebra

(The excluded neighbours.) The unconditional recurrence of 1.2 [step 1.2] for the path with labels $3, 5, 3$ gives $d_{4} = d_{3} - \cos^{2} (π / 3) d_{2} = \frac{3 - \sqrt{5}}{8} - \frac{1}{4} \cdot \frac{3}{4} = \frac{3 - 2 \sqrt{5}}{16} < 0$ because $2 \sqrt{5} > 3$ , i.e. $\sqrt{5} > 3 / 2$ , follows from $\sqrt{5} > 2$ [F2]; for the path with labels $3, 3, 5, 3$ it gives $d_{5} = d_{4} - \cos^{2} (π / 3) d_{3} = \frac{7 - 3 \sqrt{5}}{32} - \frac{1}{4} \cdot \frac{1}{2} = \frac{3 - 3 \sqrt{5}}{32} < 0$ because $\sqrt{5} > 1$ [F2]; a negative leading minor excludes positive definiteness by [F3]. For the star with centre $c$ and neighbours $a, b, d$ , edges $c a, c b$ labelled $3$ and $c d$ labelled $5$ , the vector $u = e_{c} + \frac{1}{2} (e_{a} + e_{b}) + \cos (π / 5) e_{d} \neq 0$ has non-negative coordinates and satisfies $B (u, u) = 1 + \frac{1}{4} + \frac{1}{4} + \cos^{2} (π / 5) + 2 (- \frac{1}{4} - \frac{1}{4} - \cos^{2} (π / 5)) = 1 - (\frac{1}{4} + \frac{1}{4} + \cos^{2} (π / 5)) = \frac{1 - \sqrt{5}}{8} < 0$ because $\cos^{2} (π / 5) = (3 + \sqrt{5}) / 8$ [F2]; by [F3] this star is not positive definite either.

3.1F3F5step 2.1step 2.2step 2.3algebra∎

(Conclusion.) The leading principal minors of $2 C$ computed in 2.1 and 2.2 [step 2.1, step 2.2] are all positive, so $B$ is positive definite for $H_{3}$ and for $H_{4}$ by Sylvester's criterion [F3]; by the finiteness criterion [F3] the Coxeter groups of types $H_{3}$ and $H_{4}$ are finite, agreeing with their appearance in the classification [F5]. The three diagrams of 2.3 [step 2.3] either have a negative leading principal minor or a nonzero non-negative vector with non-positive value, so none of them is positive definite and none of their Coxeter groups is finite [F3], which is (iii).

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-08Open item page →

$B_{n}$ and $C_{n}$ define the same Coxeter diagram and the same Coxeter group

Example

Let $n \geq 2$ and let $B_{n}$ and $C_{n}$ denote the Coxeter systems whose diagram is the path on $n$ vertices with labels $3, \dots, 3, 4$ (Coxeter diagrams: edges, labels, components and finite type). Then:

(i) Same diagram, same system. The Coxeter matrices of $B_{n}$ and $C_{n}$ are equal, so the two names denote the same Coxeter system, the same diagram and the same group $W$ ; the distinction between the $B$ and $C$ families belongs to root-system data (a long and a short simple root), not to the Coxeter presentation.

(ii) Positivity and determinant. With $C$ the cosine matrix, $\det (2 C) (B_{n}) = 2$ , while the leading principal minors of $2 C$ are those of the paths $A_{k}$ ( $1 \leq k \leq n - 1$ ), namely $2, 3, \dots, n$ , and the full determinant $2$ . Hence $B_{n}$ is positive definite, $B_{n}$ is of finite type, and the groups $B_{n}$ , $C_{n}$ are finite of the same order (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite).

(iii) Small ranks. $B_{2} = C_{2} = I_{2} (4)$ , and $B_{n}$ contains $A_{n - 1} = S_{n}$ as a standard parabolic; type $C_{n}$ has the same Coxeter system and the same Coxeter diagram as $B_{n}$ for every $n \geq 2$ (Classification of finite Coxeter systems, including the H and dihedral families (4)).

Facts & Assumptions

Given: $n \geq 2$ , the path $B_{n} = C_{n}$ on vertices $s_{1}, \dots, s_{n}$ with labels $m (s_{k}, s_{k + 1}) = 3$ for $k \leq n - 2$ and $m (s_{n - 1}, s_{n}) = 4$ , the Coxeter form $B$ on $V = R^{{s_{1}, \dots, s_{n}}}$ , its cosine matrix $C = (B (e_{s}, e_{t}))$ , and the presented group $W$ .

[F1]

A diagram is determined by its Coxeter matrix and conversely: two Coxeter systems with the same labelled graph have the same matrix and hence are the same diagram and the same presented group; the subdiagram on a vertex subset is the induced labelled graph (Coxeter diagrams: edges, labels, components and finite type).

[F2]

$B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ and $B (e_{s}, e_{t}) = - 1$ for $m (s, t) = \infty$ , without any positivity assumption (The real Coxeter form, its radical, reflections, and form-preserving maps).

[F3]

$\cos (π / 4) = \sqrt{2} / 2$ ; $\cos (2 x) = 2 \cos^{2} x - 1$ , $\cos (x + π) = - \cos x$ , cosine is even and strictly decreasing on $[0, π]$ , and $\cos π = - 1$ (The finite Viete cosine product and its positive nested-radical factors, Double-angle and quadratic power-reduction identities, Quarter-turn values and shifts by pi/2 and pi, Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine).

[F4]

Laplace expansion along any row or column expresses a determinant in terms of cofactors, with the empty minor assigned determinant $1$ ; scaling an $n \times n$ matrix by $2$ multiplies its determinant by $2^{n}$ ; and induction on $N$ is available (Laplace expansion computes the determinant along every row and every column over a commutative ring, Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring, The determinant is the unique normalized alternating multilinear function on the columns, The principle of mathematical induction, The natural numbers $N$ (von Neumann)).

[F5]

A symmetric real matrix is positive definite if and only if its leading principal minors are positive; $W$ is finite if and only if $B$ is positive definite (Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form, Finiteness criterion: W is finite exactly when the Coxeter form is positive definite).

[F6]

$B_{2}$ , $C_{2}$ and $I_{2} (4)$ name the two-vertex diagram with label $4$ , and the standard parabolic $W_{A_{n - 1}}$ of type $A_{n - 1}$ is isomorphic to the symmetric group $S_{n}$ (Classification of finite Coxeter systems, including the H and dihedral families (4), Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2),(4)).

Verification

1.1F1F2F3F4algebra

(The diagram and the leading minors.) The two names denote the same labelled path, hence the same Coxeter matrix, presented group and form [F1]. Let $C_{k}$ be its leading $k \times k$ submatrix and $d_{k} : = \det C_{k}$ , with $d_{0} : = 1$ and $d_{1} = 1$ . For $k \geq 2$ put $a : = \cos (π / m_{k - 1})$ . The last row has only $- a$ and $1$ as potentially nonzero entries [F2]. Its diagonal cofactor is $d_{k - 1}$ ; deleting row $k$ and column $k - 1$ leaves a matrix whose last column has only the bottom entry $- a$ , so a second Laplace expansion gives deleted-matrix determinant $- a d_{k - 2}$ . The off-diagonal cofactor has sign $- 1$ , and therefore $d_{k} = d_{k - 1} - a^{2} d_{k - 2}$ [F4], without assuming positivity. To compute the all- $3$ prefix, put $c : = \cos (π / 3)$ : [F3] gives $2 c^{2} - 1 = \cos (2 π / 3) = - c$ and $c > - 1$ , hence $(2 c - 1) (c + 1) = 0$ and $c = 1 / 2$ . Thus for $2 \leq k \leq n - 1$ the recurrence is $d_{k} = d_{k - 1} - \frac{1}{4} d_{k - 2}$ . Induction, starting with $d_{0} = d_{1} = 1$ , gives $d_{k} = (k + 1) / 2^{k}$ for $0 \leq k \leq n - 1$ , since $\frac{k}{2^{k - 1}} - \frac{k - 1}{2^{k}} = \frac{k + 1}{2^{k}}$ [F4]. At the final label- $4$ edge, [F3] gives $d_{n} = \frac{n}{2^{n - 1}} - \frac{1}{2} \frac{n - 1}{2^{n - 2}} = 2^{1 - n}$ . Consequently the leading minors of $2 C$ are $2^{k} d_{k} = k + 1$ for $1 \leq k < n$ , and $\det (2 C) = 2^{n} d_{n} = 2$ [F4]. This also covers $n = 2$ , using $d_{0} = 1$ .

1.2F1F6algebra

(Small ranks and the parabolic $A_{n - 1}$ .) For $n = 2$ the path has the single edge labelled $4$ , which is the diagram $I_{2} (4)$ , and this is the same labelled graph as $B_{2} = C_{2} = I_{2} (4)$ [F1, F6]. The subdiagram on ${s_{1}, \dots, s_{n - 1}}$ is the all- $3$ path $A_{n - 1}$ ; hence $W_{A_{n - 1}} = ⟨ s_{1}, \dots, s_{n - 1} ⟩$ is a standard parabolic subgroup of $W$ , isomorphic to the Coxeter group of type $A_{n - 1}$ , which is the symmetric group $S_{n}$ [F6]. Since the two names share the same labelled graph, type $C_{n}$ has the same Coxeter system and the same standard parabolic $A_{n - 1}$ , which is (iii).

2.1F1F5step 1.1algebra

(Positive definiteness, finiteness, and the same order.) By 1.1 [step 1.1] the leading principal minors of $2 C$ are $2, 3, \dots, n$ and the full determinant is $2$ , all positive; scaling by the positive factor $2$ does not change definiteness, so $C$ has all leading principal minors positive and is positive definite by Sylvester's criterion [F5]. By the finiteness criterion [F5] the group of $B_{n}$ is finite, and since $C_{n}$ has the same Coxeter matrix it has the same Coxeter system, the same form and the same finite group, in particular the same order [F1]. This is (i) and (ii).

3.1step 2.1step 1.2algebra∎

(Conclusion.) The Coxeter matrices of $B_{n}$ and $C_{n}$ are equal, so the two names denote the same Coxeter diagram, the same Coxeter system and the same group, and the distinction between the $B$ and $C$ families lies in root-system data rather than in the presentation; the doubled cosine determinant is $\det (2 C) (B_{n}) = 2$ with leading minors $2, 3, \dots, n$ , so $B_{n}$ and $C_{n}$ are positive definite and of finite type; and $B_{2} = C_{2} = I_{2} (4)$ while $B_{n}$ has $A_{n - 1} = S_{n}$ as a standard parabolic, with $C_{n}$ identical.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-10-08Open item page →

Path determinants $d_{k} = d_{k - 1} - \cos^{2} (π / m) d_{k - 2}$ and the three-arm inequality

Example

(i) Path recursion. Let $Γ$ be the path $s_{1} - \dots - s_{n}$ with labels $m_{1}, \dots, m_{n - 1}$ ( $n \geq 1$ ) and cosine matrix $C$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Coxeter diagrams: edges, labels, components and finite type), and let $d_{k}$ be the determinant of the leading $k \times k$ principal submatrix of $C$ . Then $\cos (π / \infty) : = 1$ is the coefficient convention. Then $d_{0} = 1$ , $d_{1} = 1$ , and $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2} (2 \leq k \leq n) .$ For the path with all labels $3$ this gives $d_{k} = (k + 1) / 2^{k}$ and $\det (2 C) = n + 1$ ; for a path whose only label $\geq 4$ is $m$ on the edge between the $i$ -th and $(i + 1)$ -st vertices, positive definiteness requires the constraint $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m)$ of Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (5)(ii), with $j = n - i$ .

(ii) The three-arm inequality. For the star with central vertex $v$ and three arms of $p, q, r \geq 1$ vertices, all edges labelled $3$ , positive definiteness is equivalent to $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} > 1.$

(iii) Numerical checks. The triples satisfying the inequality are, up to order, $(1, 1, r)$ for every $r \geq 1$ (type $D_{r + 3}$ ), $(1, 2, 2)$ (type $E_{6}$ ), $(1, 2, 3)$ (type $E_{7}$ ) and $(1, 2, 4)$ (type $E_{8}$ ); the boundary cases $(1, 2, 5)$ (type $E_{9}$ ), $(2, 2, 2)$ and $(1, 3, 3)$ give equality $1$ , and the overlong star $(1, 2, 5)$ has an explicitly non-positive vector, as treated in A cycle and an overlong arm: explicit non-positive witnesses.

Facts & Assumptions

Given: The paths of (i) and the three-arm star of (ii), with their labelled diagrams, the space $V = R^{S}$ with Coxeter form $B$ and cosine matrix $C = (B (e_{s}, e_{t}))_{s, t \in S}$ , the standard basis vectors $e_{s}$ , and the leading minors $d_{k}$ of $C$ .

[F1]

An edge of the diagram is present exactly when $m (s, t) \geq 3$ and carries the label $m (s, t)$ ; the subdiagram on a vertex subset is the induced labelled graph (Coxeter diagrams: edges, labels, components and finite type).

[F2]

$B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ ; $B (e_{s}, e_{t}) = - 1$ for $m (s, t) = \infty$ ; $B$ is symmetric and bilinear, so $B (u, w) = \sum_{s, t \in S} u (s) w (t) B (e_{s}, e_{t})$ , and the $e_{s}$ form a basis of $V$ (The real Coxeter form, its radical, reflections, and form-preserving maps).

[F4]

For every row $i$ the determinant expands as $\det A = \sum_{k} a_{i k} C_{i k} (A)$ , where $C_{i k} (A)$ is the cofactor and $A^{(i, k)}$ , the matrix with row $i$ and column $k$ deleted, is the deleted matrix (Laplace expansion computes the determinant along every row and every column over a commutative ring, Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring).

[F5]

Determinant is multilinear in the columns and normalized, so multiplying an $n \times n$ matrix by the scalar $2$ multiplies its determinant by $2^{n}$ , and the determinant of a block triangular matrix is the product of the determinants of its diagonal blocks (The determinant is the unique normalized alternating multilinear function on the columns, Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring).

[F6]

A symmetric real matrix is positive definite if and only if all its leading principal minors are positive; the form $B$ is positive definite when $B (u, u) > 0$ for every $u \neq 0$ (Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F7]

For $c : = \cos (π / 3)$ one has $\cos (2 π / 3) = 2 c^{2} - 1$ and $\cos (2 π / 3) = - \cos (π / 3) = - c$ , cosine is strictly decreasing on $[0, π]$ and $\cos π = - 1$ (Double-angle and quadratic power-reduction identities, Quarter-turn values and shifts by pi/2 and pi, Signs, monotonicity intervals, and ranges of sine and cosine, Parity and the Pythagorean identity for sine and cosine); and for a real symmetric positive definite form on a subspace, $B (u, w)^{2} \leq B (u, u) B (w, w)$ with equality exactly when $u, w$ are linearly dependent (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F8]

The strong induction principle on $N$ holds (The principle of mathematical induction, The natural numbers $N$ (von Neumann)).

[F9]

The standard irreducible diagrams of the classification are $A_{n}$ (all labels $3$ ), $D_{n}$ (arms $1, 1, n - 3$ ), $E_{6}, E_{7}, E_{8}$ (arms $1, 2, 2$ ; $1, 2, 3$ ; $1, 2, 4$ ), and the types $D_{r + 3}$ , $E_{6}, E_{7}, E_{8}$ carry those diagrams (Classification of finite Coxeter systems, including the H and dihedral families (1)).

[F10]

The star with arms $1, 2, 5$ has the explicit non-positive vector of A cycle and an overlong arm: explicit non-positive witnesses (ii), and the triple $(1, 2, 5)$ fails the inequality of (ii) with equality.

Verification

1.1F1F2F4algebra

(The recursion by expansion.) Set $d_{0} : = 1$ ; the leading $1 \times 1$ matrix is $(1)$ , so $d_{1} = 1$ [F2]. For $2 \leq k \leq n$ let $C_{k}$ be the leading $k \times k$ submatrix and put $a : = \cos (π / m_{k - 1})$ , with $a = 1$ for an infinite label. By [F1, F2] its diagonal entries are $1$ , its adjacent off-diagonal entries are $- \cos (π / m_{j})$ and all others are $0$ . In the last-row Laplace expansion [F4], the diagonal entry contributes $d_{k - 1}$ . Deleting row $k$ and column $k - 1$ leaves a matrix whose last column has only its bottom entry $- a$ ; expanding that column gives determinant $- a d_{k - 2}$ , also for $k = 2$ with the empty minor. The last-row cofactor sign at $(k, k - 1)$ is $- 1$ , so the other contribution is $- a^{2} d_{k - 2}$ . Therefore $d_{k} = d_{k - 1} - a^{2} d_{k - 2}$ without any positivity assumption.

1.2algebra

(The integer cases.) Let $1 \leq p \leq q \leq r$ satisfy the inequality of (ii). If $p \geq 2$ then $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} \leq 3 \cdot \frac{1}{3} = 1$ with equality only for $p = q = r = 2$ , so $p = 1$ . Then $\frac{1}{q + 1} + \frac{1}{r + 1} > \frac{1}{2}$ : for $q = 1$ this holds for every $r \geq 1$ ; for $q = 2$ it says $\frac{1}{r + 1} > \frac{1}{6}$ , i.e. $r < 5$ , so $r \in {2, 3, 4}$ ; and for $q \geq 3$ the sum is at most $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ , a contradiction. Hence the triples are $(1, 1, r)$ for $r \geq 1$ , $(1, 2, 2)$ , $(1, 2, 3)$ and $(1, 2, 4)$ up to order. The equality cases $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} = 1$ are found the same way: $p \geq 3$ gives a sum $\leq \frac{3}{4} < 1$ ; $p = 2$ forces $q = r = 2$ ; and $p = 1$ forces $\frac{1}{q + 1} + \frac{1}{r + 1} = \frac{1}{2}$ , i.e. $q = 2, r = 5$ or $q = 3, r = 3$ . Thus $(2, 2, 2)$ , $(1, 2, 5)$ and $(1, 3, 3)$ are exactly the boundary triples.

2.1F5F6F7F8step 1.1algebra

(The all- $3$ path.) First $\cos (π / 3) = 1 / 2$ : putting $c = \cos (π / 3)$ , [F7] gives $2 c^{2} - 1 = \cos (2 π / 3) = - c$ , so $(2 c - 1) (c + 1) = 0$ , and $c > - 1$ because $0 < π / 3 < π$ , cosine is strictly decreasing on $[0, π]$ and $\cos π = - 1$ [F7], hence $c = 1 / 2$ . If every label is $3$ , then $\cos (π / m_{j}) = 1 / 2$ , so the recursion of 1.1 reads $d_{k} = d_{k - 1} - \frac{1}{4} d_{k - 2}$ with $d_{0} = d_{1} = 1$ ; by the induction principle [F8] the formula $d_{k} = (k + 1) / 2^{k}$ holds for all $k \geq 0$ , since $\frac{k}{2^{k - 1}} - \frac{1}{4} \cdot \frac{k - 1}{2^{k - 2}} = \frac{k + 1}{2^{k}}$ . Hence $d_{k} > 0$ for every $k$ , the leading principal minors of $2 C$ are $2^{k} d_{k} = k + 1 > 0$ by [F5], and $2 C$ and $C$ are positive definite by [F6]; in particular $\det (2 C) = 2^{n} d_{n} = n + 1$ .

3.1F4F6F8step 1.1step 2.1algebra

(One large edge: the two-subpath formula.) Let the path have all labels $3$ except one edge labelled $m$ between the $i$ -th and $(i + 1)$ -st vertices, and put $c : = \cos (π / m)$ , $j : = n - i \geq 1$ . Write $δ_{k} : = (k + 1) / 2^{k}$ for the determinant of an all- $3$ path on $k$ vertices, including $δ_{0} = 1$ , as proved in 2.1 [step 2.1]; these are distinct from the leading minors $d_{k}$ of the labelled path. Then $\det C = δ_{i} δ_{j} - c^{2} δ_{i - 1} δ_{j - 1} .$ Proof by induction on $j$ [F8], writing $C (i, j)$ for this matrix: for $j = 1$ the large edge is the last one and expansion along the last row [F4] gives $\det C (i, 1) = δ_{i} - c^{2} δ_{i - 1} = δ_{i} δ_{1} - c^{2} δ_{i - 1} δ_{0}$ ; for $j = 2$ the last edge has label $3$ , so the recurrence of 1.1 [step 1.1] gives $\det C (i, 2) = \det C (i, 1) - \frac{1}{4} δ_{i} = δ_{i} (δ_{1} - \frac{1}{4} δ_{0}) - c^{2} δ_{i - 1}$ , which is $δ_{i} δ_{2} - c^{2} δ_{i - 1} δ_{1}$ since $δ_{2} = \frac{3}{4}$ ; for $j \geq 3$ the last edge again has label $3$ , so $\det C (i, j) = \det C (i, j - 1) - \frac{1}{4} \det C (i, j - 2)$ , and substituting the induction hypothesis and the recursion $δ_{j} = δ_{j - 1} - \frac{1}{4} δ_{j - 2}$ of 2.1 [step 2.1] gives the formula. With $δ_{k} = (k + 1) / 2^{k}$ of 2.1 [step 2.1] this is $\det C = \frac{(i + 1) (j + 1) - 4 i j c^{2}}{2^{n}};$ and since positive definiteness of the path would give $\det C > 0$ by [F6], such a path satisfies $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m)$ , which is the constraint of Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (5)(ii).

3.2F1F2F6step 2.1algebra

(The arm vectors.) For the star of (ii) with centre $v$ and arm spans $A_{1}, A_{2}, A_{3}$ on the three arms, $V = R e_{v} \oplus A_{1} \oplus A_{2} \oplus A_{3}$ and the arm spans are pairwise $B$ -orthogonal, because distinct arms share no edge [F1, F2]. On the arm with vertices $s_{1}, \dots, s_{p}$ ( $s_{p}$ adjacent to $v$ ) put $w : = \sum_{k = 1}^{p} k e_{s_{k}}$ . Then $B (w, w) = \sum_{k = 1}^{p} k^{2} - \sum_{k = 1}^{p - 1} k (k + 1) = p^{2} - \sum_{k = 1}^{p - 1} k = \frac{p (p + 1)}{2}$ and $B (e_{v}, w) = - \frac{p}{2}$ by [F2], and for every $u = \sum_{k = 1}^{p} u_{k} e_{s_{k}}$ the identity $B (w, u) = \frac{p + 1}{2} u_{p}$ holds, because the coefficient of $u_{k}$ in $B (w, u)$ is $k - \frac{k - 1}{2} - \frac{k + 1}{2} = 0$ for $1 < k < p$ (also for $k = 1$ when $p > 1$ ), while the coefficient of $u_{p}$ is $p - \frac{p - 1}{2} = \frac{p + 1}{2}$ ; for $p = 1$ the coefficient is $1 = (p + 1) / 2$ directly. In particular $B (e_{v}, u) = - \frac{u_{p}}{2} = - \frac{B (w, u)}{p + 1}$ on the arm span. By 2.1 and [F6] the form restricted to each arm span (an all- $3$ path) is positive definite; hence $B$ is positive definite on all of $A = A_{1} \oplus A_{2} \oplus A_{3}$ , since a nonzero element $z_{1} + z_{2} + z_{3}$ has some nonzero component and $B (z, z) = \sum_{i} B (z_{i}, z_{i}) \geq B (z_{j}, z_{j}) > 0$ .

4.1F2F6F7step 3.2algebra

(Completing the square: equivalence of positive definiteness and the inequality.) Keep the notation of 3.2 [step 3.2] with arms of $p_{1}, p_{2}, p_{3}$ vertices and vectors $w_{1}, w_{2}, w_{3}$ , put $μ : = \sum_{i} \frac{p_{i}}{2 (p_{i} + 1)}$ and $\overset{ˉ}{w} : = - \sum_{i} \frac{w_{i}}{p_{i} + 1} \in A : = A_{1} \oplus A_{2} \oplus A_{3}$ , so that $\overset{ˉ}{w} \neq 0$ because its components in the distinct summands are the nonzero multiples $- w_{i} / (p_{i} + 1)$ . Then $B (\overset{ˉ}{w}, u) = B (e_{v}, u)$ for every $u \in A$ by the last identity of 3.2 [step 3.2], and $B (\overset{ˉ}{w}, \overset{ˉ}{w}) = \sum_{i} \frac{p_{i} (p_{i} + 1) / 2}{(p_{i} + 1)^{2}} = μ$ . Since $\overset{ˉ}{w}$ is a nonzero vector of $A$ , on which $B$ is positive definite [step 3.2], every $u \in V$ has a unique form $u = α e_{v} + γ \overset{ˉ}{w} + z$ with $α, γ \in R$ , $z \in A$ and $B (\overset{ˉ}{w}, z) = 0$ : the $e_{v}$ -coordinate $α$ is forced, and $u - α e_{v} \in A$ has the unique orthogonal decomposition $γ \overset{ˉ}{w} + z$ along $\overset{ˉ}{w}$ in the inner product space $(A, B)$ , with $γ = B (u - α e_{v}, \overset{ˉ}{w}) / B (\overset{ˉ}{w}, \overset{ˉ}{w})$ and $z = u - α e_{v} - γ \overset{ˉ}{w}$ . Then $B (u, u) = α^{2} + 2 α (γ B (e_{v}, \overset{ˉ}{w}) + B (e_{v}, z)) + γ^{2} B (\overset{ˉ}{w}, \overset{ˉ}{w}) + B (z, z) = α^{2} (1 - μ) + μ (γ + α)^{2} + B (z, z),$ because $B (e_{v}, \overset{ˉ}{w}) = B (\overset{ˉ}{w}, \overset{ˉ}{w}) = μ$ and $B (e_{v}, z) = B (\overset{ˉ}{w}, z) = 0$ [F2]. If $1 - μ > 0$ then $B (u, u)$ is a sum of three terms that are $\geq 0$ and is $0$ only when $α = 0$ , $γ = - α = 0$ and $z = 0$ , i.e. only for $u = 0$ ; conversely, if $1 - μ \leq 0$ , then $u = e_{v} - \overset{ˉ}{w} \neq 0$ (its $e_{v}$ -coordinate is $1$ ) has $B (u, u) = 1 - μ \leq 0$ , so $B$ is not positive definite [F6]. Hence $B$ is positive definite if and only if $1 - μ > 0$ , and $1 - μ > 0$ is equivalent to $\sum_{i} \frac{p_{i}}{p_{i} + 1} < 2$ , i.e. to $3 - \sum_{i} \frac{1}{p_{i} + 1} < 2$ , which is the inequality of (ii). When $B$ is positive definite, the same identity exhibits the strict Cauchy-Schwarz bound $B (\overset{ˉ}{w}, \overset{ˉ}{w}) < B (e_{v}, e_{v}) = 1$ of the projection of $e_{v}$ onto $A$ used in Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (6), strict because $e_{v} \notin A$ [F7].

5.1F9F10step 1.1step 1.2step 2.1step 3.1step 3.2step 4.1∎

(Conclusion.) The recursion and the all- $3$ values of (i) are steps 1.1 and 2.1 [step 1.1, step 2.1], and the two-subpath determinant formula giving the constraint $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m)$ is step 3.1 [step 3.1]. The equivalence of (ii) is step 4.1 [step 4.1], proved by completing the square along the three positive definite arms of 3.2 [step 3.2]. The list of triples of (iii) and the boundary triples are step 1.2 [step 1.2], and the boundary star $(1, 2, 5)$ is the one with the non-positive vector of A cycle and an overlong arm: explicit non-positive witnesses [F10]; the names $D_{r + 3}, E_{6}, E_{7}, E_{8}$ of the surviving triples are those of the classification [F9].

Finite Coxeter Diagrams and Complete Classification — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A cycle and an overlong arm: explicit non-positive witnesses

Example

Facts & Assumptions

Verification

Dihedral diagrams I2(m)I2​(m): Gram determinants, the infinite case, and the low-rank coincidences

Example

Facts & Assumptions

Verification

Gram determinants and principal minors of the non-crystallographic types H3H3​ and H4H4​

Example

Facts & Assumptions

Verification

BnBn​ and CnCn​ define the same Coxeter diagram and the same Coxeter group

Example

Facts & Assumptions

Verification

Path determinants dk=dk−1−cos⁡2(π/m)dk−2dk​=dk−1​−cos2(π/m)dk−2​ and the three-arm inequality

Example

Facts & Assumptions

Verification

Sources

Dihedral diagrams $I_{2} (m)$ : Gram determinants, the infinite case, and the low-rank coincidences

Gram determinants and principal minors of the non-crystallographic types $H_{3}$ and $H_{4}$

$B_{n}$ and $C_{n}$ define the same Coxeter diagram and the same Coxeter group

Path determinants $d_{k} = d_{k - 1} - \cos^{2} (π / m) d_{k - 2}$ and the three-arm inequality