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Finite Coxeter Diagrams and Complete Classification

1 · Prerequisites

2 · Summary

Finiteness of a Coxeter group is equivalent to positive definiteness of its canonical form, so the classification of finite Coxeter systems becomes a finite matrix problem. The route taken here includes the noncrystallographic diagrams rather than restricting to the Dynkin classification.

Coxeter diagrams: edges, labels, components and finite type sets up the labelled diagram of a Coxeter matrix - an edge exactly when $m (s, t) \geq 3$ , the label $3$ omitted and $m (s, t) = 2$ drawing no edge - together with its components and the notions of irreducibility and finite type. The definition deliberately builds in no list of diagrams and no positivity, so the classification below is a theorem and not a convention. Exclusions for positive definite diagrams: trees, valency, labels, chains and arms then extracts the necessary conditions a connected positive definite diagram must satisfy: it has no cycle; the neighbour inequality $\sum_{t \in N (s)} c (s, t)^{2} < 1$ holds with its consequences for valency and labels; there is at most one vertex of degree $3$ and at most one edge of label $\geq 4$ ; the leading minors of a path satisfy $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ , which forces $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m)$ for the edge of label $m \geq 4$ splitting the path into subpaths of $i$ and $j$ vertices; and a branch vertex with arms of $p, q, r$ vertices forces $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} > 1$ . The lemma closes with the complete list of connected positive definite diagrams.

Disconnected diagrams, direct products, and comparison of invariant forms proves first that a disconnected diagram presents $W$ as the direct product of the standard parabolics of its components, with the length function adding, that the Coxeter form decomposes orthogonally over the components, and that every $ρ (W)$ -invariant symmetric bilinear form is a componentwise multiple of $B$ ; averaging an explicit inner product over a finite $W$ yields one direction of Finiteness criterion: W is finite exactly when the Coxeter form is positive definite: finite $W$ has positive definite form. For the converse the theorem builds the dual form $B^{*}$ , isolates the identity in the image of the dual action through the open chamber interiors and the collision theorem of tits-cones-chambers-and-parabolic-stabilizers, and concludes with compactness of the orthogonal group and a choice-free Heine-Borel argument, so $W$ is finite if and only if $B$ is positive definite. Classification of finite Coxeter systems, including the H and dihedral families assembles the classification: the irreducible finite Coxeter systems are exactly those of types $A_{n}$ ( $n \geq 1$ ), $B_{n}$ ( $n \geq 2$ ), $D_{n}$ ( $n \geq 4$ ), $E_{6}, E_{7}, E_{8}$ , $F_{4}$ , $H_{3}$ , $H_{4}$ and $I_{2} (m)$ ( $3 \leq m < \infty$ ), positivity of every surviving diagram being verified by the doubled determinants $\det (2 C)$ together with Sylvester's criterion, and the reducible finite systems are exactly the direct products of these, with the low-rank coincidences $A_{2} = I_{2} (3)$ , $B_{2} = C_{2} = I_{2} (4)$ , $G_{2} = I_{2} (6)$ , $H_{2} = I_{2} (5)$ recorded explicitly.

The noncrystallographic types $H_{3}$ and $H_{4}$ and the arbitrary dihedral family $I_{2} (m)$ are included on purpose; crystallographic integrality and the affine semidefinite classification are not treated here, the latter being the subject of affine-coxeter-diagrams-and-semidefinite-classification. Earlier pages: tits-cones-chambers-and-parabolic-stabilizers supplies the dual chambers, their nonempty interiors and the collision theorem $w \cdot f = g \Rightarrow f = g$ , $w \in W_{S (f)}$ used to isolate the identity; coxeter-presentations-exchange-and-reduced-word-theorems supplies the presentation, the length function and the intrinsic parabolic presentations. The companion finite-coxeter-diagrams-and-complete-classification-examples carries the determinant computations in $I_{2} (m)$ , $H_{3}$ and $H_{4}$ , the comparison of $B_{n}$ and $C_{n}$ and the explicit non-positive witnesses for the excluded diagrams.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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

Coxeter diagrams: edges, labels, components and finite type

Definition

Let $S$ be a finite set and let $m : S \times S \to {1, 2, 3, \dots} \cup {\infty}$ be a Coxeter matrix on $S$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), with presented group $W$ , length function $ℓ$ and standard parabolics $W_{T} = ⟨ s : s \in T ⟩$ for $T \subseteq S$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, Group and abelian group).

(1) The diagram. The Coxeter diagram $Γ (W, S)$ of $(W, S)$ has vertex set $S$ , and two distinct vertices $s \neq t$ are joined by an edge exactly when $m (s, t) \geq 3$ ; that edge carries the label $m (s, t) \in {3, 4, 5, \dots} \cup {\infty}$ . By convention the label $3$ is omitted (an unlabelled edge has label $3$ ) and no edge is drawn when $m (s, t) = 2$ . Thus $Γ$ is a finite simple graph whose edges are labelled in ${3, 4, \dots} \cup {\infty}$ (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree), and $m$ is recovered from the pair $(S, Γ)$ by $m (s, s) = 1$ ; $m (s, t) = m (t, s) = 2$ if ${s, t}$ is not an edge; $m (s, t) = 3$ if ${s, t}$ is an unlabelled edge; and $m (s, t) =$ the label otherwise. In particular $m \mapsto Γ$ is injective on the Coxeter matrices on $S$ . For $T \subseteq S$ the subdiagram $Γ_{T}$ is the induced labelled graph on $T$ , i.e. the diagram of the restricted matrix $m ∣_{T \times T}$ .

(2) Graph-theoretic vocabulary. A cycle of $Γ$ is a cycle of the underlying simple graph (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges); a path (or chain) is a diagram whose underlying graph is a path. $Γ$ is connected when its underlying graph is connected (Connected graphs and connected components defined by the existence of vertex paths); its components are the connected components of the underlying graph, and their vertex sets are nonempty and partition $S$ . For $s \in S$ the neighbours of $s$ are $N (s) : = {t \in S : t \neq s, m (s, t) \geq 3}$ and the degree of $s$ is $∣ N (s) ∣$ (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

(3) Irreducibility. $(W, S)$ is irreducible when $Γ$ is connected, and reducible otherwise. (For $S = \emptyset$ the diagram is empty; the trivial system is not called irreducible.)

(4) Finite type. $(W, S)$ - and, by extension, $Γ$ - is of finite type, or spherical, when the group $W$ is finite. This is a property of $W$ itself: no list of diagrams is part of the definition, and no positivity, definiteness or nondegeneracy of any bilinear form, and no geometric realization, is asserted here.

(5) Invariance and abstentions. By the conventions of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, $m (s, t)$ is the order of $s t$ in $W$ ; consequently an isomorphism of Coxeter systems $(W, S) \to (W^{'}, S^{'})$ (a group isomorphism carrying $S$ onto $S^{'}$ ) matches the two diagrams, so connectedness, irreducibility and finite type depend only on the isomorphism type of $(W, S)$ . It is not asserted here that $W$ is the direct product of the standard parabolics of its components - that is the content of Disconnected diagrams, direct products, and comparison of invariant forms - nor that a diagram of finite type is one of the diagrams classified in Classification of finite Coxeter systems, including the H and dihedral families.

Remarks

The diagram is a complete record of the matrix. The reconstruction rule displayed in (1) is a dictionary, not a theorem about groups: it lists the value of $m$ on every ordered pair of vertices in terms of $(S, Γ)$ , and therefore shows simultaneously that distinct Coxeter matrices on $S$ give distinct labelled graphs, and that the induced labelled subgraph on $T$ is the diagram of the restricted matrix $m ∣_{T \times T}$ .
Conventions used consistently on this page. Labels lie in ${3, 4, \dots} \cup {\infty}$ ; a label $3$ edge is drawn unlabelled; a pair with $m (s, t) = 2$ is not joined at all. Consequently the edge set of $Γ$ is ${{s, t} : s \neq t, m (s, t) \geq 3}$ , and $N (s)$ of (2) is exactly the open neighbourhood of $s$ in the underlying simple graph.
What finite type does not mean here. In (4) "spherical" is a synonym for finiteness of $W$ only. The equivalence with positive definiteness of the Coxeter form is a theorem (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite ↗), and the explicit list of finite type diagrams is the content of Classification of finite Coxeter systems, including the H and dihedral families; neither is built into the definition.

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

Exclusions for positive definite diagrams: trees, valency, labels, chains and arms

Statement

Let $S$ be a finite set with Coxeter matrix $m$ and diagram $Γ$ (Coxeter diagrams: edges, labels, components and finite type), let $V = R^{S}$ and let $B$ be the Coxeter form (The real Coxeter form, its radical, reflections, and form-preserving maps). For $s \neq t$ in $S$ write $c (s, t) : = - B (e_{s}, e_{t}) = \cos (π / m (s, t)) (finite m (s, t)), c (s, t) : = 1 (m (s, t) = \infty),$ so that $c (s, t) \in [0, 1]$ , $c (s, t) = 0$ exactly when $m (s, t) = 2$ , and $c (s, t) \geq 1 / 2$ whenever $m (s, t) \geq 3$ . The cosine matrix is $C : = (B (e_{s}, e_{t}))_{s, t \in S}$ , so $C$ has diagonal entries $1$ and off-diagonal entries $- c (s, t)$ . Assume that $Γ$ is connected and that $B$ is positive definite (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form). Then:

(1) Witness principle. If $T \subseteq S$ and there is $0 \neq u \in V_{T} = s p a n {e_{s} : s \in T}$ with $B (u, u) \leq 0$ , then $B$ is not positive definite, because such a $u$ is a nonzero vector of $V$ with $B (u, u) \leq 0$ . Moreover, if all coordinates of $u$ in the basis $(e_{s})_{s \in T}$ are $\geq 0$ and a labelled graph $Γ_{0}$ on $T$ has all labels at most the corresponding labels of $Γ_{T}$ (a non-edge having label $2$ ), then $B_{T} (u, u) \leq B_{0} (u, u)$ , where $B_{0}$ is the cosine form of $Γ_{0}$ ; so a non-positive value of $B_{0}$ on a non-negative vector excludes positive definiteness of $B$ . The explicit non-negative witnesses below use this comparison.

(2) No cycles. $Γ$ contains no cycle: if $s_{1}, \dots, s_{r}$ ( $r \geq 3$ ) are distinct vertices whose consecutive pairs ${s_{i}, s_{i + 1}}$ ( $i$ modulo $r$ ) are edges, then $u = e_{s_{1}} + \dots + e_{s_{r}}$ satisfies $B (u, u) \leq r - 2 r \cdot \frac{1}{2} = 0$ , because the $r$ consecutive pairs contribute $c \geq \frac{1}{2}$ each and all other pairs contribute $c \geq 0$ .

(3) Valency and local labels. For every $s \in S$ , $\sum_{t \in N (s)} c (s, t)^{2} < 1.$ In particular: no label is $\infty$ ; no vertex has four or more neighbours; if a vertex has exactly three neighbours then its three edges all have label $3$ ; and if a vertex has exactly two neighbours with labels $m_{1} \leq m_{2} < \infty$ , then $m_{1} = 3$ and $m_{2} \leq 5$ ; in particular the pairs $(4, 4)$ and $(3, 6)$ are forbidden at a vertex.

(4) At most one branch vertex, and at most one large label.

(i) $Γ$ has at most one vertex of degree $3$ (hence, with (3), at most one vertex of degree $\geq 3$ ).

(ii) $Γ$ has at most one edge whose label is $\geq 4$ ; and if such an edge exists then $Γ$ has no vertex of degree $3$ , so by (3) $Γ$ is a path.

(5) Paths. Suppose $Γ$ is a path on $n$ vertices, with labels $m_{1}, \dots, m_{n - 1}$ along the path. In formulas involving labels, $\cos (π / \infty)$ denotes the coefficient $1$ .

(i) If $d_{k}$ is the determinant of the leading $k \times k$ principal submatrix of the cosine matrix $C$ , then $d_{0} = 1$ , $d_{1} = 1$ and $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ for $2 \leq k \leq n$ ; for the path with all labels $3$ one gets $d_{k} = (k + 1) / 2^{k} > 0$ for every $k$ , and $\det (2 C) = n + 1$ .

(ii) If the labels are all $3$ except one edge labelled $m \geq 4$ , and that edge splits the path into two subpaths with $i$ and $j$ vertices ( $i + j = n$ , $i \leq j$ , $i, j \geq 1$ ), then $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m) .$ Consequently: if $m \geq 6$ then $i = j = 1$ ; if $m = 5$ then $(i, j) \in {(1, 1), (1, 2), (1, 3)}$ ; if $m = 4$ then $i = 1$ , or $(i, j) = (2, 2)$ .

(6) Three arms. Suppose $Γ$ has a (unique) vertex $v$ of degree $3$ and all its edges have label $3$ , and let $p, q, r \geq 1$ be the numbers of vertices in the three components of $Γ - v$ (each of which is a path). Then $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} > 1.$ Consequently, up to permutation, $(p, q, r) = (1, 1, r)$ for some $r \geq 1$ , or $(p, q, r) \in {(1, 2, 2), (1, 2, 3), (1, 2, 4)}$ .

(7) Conclusion. Every connected positive definite Coxeter diagram is isomorphic as a labelled graph to one of: $A_{n}$ ( $n \geq 1$ ; a path, all labels $3$ ), $B_{n}$ ( $n \geq 2$ ; a path, labels $3, \dots, 3, 4$ ), $D_{n}$ ( $n \geq 4$ ; the star with arms of $1, 1, n - 3$ vertices), $E_{6}, E_{7}, E_{8}$ (the stars with arms $1, 2, 2$ ; $1, 2, 3$ ; $1, 2, 4$ ), $F_{4}$ (the path with labels $3, 4, 3$ ), $H_{3}$ (the path with labels $3, 5$ ), $H_{4}$ (the path with labels $3, 3, 5$ ), or $I_{2} (m)$ ( $m \geq 3$ ; two vertices joined by one edge labelled $m$ ).

Facts & Assumptions

Given: A finite set $S$ with Coxeter matrix $m$ , its diagram $Γ$ , the space $V = R^{S}$ with the Coxeter form $B$ , the cosine numbers $c (s, t)$ of the statement, and the hypothesis that $Γ$ is connected and $B$ is positive definite.

[F1]

In the diagram $Γ$ , distinct vertices $s \neq t$ are joined by an edge exactly when $m (s, t) \geq 3$ , and the neighbours of $s$ are $N (s) = {t \in S : t \neq s, m (s, t) \geq 3}$ ; the components of $Γ$ partition $S$ , and a cycle of $Γ$ is a cycle of the underlying simple graph (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 $m (s, t) = \infty$ ; in particular $C$ is a symmetric matrix and $B (u, w) = \sum_{s, t \in S} u (s) w (t) B (e_{s}, e_{t})$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms).

[F3]

$B$ positive definite means $B (u, u) > 0$ for every $u \neq 0$ (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F4]

The addition formulas and the Pythagorean identity hold for all reals: $\cos (x + y) = \cos x \cos y - \sin x \sin y$ , $\cos^{2} x + \sin^{2} x = 1$ , and $\cos$ is even (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine).

[F5]

$\cos (π / 2) = 0$ , $\sin (π / 2) = 1$ and $\cos π = - 1$ (Quarter-turn values and shifts by pi/2 and pi).

[F6]

For every real $θ$ one has $T_{5} (\cos θ) = \cos (5 θ)$ , and the Chebyshev polynomials of the first kind satisfy $T_{0} = 1$ , $T_{1} = t$ and $T_{n + 2} = 2 t T_{n + 1} - T_{n}$ ( $T_{n} (\cos θ) = \cos (n θ)$ 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).

[F7]

Cosine is strictly decreasing on $[0, π]$ , $π / 2$ is the smallest positive zero of cosine, and $\cos$ and $\sin$ are defined by their power series (Signs, monotonicity intervals, and ranges of sine and cosine, Pi as twice the smallest positive zero of cosine, Sine and cosine defined by their real power series).

[F8]

Every $a \geq 0$ has a unique $\sqrt{a} \geq 0$ with ${\sqrt{a}}^{2} = a$ , and squaring is strictly increasing on the nonnegative reals (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Squaring is monotone on the nonnegatives).

[F9]

If $W$ is a subspace of a finite-dimensional inner product space $V$ , then $V = W \oplus W^{⊥}$ , the orthogonal projection $P_{W}$ is the map with $v - P_{W} v \in W^{⊥}$ , and if $∥ u ∥^{2} = ⟨ u, u ⟩$ then $∥ v ∥^{2} = ∥ P_{W} v ∥^{2} + ∥ v - P_{W} v ∥^{2}$ with $P_{W} v \neq v$ precisely when $v \notin W$ (Real and complex inner-product spaces and their induced length, The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$ , For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$ ).

[F10]

For all vectors $u, v$ , $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality if and only if $u, v$ are linearly dependent (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F11]

The minor $M_{i j} (A)$ is the determinant of the matrix obtained by deleting row $i$ and column $j$ , the cofactor is $C_{i j} (A) = (- 1)^{i + j} M_{i j} (A)$ , and determinant is the unique normalized alternating column-multilinear function of the matrix (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, Laplace expansion computes the determinant along every row and every column over a commutative ring).

[F12]

The $e_{s}$ form a basis of $V$ , so vectors supported on disjoint subsets of $S$ are linearly independent unless one of them is $0$ , $s p a n$ and subspaces are the published notions, and $∣ N (s) ∣$ is a finite cardinality (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Linear subspace of a vector space, The cardinality $∣ A ∣$ of a finite set).

[F13]

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

Proof

technique · direct, by explicit witnesses and two determinant/inequality computations

1.1F4F5F6F7F8algebra

(Trigonometric values and comparisons.) From [F4] one gets the double-angle formula $\cos 2 x = 2 \cos^{2} x - 1$ and the triple-angle formula $\cos 3 x = 4 \cos^{3} x - 3 \cos x$ for every real $x$ . Cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0 > \cos π = - 1$ and is positive on $[0, π / 2)$ [F5, F7]: (i) $\cos (π / 4) = \sqrt{2} / 2$ because $\cos^{2} (π / 4) = \frac{1 + \cos (π / 2)}{2} = \frac{1}{2}$ and $\cos (π / 4) > 0$ [F8]; (ii) writing $c_{3} = \cos (π / 3)$ , the triple-angle formula at $x = π / 3$ gives $4 c_{3}^{3} - 3 c_{3} + 1 = 0 = (c_{3} + 1) (2 c_{3} - 1)^{2}$ , and $c_{3} > 0$ because $0 < π / 3 < π / 2$ forces $c_{3} = 1 / 2$ ; (iii) $\cos (π / 6) = \sqrt{3} / 2$ because $\cos^{2} (π / 6) = \frac{1 + \cos (π / 3)}{2} = \frac{3}{4}$ and $\cos (π / 6) > 0$ [F8]; (iv) since $\cos (π / m)$ is strictly increasing in $m \in {2, 3, 4, \dots}$ : for $m \geq 3$ one has $c (s, t) = \cos (π / m) \geq 1 / 2$ , for $m \geq 4$ one has $c (s, t) \geq \sqrt{2} / 2$ , and for $m \geq 6$ one has $c (s, t) \geq \cos (π / 6) = \sqrt{3} / 2$ . Also $\cos (π / 5) = (1 + \sqrt{5}) / 4$ : with $c_{5} = \cos (π / 5)$ , iterating the recurrence of [F6] gives $T_{2} = 2 t^{2} - 1$ , $T_{3} = 4 t^{3} - 3 t$ , $T_{4} = 8 t^{4} - 8 t^{2} + 1$ and $T_{5} = 16 t^{5} - 20 t^{3} + 5 t$ , so $T_{5} (c_{5}) = \cos (5 \cdot π / 5) = \cos π = - 1$ [F5, F6], that is $16 c_{5}^{5} - 20 c_{5}^{3} + 5 c_{5} + 1 = (c_{5} + 1) (4 c_{5}^{2} - 2 c_{5} - 1)^{2} = 0$ by expansion; since $0 < π / 5 < π / 2$ gives $0 < c_{5} < 1$ [F5, F7], one has $c_{5} \neq - 1$ and $4 c_{5}^{2} - 2 c_{5} - 1 = 0$ , i.e. $(c_{5} - \frac{1}{4})^{2} = \frac{5}{16}$ ; by uniqueness of the nonnegative square root $c_{5} - \frac{1}{4} = \pm \frac{\sqrt{5}}{4}$ , and $c_{5} > 0$ excludes the negative alternative, which is $< 0$ because $\sqrt{5} > 1$ by $1 < 5$ and [F8]; hence $c_{5} = (1 + \sqrt{5}) / 4$ and $4 \cos^{2} (π / 5) = (3 + \sqrt{5}) / 2$ satisfies $5 / 2 < 4 \cos^{2} (π / 5) < 3$ because $2 < \sqrt{5} < 3$ [F8].

1.2F1F2F3F12algebra

(Witness principle and form comparison.) By [F3], $B$ positive definite means $B (u, u) > 0$ for every nonzero $u$ ; a nonzero $u \in V_{T} \subseteq V$ with $B (u, u) \leq 0$ therefore contradicts positive definiteness, which is the first assertion of (1). For the comparison, let $u = \sum_{s \in T} u_{s} e_{s}$ have all $u_{s} \geq 0$ and let $Γ_{0}$ be a labelled graph on $T$ whose labels are at most the corresponding labels of $Γ_{T}$ ; for distinct $s, t$ the comparison of labels gives $c_{0} (s, t) \leq c_{T} (s, t) \leq 1$ (with value $0$ exactly for the non-edges and $c_{0} \geq 0$ throughout), so $B_{T} (u, u) - B_{0} (u, u) = - 2 \sum_{s < t} (c_{T} (s, t) - c_{0} (s, t)) u_{s} u_{t} \leq 0,$ each term being $\leq 0$ ; hence $B_{T} (u, u) \leq B_{0} (u, u)$ , and if $B_{0} (u, u) \leq 0$ then $B (u, u) = B_{T} (u, u) \leq 0$ with $u \neq 0$ , excluding positive definiteness.

1.3F2F11algebra

(Path determinant recursion.) Let $Γ$ be a path with vertices $s_{1}, \dots, s_{n}$ and $m_{k} = m (s_{k}, s_{k + 1})$ , and let $C$ be the cosine matrix $C = (B (e_{s}, e_{t}))$ . Its leading $k \times k$ submatrix $C_{k}$ has diagonal entries $1$ , sub- and super-diagonal entries $- c (s_{k}, s_{k + 1}) = - \cos (π / m_{k})$ , and all other entries $0$ . Put $d_{k} = \det C_{k}$ , with $d_{0} = 1$ and $d_{1} = \det (1) = 1$ . Expanding $\det C_{k}$ along its last row $(0, \dots, 0, - \cos (π / m_{k - 1}), 1)$ for $k \geq 2$ : the last entry contributes $\det C_{k - 1}$ , and writing $c = \cos (π / m_{k - 1})$ , the other entry contributes $(- c) (- 1)^{2 k - 1} \det M$ , where $M$ is obtained by deleting row $k$ and column $k - 1$ . Its last column has the sole nonzero entry $- c$ in its last row, so expansion gives $\det M = - c \det C_{k - 2}$ ; this contribution is therefore $- c^{2} d_{k - 2}$ , giving $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2} (k \geq 2) .$

2.1F2F3F9F10F12step 1.1algebra

(Chain inequality.) Let $Γ$ be a path, all of whose labels are $3$ except one edge ${s_{i}, s_{i + 1}}$ labelled $m \geq 4$ , with $1 \leq i \leq j = n - i$ ; by [F3] and [F9] the form $B$ makes $V$ a finite-dimensional inner product space. Put $u = \sum_{k = 1}^{i} k e_{s_{k}}$ and $v = \sum_{k = 1}^{j} k e_{s_{n + 1 - k}}$ , so that $u$ is supported on ${s_{1}, \dots, s_{i}}$ , $v$ on ${s_{i + 1}, \dots, s_{n}}$ , and the coefficients of the two endpoints $s_{i}, s_{i + 1}$ of the large edge are $i$ and $j$ . All internal edges of the two chains have label $3$ , and $B (e_{s_{k}}, e_{s_{k + 1}}) = - \frac{1}{2}$ by step 1.1, so the diagonal terms and the two symmetric terms for each internal edge give $B (u, u) = \sum_{k = 1}^{i} k^{2} - \sum_{k = 1}^{i - 1} k (k + 1) = i^{2} - \sum_{k = 1}^{i - 1} k = \frac{i (i + 1)}{2}, B (v, v) = \frac{j (j + 1)}{2},$ with the same computation for $v$ , while every mixed pair contributes $0$ except ${s_{i}, s_{i + 1}}$ , giving $B (u, v) = - i j \cos (π / m)$ . Since $u, v$ are nonzero and supported on disjoint subsets of the basis $(e_{s})_{s \in S}$ , they are linearly independent [F12], so [F10] is strict, $B (u, v)^{2} < B (u, u) B (v, v)$ , that is $i^{2} j^{2} \cos^{2} (π / m) < \frac{i (i + 1)}{2} \cdot \frac{j (j + 1)}{2}$ , which gives $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m)$ .

2.2F1F2step 1.1step 1.2algebra

(No cycles.) Let $s_{1}, \dots, s_{r}$ $(r \geq 3)$ be distinct vertices whose consecutive pairs are edges, and put $u = e_{s_{1}} + \dots + e_{s_{r}} \neq 0$ , a vector with all coordinates $\geq 0$ in the basis $(e_{s})$ . Every consecutive pair 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 [1.1], and every other pair contributes $2 B \leq 0$ (the value $0$ for non-edges, $\leq - 1$ for the remaining edges), so $B (u, u) \leq r - r = 0$ ; by the witness principle [1.2] this contradicts positive definiteness of $B$ . Hence $Γ$ contains no cycle. Connectedness gives a path between any two vertices, and two different simple paths would give a cycle between their first divergence and subsequent reunion; thus that path is unique. Removing a vertex separates its neighbours into distinct components, since a path between two neighbours avoiding the removed vertex would create a cycle. Finally, a finite connected acyclic graph of maximum degree at most $2$ is a path: a longest simple path has no extension at either end, and an additional vertex would have a path to it meeting an internal vertex (creating degree at least $3$ ) or an endpoint (extending it). These elementary consequences will be used below.

2.3step 1.3F13algebra

(Path with all labels $3$ .) For the path with every label $3$ the recursion of [1.3] reads $d_{k} = d_{k - 1} - \frac{1}{4} d_{k - 2}$ ; by induction on $k$ [F13] one has $d_{k} = (k + 1) / 2^{k}$ for all $k \geq 0$ , since $d_{0} = 1$ , $d_{1} = 1$ and $\frac{k}{2^{k - 1}} - \frac{1}{4} \cdot \frac{k - 1}{2^{k - 2}} = \frac{2 k - (k - 1)}{2^{k}} = \frac{k + 1}{2^{k}}$ . In particular $d_{k} > 0$ , and multiplying the $n \times n$ matrix $C$ by $2$ scales its determinant by $2^{n}$ , so $\det (2 C) = 2^{n} d_{n} = n + 1$ .

3.1F2step 1.1step 2.1algebra

(Integer consequences of the chain inequality.) Write $c = \cos (π / m)$ with $m \geq 4$ , where $m = \infty$ means $c = 1$ [F2]. If $i \geq 2$ , then $j \geq i \geq 2$ and $(i + 1) (j + 1) = i j + i + j + 1 \leq 3 i j$ , because $3 i j - (i + 1) (j + 1) = 2 i j - i - j - 1 = (i - 1) (j - 1) + i j - 2 \geq 1 + 4 - 2 = 3 > 0$ . If $m \geq 6$ , including $m = \infty$ , then $4 c^{2} \geq 4 \cos^{2} (π / 6) = 3$ by [1.1], so $(i + 1) (j + 1) \leq 3 i j \leq 4 c^{2} i j$ and the inequality $(i + 1) (j + 1) > 4 c^{2} i j$ forces $i = 1$ ; then it reads $2 (j + 1) > 4 c^{2} j$ , i.e. $1 > j (2 c^{2} - 1) \geq j / 2$ because $2 c^{2} \geq 3 / 2$ , and hence $j = 1$ (for $m = \infty$ it reads $4 > 4$ , false, so no infinite label occurs). If $m = 5$ , then $4 c^{2} = (3 + \sqrt{5}) / 2$ and $4 c^{2} - 1 = \frac{1 + \sqrt{5}}{2}$ by [1.1]; for $i \geq 2$ the inequality fails because $4 c^{2} i j - (i + 1) (j + 1) = (4 c^{2} - 1) i j - i - j - 1 \geq \frac{1 + \sqrt{5}}{2} \cdot 2 j - 2 j - 1 = (\sqrt{5} - 1) j - 1 > 0$ for $j \geq 2$ , so $i = 1$ , and then $2 (j + 1) > 4 c^{2} j$ reads $j < 2 / (4 c^{2} - 2) = 4 / (\sqrt{5} - 1) = \sqrt{5} + 1$ , so $j \leq 3$ . If $m = 4$ , then $4 c^{2} = 2$ by [1.1] and the inequality reads $i + j + 1 > i j$ : for $i = 1$ this holds for every $j \geq 1$ , for $i = 2$ it reads $j < 3$ , so $j = 2$ , and for $i \geq 3$ it fails since $i j - i - j - 1 = (i - 1) (j - 1) - 2 \geq 2 > 0$ .

3.2F1F2step 1.1step 1.2step 2.2algebra

(Two large labels are impossible.) Suppose two distinct edges of $Γ$ carry labels $\geq 4$ . Since [2.2] shows that $Γ$ is acyclic, the two edges are joined by a unique simple path; write its vertices as $x_{0}, x_{1}, \dots, x_{r}$ so that $x_{0}, x_{1}$ and $x_{r - 1}, x_{r}$ are the two large edges, $r \geq 2$ . Put $u = e_{x_{0}} + \sqrt{2} \sum_{h = 1}^{r - 1} e_{x_{h}} + e_{x_{r}}$ (all coordinates $\geq 0$ ). The diagonal contribution is $1 + 2 (r - 1) + 1 = 2 r$ . The two large edges contribute $2 (- \cos (π / m)) \sqrt{2} \leq - 2 \sqrt{2} \cdot \frac{\sqrt{2}}{2} = - 2$ each, by [1.1] and $m \geq 4$ ; each of the $r - 2$ remaining path edges contributes $2 (- \cos (π / m_{h})) \cdot 2 \leq - 2$ ; and all other pairs contribute $\leq 0$ . Hence $B (u, u) \leq 2 r - 4 - 2 (r - 2) = 0$ , so by the witness principle [1.2] $B$ is not positive definite, a contradiction. Therefore at most one edge of $Γ$ has label $\geq 4$ .

3.3F1F2F3F9F12step 2.2algebra

(The neighbour inequality.) Fix $s \in S$ and let $t \neq t^{'}$ be neighbours of $s$ . If $t, t^{'}$ were joined by an edge, then $s, t, t^{'}$ would be three distinct vertices whose consecutive pairs are edges, i.e. a cycle, which [2.2] excludes; so $B (e_{t}, e_{t^{'}}) = 0$ for distinct neighbours. By [F2] each $B (e_{t}, e_{t}) = 1$ , and the restriction of $B$ to the subspace $P = s p a n {e_{t} : t \in N (s)}$ is an inner product, because $B$ is positive definite on $V$ [F3] and restricts to $V \times V$ . Let $e_{s} = p + q$ with $p \in P$ , $q \in P^{⊥}$ be the orthogonal decomposition of [F9]; then $p = \sum_{t \in N (s)} B (e_{s}, e_{t}) e_{t} = - \sum_{t \in N (s)} c (s, t) e_{t}$ and $∥ p ∥^{2} = \sum_{t \in N (s)} c (s, t)^{2}$ . Since $e_{s}$ together with the $e_{t}$ , $t \in N (s)$ , consists of distinct vectors of the basis $(e_{s})_{s \in S}$ , the vector $e_{s}$ is not in $P$ [F12], so $q \neq 0$ and $\sum_{t \in N (s)} c (s, t)^{2} = ∥ p ∥^{2} = ∥ e_{s} ∥^{2} - ∥ q ∥^{2} < ∥ e_{s} ∥^{2} = B (e_{s}, e_{s}) = 1.$

3.4F2F9F12step 2.1step 2.2algebra

(Three-arm inequality.) Suppose $v$ has degree $3$ and every edge of $Γ$ has label $3$ , and the three components of $Γ - v$ are paths with $p, q, r \geq 1$ vertices. Weight each arm from its far end toward $v$ : if the arm of $v$ with $p$ vertices is $a_{1} - a_{2} - \dots - a_{p}$ with $a_{p}$ adjacent to $v$ , put $u = \sum_{h = 1}^{p} h e_{a_{h}}$ , and define $w, z$ for the other two arms cyclically. Then $B (u, u) = p (p + 1) / 2$ by the same computation as in [2.1], $B (e_{v}, u) = p \cdot (- \frac{1}{2}) = - \frac{p}{2}$ since only the pair ${v, a_{p}}$ contributes, and $u, w, z$ are pairwise orthogonal because their supports lie in different components of $Γ - v$ , so no edge joins two of them [2.2]. Moreover $e_{v} \notin s p a n {u, w, z}$ , as $u, w, z$ involve only basis vectors different from $e_{v}$ ; hence the orthogonal decomposition of $e_{v}$ with respect to that subspace is strict and [F9] gives $1 = ∥ e_{v} ∥^{2} > \frac{B (e_{v}, u)^{2}}{B (u, u)} + \frac{B (e_{v}, w)^{2}}{B (w, w)} + \frac{B (e_{v}, z)^{2}}{B (z, z)} = \sum_{p} \frac{p^{2} / 4}{p (p + 1) / 2} = \frac{1}{2} \sum_{p} \frac{p}{p + 1},$ i.e. $\sum 1 / (p + 1) > 1$ .

4.1F2step 1.1step 3.3algebra

(Local consequences.) Fix $s \in S$ and use $\sum_{t \in N (s)} c (s, t)^{2} < 1$ [3.3] together with $c (s, t) \geq \frac{1}{2}$ for every neighbour and $c \leq 1$ . A neighbour with $m (s, t) = \infty$ would have $c (s, t) = 1$ and hence a sum $\geq 1$ , so no label is $\infty$ ; four neighbours would give a sum $\geq 4 \cdot \frac{1}{4} = 1$ , so $∣ N (s) ∣ \leq 3$ ; if there are three neighbours and one of their edges has label $\geq 4$ , that edge contributes at least $\frac{1}{2}$ while the other two each contribute at least $\frac{1}{4}$ by step 1.1, giving a sum $\geq 1$ , a contradiction; hence all three edges have label $3$ ; and two neighbours with labels $m_{1} \leq m_{2}$ and $c_{j} = \cos (π / m_{j})$ satisfy $c_{1}^{2} + c_{2}^{2} < 1$ : if $m_{1} \geq 4$ then $c_{1}^{2} \geq \frac{1}{2}$ and $c_{2}^{2} \geq c_{1}^{2} \geq \frac{1}{2}$ (cosine increases with $m$ [1.1]), a contradiction, so $m_{1} = 3$ and then $c_{2}^{2} < \frac{3}{4}$ ; since $\cos^{2} (π / 6) = \frac{3}{4}$ and cosine increases with $m$ , $m_{2} \geq 6$ would give $c_{2}^{2} \geq \frac{3}{4}$ , so $m_{2} \leq 5$ . In particular $(m_{1}, m_{2}) = (4, 4)$ gives $c_{1}^{2} + c_{2}^{2} = 1$ and $(3, 6)$ gives $\frac{1}{4} + \frac{3}{4} = 1$ , both excluded.

4.2step 3.4algebra

(Integer consequences of the three-arm inequality.) Let $1 \leq p \leq q \leq r$ with $1 / (p + 1) + 1 / (q + 1) + 1 / (r + 1) > 1$ . If $p \geq 2$ , then $p + 1, q + 1, r + 1 \geq 3$ and the sum is at most $1$ , so $p = 1$ . Then $1 / (q + 1) + 1 / (r + 1) > \frac{1}{2}$ : if $q = 1$ this holds for every $r \geq 1$ ; if $q = 2$ it reads $1 / (r + 1) > \frac{1}{6}$ , i.e. $r < 5$ , so $r \in {2, 3, 4}$ ; and if $q \geq 3$ the sum is at most $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ , a contradiction. Hence, up to permutation, $(p, q, r) = (1, 1, r)$ for some $r \geq 1$ , or $(p, q, r) \in {(1, 2, 2), (1, 2, 3), (1, 2, 4)}$ .

5.1F1F2step 1.1step 1.2step 2.2step 4.1algebra

(At most one vertex of degree $3$ .) Suppose $s \neq t$ are two vertices of degree $\geq 3$ . By [4.1] both have degree exactly $3$ . By [2.2] the diagram is acyclic, so $s$ and $t$ are joined by a unique simple path $s = v_{0}, v_{1}, \dots, v_{L} = t$ with $L \geq 1$ , the remaining neighbours $a, a^{'}$ of $s$ and $b, b^{'}$ of $t$ lie outside this path, and all of $a, a^{'}, b, b^{'}$ are pairwise distinct (two of them equal would create a second $s$ - $t$ path, or a triangle when $L = 1$ ). Put $u = \sum_{h = 0}^{L} e_{v_{h}} + \frac{1}{2} (e_{a} + e_{a^{'}} + e_{b} + e_{b^{'}})$ , a nonzero vector with all coordinates $\geq 0$ : its diagonal contribution is $(L + 1) + 4 \cdot \frac{1}{4} = L + 2$ , the $L$ path edges contribute $2 (- c) \leq - 1$ each, the four pendant edges contribute $2 (- c) \cdot \frac{1}{2} = - c \leq - \frac{1}{2}$ each, and all other pairs contribute $\leq 0$ ; hence $B (u, u) \leq (L + 2) - L - 2 = 0$ , contradicting positive definiteness by the witness principle [1.2]. So at most one vertex of degree $3$ exists, and by [4.1] at most one of degree $\geq 3$ .

5.2F1F2step 1.1step 1.2step 2.2step 4.1algebra

(A degree- $3$ vertex excludes every large label.) Suppose $v$ has degree $3$ and some edge has label $m \geq 4$ . Removing $v$ from the acyclic graph of [2.2] leaves three components, and the large edge lies in one of them: write $x_{0} = v, x_{1}, \dots, x_{k}, y$ in order along a path, so that ${x_{k}, y}$ is the large edge (possibly $k = 0$ with $y$ a neighbour of $v$ ), and let $b, b^{'}$ be the neighbours of $v$ in the other two components. Then $b, b^{'}$ and $x_{0}, \dots, x_{k}, y$ are pairwise distinct except for the described edges, and $u = \sum_{h = 0}^{k} e_{x_{h}} + \cos (π / m) e_{y} + \frac{1}{2} (e_{b} + e_{b^{'}})$ is nonzero with all coordinates $\geq 0$ ; its diagonal contribution is $(k + 1) + c^{2} + \frac{1}{2}$ with $c = \cos (π / m)$ , the $k$ path edges $v = x_{0}, \dots, x_{k}$ contribute $2 (- c_{h}) \leq - 1$ each, the large edge contributes $- 2 c^{2}$ , the two edges at $v$ toward $b, b^{'}$ contribute $- c (v, b) \leq - \frac{1}{2}$ each, and all other pairs contribute $\leq 0$ . Hence $B (u, u) \leq (k + 1) + c^{2} + \frac{1}{2} - k - 2 c^{2} - 1 = \frac{1}{2} - c^{2} \leq 0$ because $m \geq 4$ gives $c \geq \frac{\sqrt{2}}{2}$ and $c^{2} \geq \frac{1}{2}$ by [1.1]; by the witness principle [1.2] this contradicts positive definiteness. So if a large label exists, no vertex of degree $3$ exists; combined with [4.1] every degree is $\leq 2$ and the acyclic connected $Γ$ is a path.

6.1step 2.1step 2.2step 2.3step 3.1step 3.2step 4.1step 4.2step 5.1step 5.2algebra∎

(Conclusion.) Let $Γ$ be connected and positive definite. By [2.2] it is acyclic, hence a tree. If $Γ$ has no vertex of degree $3$ , then by [4.1] all degrees are $\leq 2$ , so $Γ$ is a path: with all labels $3$ it is $A_{n}$ ( $n \geq 1$ ) [2.3]; otherwise by [3.2] exactly one edge has label $m \geq 4$ and [5.2] applies, and splitting the path at that edge into subpaths of $i \leq j$ vertices, the inequality of [2.1] and its case analysis [3.1] give $(i, j) = (1, 1)$ with $m \geq 6$ , giving the single edge $I_{2} (m)$ ; $(i, j) = (1, 1)$ with $m = 4$ or $m = 5$ , giving $I_{2} (4) = B_{2}$ and $I_{2} (5)$ ; $(i, j) = (1, j)$ with $j \geq 2$ and $m = 4$ , giving the paths with labels $3, \dots, 3, 4$ (type $B_{j + 1}$ , the label- $4$ edge at an end); $(i, j) = (1, 2)$ and $(1, 3)$ with $m = 5$ , giving the paths with labels $3, 5$ (type $H_{3}$ ) and $3, 3, 5$ (type $H_{4}$ ); and $(i, j) = (2, 2)$ with $m = 4$ , giving the path with labels $3, 4, 3$ (type $F_{4}$ ). If $Γ$ has a vertex of degree $3$ , it is unique by [5.1], its edges have label $3$ and no edge has label $\geq 4$ by [5.2] and [4.1], and the three arms have $p, q, r \geq 1$ vertices satisfying the inequality of [3.4]; by [4.2] they are $(1, 1, r)$ with $r \geq 1$ , giving the star with arms $1, 1, r$ , i.e. $D_{r + 3}$ ( $n = r + 3 \geq 4$ ), or $(1, 2, 2)$ , $(1, 2, 3)$ , $(1, 2, 4)$ , giving $E_{6}$ , $E_{7}$ , $E_{8}$ . Thus every connected positive definite diagram is one of $A_{n}$ ( $n \geq 1$ ), $B_{n}$ ( $n \geq 2$ ), $D_{n}$ ( $n \geq 4$ ), $E_{6}, E_{7}, E_{8}$ , $F_{4}$ , $H_{3}$ , $H_{4}$ , $I_{2} (m)$ ( $m \geq 3$ ) with the labels displayed in (7), which is the asserted list.

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

Disconnected diagrams, direct products, and comparison of invariant forms

Statement

Let $S$ be a finite set with Coxeter matrix $m$ , presented group $W$ and length function $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), with diagram $Γ$ (Coxeter diagrams: edges, labels, components and finite type), and let $V = R^{S}$ carry the Coxeter form $B$ with canonical reflection homomorphism $ρ : W \to G L (V)$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone). Let the connected components of $Γ$ have the nonempty pairwise disjoint vertex sets $S_{1}, \dots, S_{k}$ of $S$ ; put $W_{i} : = W_{S_{i}}$ and $V_{i} : = s p a n {e_{s} : s \in S_{i}}$ . Allow $k = 1$ when $Γ$ is connected and $k = 0$ when $S = \emptyset$ , with the empty product equal to the trivial group and the empty direct sum equal to ${0}$ .

(1) Direct product and length. The subgroups $W_{i}$ commute elementwise, $W_{i} \cap W_{j} = {1}$ for $i \neq j$ , and the multiplication map $μ : W_{1} \times \dots \times W_{k} \to W$ , $μ (w_{1}, \dots, w_{k}) = w_{1} \dots w_{k}$ , is an isomorphism of groups (Group isomorphisms, automorphisms and the set $Aut (G)$ , The external direct product $G \times H$ with componentwise multiplication). Moreover $ℓ (w_{1} \dots w_{k}) = ℓ (w_{1}) + \dots + ℓ (w_{k})$ for all $w_{i} \in W_{i}$ , the lengths on the right being those of the factors, which agree with the restriction of $ℓ$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2)).

(2) Orthogonal decomposition. $B (v_{i}, v_{j}) = 0$ for all $v_{i} \in V_{i}$ , $v_{j} \in V_{j}$ , $i \neq j$ ; hence $V = V_{1} \oplus \dots \oplus V_{k}$ is a $B$ -orthogonal direct sum, each $ρ (W_{i})$ preserves $V_{i}$ and fixes every $V_{j}$ ( $j \neq i$ ) pointwise, and with $B_{i} : = B ∣_{V_{i} \times V_{i}}$ the form is $B = B_{1} \oplus \dots \oplus B_{k}$ .

(3) Invariant forms. Let $β$ be a symmetric bilinear form on $V$ invariant under $ρ (W)$ , i.e. $β (ρ (w) u, ρ (w) v) = β (u, v)$ for all $w \in W$ and $u, v \in V$ .

(i) For every $s \in S$ one has $β (e_{s}, \cdot) = λ_{s} B (e_{s}, \cdot)$ with $λ_{s} : = β (e_{s}, e_{s})$ ; in particular $β (e_{s}, e_{t}) = λ_{s} B (e_{s}, e_{t})$ for all $s, t \in S$ .

(ii) $λ_{s} = λ_{t}$ whenever $s$ and $t$ lie in the same component of $Γ$ . Consequently there are $λ_{1}, \dots, λ_{k} \in R$ with $β ∣_{V_{i} \times V_{i}} = λ_{i} B_{i}$ for every $i$ , that is, $β = λ_{1} B_{1} \oplus \dots \oplus λ_{k} B_{k}$ ; if $Γ$ is connected then $β = λ B$ for a single $λ \in R$ .

(iii) If in addition $β$ is positive definite (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form), then $λ_{i} = β (e_{s}, e_{s}) > 0$ for every $i$ and every $s \in S_{i}$ , and $B$ is positive definite.

(4) Finite groups have positive definite form. If $W$ is finite then $B$ is positive definite.

Facts & Assumptions

Given: A finite set $S$ with Coxeter matrix $m$ , the presented group $W$ with length $ℓ$ , the diagram $Γ$ with components $S_{1}, \dots, S_{k}$ , the space $V = R^{S}$ with the Coxeter form $B$ and the canonical reflection homomorphism $ρ$ ; and, when a form $β$ is mentioned, a symmetric bilinear form $β$ invariant under $ρ (W)$ .

[F1]

The relators of the presentation are $s^{2}$ ( $s \in S$ ) and $(s t)^{m (s, t)}$ ( $s \neq t$ , $m (s, t) < \infty$ ); every map $S \to G$ into a group sending these relators to $1$ extends uniquely to a homomorphism $W \to G$ . For $J \subseteq S$ , $W_{J} = ⟨ J ⟩$ is the group presented by the restricted matrix $m ∣_{J \times J}$ under the canonical map, and $W_{J} = {w : S (w) \subseteq J}$ , where $S (w) \subseteq J$ means that $w$ has a reduced expression with all letters in $J$ ; hence $W_{I} \cap W_{J} = W_{I \cap J}$ and $W_{\emptyset} = {1}$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification).

[F2]

The components of $Γ$ partition $S$ and are connected; distinct components are joined by no edge, so for $s \in S_{i}$ , $t \in S_{j}$ with $i \neq j$ one has $m (s, t) = 2$ , and within a component two vertices are joined exactly when $m (s, t) \geq 3$ (Coxeter diagrams: edges, labels, components and finite type).

[F3]

$B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ , while $B (e_{s}, e_{t}) = - 1$ when $m (s, t) = \infty$ ; also $\cos (π / 2) = 0$ , and $\cos (π / m) > 0$ for finite $m \geq 3$ by strict decrease of cosine on $[0, π]$ (Signs, monotonicity intervals, and ranges of sine and cosine). For $a \in V$ with $B (a, a) \neq 0$ the reflection $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ is linear, $r_{a}^{2} = {i d}_{V}$ , $r_{a} (a) = - a$ , $\ker B (-, a)$ is a hyperplane fixed pointwise by $r_{a}$ , and $B (r_{a} u, r_{a} w) = B (u, w)$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, Quarter-turn values and shifts by pi/2 and pi).

[F4]

$ρ : W \to G L (V)$ is a group homomorphism with $ρ (s) = r_{e_{s}}$ for every $s \in S$ , and $B (ρ (w) u, ρ (w) w^{'}) = B (u, w^{'})$ for all $w \in W$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F5]

The external direct product $W_{1} \times \dots \times W_{k}$ is a group under componentwise operations; a homomorphism on each factor with pairwise commuting images defines a homomorphism of the product, and a bijective homomorphism is an isomorphism. For $J \subseteq S$ , the finite words in $J$ form a subgroup (inverses reverse words because $s^{- 1} = s$ ) containing $J$ and contained in every subgroup containing $J$ ; thus they constitute $W_{J}$ (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, Monoid homomorphism and group homomorphism, Group isomorphisms, automorphisms and the set $Aut (G)$ , The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, Group and abelian group, Internal direct products of finitely many normal subgroups).

[F6]

The $e_{s}$ $(s \in S)$ form a basis of $V$ , so every $v \in V$ is a unique finite linear combination $\sum_{s} v (s) e_{s}$ ; the span of a subset and a subspace are the published notions; the sum $V_{1} + \dots + V_{k}$ of subspaces is direct when every vector has a unique decomposition, and a bilinear form on a direct sum is the orthogonal sum of its restrictions when the summands are pairwise orthogonal for it; a linear map with a nonzero functional $B (-, a)$ has kernel ${v : B (v, a) = 0}$ (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Linear subspace of a vector space, Internal direct sum $V = ⨁_{i < n} U_{i}$ : the sum is everything and each summand meets the sum of the others only in $0_{V}$ , The sum $U + W$ of two linear subspaces and the sum $\sum_{i < n} U_{i}$ of a finite family, Linear map between vector spaces over the same field, Kernel and image of a linear map, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis, Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V \to V^{*}$ ).

[F7]

A symmetric bilinear form is positive definite when its quadratic form is $> 0$ on every nonzero vector; a positive multiple of a positive definite form is positive definite, as is its restriction to a subspace (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F8]

The standard inner product $β_{0} (u, v) = \sum_{s \in S} u (s) v (s)$ is a symmetric positive definite bilinear form on $V$ , every $u \neq 0$ has $β_{0} (u, u) > 0$ , finite sums may be reindexed by a bijection of the finite index set (enumerate that set; adjacent swaps preserve the sum by associativity and commutativity, and every finite permutation is obtained by such swaps), and a finite set has a cardinality (Real and complex inner-product spaces and their induced length, Finite sums and finite products, by recursion, Laws of finite sums and finite products, The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct; universal properties for the group statements and a proportionality argument for the forms

1.1F1F2F5algebra

(Commuting factors and trivial intersections.) If $S = \emptyset$ , all four clauses hold: $W = {1}$ , $V = {0}$ , the product and sums are empty, and positive definiteness is vacuous. Hence assume $S \neq \emptyset$ for the remaining argument. Let $s \in S_{i}$ , $t \in S_{j}$ with $i \neq j$ ; by [F2] $m (s, t) = 2$ , so $(s t)^{2}$ is a relator and $s t = t s$ in $W$ by [F1]; since the $s \in S_{i}$ generate $W_{i}$ [F1, F5], the subgroups $W_{i}, W_{j}$ commute elementwise. For the intersection, $W_{i} \cap W_{j} = W_{S_{i}} \cap W_{S_{j}} = W_{S_{i} \cap S_{j}} = W_{\emptyset} = {1}$ by the support description of [F1], because $S_{i} \cap S_{j} = \emptyset$ [F2].

1.2F2F3F4F5F6algebra

(Orthogonal decomposition.) For $s \in S_{i}$ and $t \in S_{j}$ with $i \neq j$ , $m (s, t) = 2$ [F2] and hence $B (e_{s}, e_{t}) = - \cos (π / 2) = 0$ by [F3]; by bilinearity [F6] this gives $B (V_{i}, V_{j}) = 0$ . Since the $e_{s}$ form a basis of $V$ and the $S_{i}$ partition it, grouping the unique basis expansion by its supports $S_{i}$ gives a unique decomposition into vectors of the $V_{i}$ , so $V = V_{1} \oplus \dots \oplus V_{k}$ is a $B$ -orthogonal direct sum and $B = B_{1} \oplus \dots \oplus B_{k}$ with $B_{i} = B ∣_{V_{i} \times V_{i}}$ [F6]. For the action, the reflection formula gives $r_{e_{s}} e_{t} = e_{t} - 2 B (e_{t}, e_{s}) e_{s}$ : for $t \in S_{i}$ this lies in $V_{i}$ , while for $t \in S_{j}$ , $j \neq i$ , it equals $e_{t}$ because $B (e_{t}, e_{s}) = B (e_{s}, e_{t}) = 0$ ; hence $ρ (s)$ preserves $V_{i}$ and fixes each $V_{j}$ with $j \neq i$ pointwise, and the same holds for every $ρ (w)$ with $w \in W_{i}$ since these are products of such generators [F4, F5].

1.3F3F4F6algebra

(Proportionality on one generator.) Fix $s \in S$ and put $H = {v : B (v, e_{s}) = 0}$ ; by [F3] $H$ is a hyperplane fixed pointwise by $r_{e_{s}} = ρ (s)$ and $r_{e_{s}} e_{s} = - e_{s}$ . For $v \in H$ , invariance of $β$ under $ρ (s)$ gives $β (e_{s}, v) = β (ρ (s) e_{s}, ρ (s) v) = β (- e_{s}, v) = - β (e_{s}, v)$ , so $β (e_{s}, v) = 0$ ; thus the linear functional $β (e_{s}, \cdot)$ vanishes on $H$ , as does $B (e_{s}, \cdot)$ , which is nonzero because $B (e_{s}, e_{s}) = 1$ [F3]. Any linear functional $ψ$ vanishing on $H = \ker φ$ is a multiple of the nonzero functional $φ$ : if $φ (v_{0}) \neq 0$ , then $u - (φ (u) / φ (v_{0})) v_{0} \in H$ for every $u$ , so $ψ (u) = (ψ (v_{0}) / φ (v_{0})) φ (u)$ . Hence $β (e_{s}, \cdot) = λ_{s} B (e_{s}, \cdot)$ with $λ_{s} : = β (e_{s}, e_{s})$ , and evaluating at $e_{t}$ gives $β (e_{s}, e_{t}) = λ_{s} B (e_{s}, e_{t})$ for all $t$ .

2.1F1F4F5step 1.1algebra

(The multiplication map is an isomorphism.) Every relator of $(S, m)$ is mapped to $1$ by the assignment $s \mapsto (1, \dots, s, \dots, 1) \in W_{1} \times \dots \times W_{k}$ placing $s$ in the factor $W_{i}$ with $s \in S_{i}$ : the relators $s^{2}$ and $(s t)^{m (s, t)}$ with $s, t$ in one component hold in that factor because they hold in $W$ , and for $s \in S_{i}$ , $t \in S_{j}$ with $i \neq j$ the two images have disjoint supports, hence commute and are involutions, so the image of $(s t)^{2}$ is $1$ ; by the universal property [F1] there is a homomorphism $φ : W \to W_{1} \times \dots \times W_{k}$ with $φ (s) = (1, \dots, s, \dots, 1)$ . Conversely the inclusions $W_{i} \to W$ are homomorphisms [F1] with pairwise commuting images by step 1.1, so $(w_{1}, \dots, w_{k}) \mapsto w_{1} \dots w_{k}$ is a homomorphism $ψ : W_{1} \times \dots \times W_{k} \to W$ [F5]. The two are mutually inverse: $ψ φ$ and the identity of $W$ are homomorphisms agreeing on the generating set $S$ , and $φ ψ$ and the identity of $W_{1} \times \dots \times W_{k}$ are homomorphisms agreeing on each coordinate generating set $W_{i}$ (a generator $s \in S_{i}$ of the $i$ -th factor is sent by $φ ψ$ to $φ (s) = (1, \dots, s, \dots, 1)$ ); hence $μ = ψ$ is an isomorphism [F5].

2.2F2F3F6step 1.3algebra

(The scalars are constant on components.) For $s \neq t$ in the same component, step 1.3 gives $β (e_{s}, e_{t}) = λ_{s} B (e_{s}, e_{t})$ and, by symmetry of $β$ and $B$ , $β (e_{s}, e_{t}) = β (e_{t}, e_{s}) = λ_{t} B (e_{t}, e_{s}) = λ_{t} B (e_{s}, e_{t})$ ; since $s, t$ lie in one component and are joined by a path, it suffices to treat adjacent pairs, where $B (e_{s}, e_{t}) \neq 0$ : indeed for finite labels the cosine is positive when $m (s, t) \geq 3$ , while an infinite label has $B (e_{s}, e_{t}) = - 1$ [F3]; thus $B (e_{s}, e_{t}) = 0$ forces $m (s, t) = 2$ , i.e. no edge [F2]. For such a pair $(λ_{s} - λ_{t}) B (e_{s}, e_{t}) = 0$ gives $λ_{s} = λ_{t}$ , and equality propagates along the edges of the connected component [F2], so there is $λ_{i}$ with $λ_{s} = λ_{i}$ for all $s \in S_{i}$ . Evaluating $β$ on pairs of basis vectors of $V_{i}$ step 1.3 then gives $β ∣_{V_{i} \times V_{i}} = λ_{i} B_{i}$ for every $i$ , that is, $β = λ_{1} B_{1} \oplus \dots \oplus λ_{k} B_{k}$ [F6]; if $k = 1$ this is $β = λ B$ .

3.1F1F5step 1.1step 2.1algebra

(Length additivity.) Let $w_{i} \in W_{i}$ . Choosing a reduced expression of each $w_{i}$ , concatenation represents $w_{1} \dots w_{k}$ with $\sum_{i} ℓ (w_{i})$ letters, so $ℓ (w_{1} \dots w_{k}) \leq \sum_{i} ℓ (w_{i})$ . For the reverse inequality let $w_{1} \dots w_{k} = s_{1} s_{2} \dots s_{ℓ}$ be a reduced expression of length $ℓ = ℓ (w_{1} \dots w_{k})$ , with letters $s_{j} \in S$ ; letters lying in distinct components commute in $W$ by step 1.1, so we may reorder the $s_{j}$ within this word so that the letters of each $S_{i}$ become consecutive (the value in $W$ is unchanged), obtaining $w_{1} \dots w_{k} = w_{1}^{'} \dots w_{k}^{'}$ with $w_{i}^{'}$ a product of $n_{i}$ letters from $S_{i}$ and $\sum_{i} n_{i} = ℓ$ . By the isomorphism of step 2.1 the projection $W \to W_{i}$ is the restriction of the inverse map and is a homomorphism [F5], so it sends $w_{1} \dots w_{k}$ to $w_{i}$ and $w_{1}^{'} \dots w_{k}^{'}$ to $w_{i}^{'}$ ; hence $w_{i}^{'} = w_{i}$ and $ℓ (w_{i}) \leq n_{i}$ for each $i$ ; summing, $\sum_{i} ℓ (w_{i}) \leq ℓ$ .

3.2F6F7step 2.2algebra

(Positive definite invariant forms give positive definite $B$ .) Assume $β$ positive definite. By step 2.2 $β = λ_{1} B_{1} \oplus \dots \oplus λ_{k} B_{k}$ , and each $λ_{i} = β (e_{s}, e_{s}) > 0$ for $s \in S_{i}$ because $e_{s} \neq 0$ and $β$ is positive definite [F7]. Hence $B_{i} = λ_{i}^{- 1} β ∣_{V_{i} \times V_{i}}$ is a positive multiple of the restriction of a positive definite form and is positive definite [F7]; a $B$ -orthogonal direct sum of positive definite forms is positive definite, since a nonzero vector has some nonzero component $v_{i}$ and $B (v, v) \geq B_{i} (v_{i}, v_{i}) > 0$ [F6, F7]. Thus $B = B_{1} \oplus \dots \oplus B_{k}$ is positive definite, which is (3)(iii).

4.1F4F8step 3.2algebra∎

(Finite $W$ has positive definite $B$ .) Assume $W$ finite and define $β (u, v) : = \sum_{w \in W} β_{0} (ρ (w) u, ρ (w) v)$ , a finite sum over the finite set $W$ [F8] of symmetric bilinear terms, hence a symmetric bilinear form. It is $ρ (W)$ -invariant: for $g \in W$ , substituting $w^{'} = w g$ and using that $w \mapsto w g$ is a bijection of the finite set $W$ [F8] gives $β (ρ (g) u, ρ (g) v) = \sum_{w} β_{0} (ρ (w g) u, ρ (w g) v) = \sum_{w^{'}} β_{0} (ρ (w^{'}) u, ρ (w^{'}) v) = β (u, v)$ . It is positive definite: every summand is $\geq 0$ by [F8] and the summand with $w = 1$ equals $β_{0} (u, u) > 0$ for $u \neq 0$ , so $β (u, u) > 0$ . Applying step 3.2 to this $β$ gives that $B$ is positive definite, which is (4).

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

Finiteness criterion: W is finite exactly when the Coxeter form is positive definite

Statement

Let $S$ be a finite set with Coxeter matrix $m$ , presented group $W$ , length function $ℓ$ and diagram $Γ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Coxeter diagrams: edges, labels, components and finite type); let $V = R^{S}$ carry the Coxeter form $B$ with reflections $r_{a}$ (The real Coxeter form, its radical, reflections, and form-preserving maps), let $ρ : W \to G L (V)$ be the canonical reflection homomorphism (The canonical reflection homomorphism, roots, reflections, and the positive cone), and let $V^{*}$ be the algebraic dual with its dual action, the closed chamber $C$ and its interior $C^{\circ}$ (The dual action, chambers, faces, and root hyperplanes, The dual action, the faces, and the rank-two chamber tiling).

(1) Finiteness criterion. $W$ is finite if and only if $B$ is positive definite (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

(2) The dual form. Assume $B$ is positive definite. Then $b : V \to V^{*}$ , $b (v) : = B (v, \cdot)$ , is a linear isomorphism, and $B^{*} (f, g) : = B (b^{- 1} f, b^{- 1} g)$ defines a positive definite symmetric bilinear form $B^{*}$ on $V^{*}$ ; for every $w \in W$ the dual map $ρ^{*} (w)$ preserves $B^{*}$ : $B^{*} (w \cdot f, w \cdot g) = B^{*} (f, g)$ for all $f, g \in V^{*}$ .

(3) Isolation of the identity and discreteness. For every $f \in C^{\circ}$ the set $Ω_{f} : = {h \in G L (V^{*}) : h \cdot f \in C^{\circ}}$ is an open neighbourhood of ${i d}_{V^{*}}$ in $G L (V^{*})$ and $Ω_{f} \cap ρ^{*} (W) = {i d}$ . Consequently $ρ^{*} (W)$ is a discrete subgroup of $G L (V^{*})$ , and $ρ (W)$ is a discrete subgroup of $G L (V)$ . This clause uses no positive definiteness of $B$ .

Facts & Assumptions

Given: A finite set $S$ with Coxeter matrix $m$ , the presented group $W$ with length $ℓ$ , the diagram $Γ$ , the space $V = R^{S}$ with Coxeter form $B$ , the canonical homomorphism $ρ$ , the dual space $V^{*}$ with the dual action, the chambers $C, C^{\circ}$ and the sets $S (f) = {s \in S : f (e_{s}) = 0}$ .

[F1]

If $W$ is finite then $B$ is positive definite (Disconnected diagrams, direct products, and comparison of invariant forms (4)).

[F2]

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

[F3]

$ρ : W \to G L (V)$ is a homomorphism with $ρ (s) = r_{e_{s}}$ , and $B (ρ (w) u, ρ (w) u^{'}) = B (u, u^{'})$ for all $w \in W$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F4]

The dual action is $(w \cdot f) (v) = f (ρ (w)^{- 1} v)$ ; the closed chamber is $C = {f : f (e_{s}) \geq 0 \forall s}$ , its interior is $C^{\circ} = {f : f (e_{s}) > 0 \forall s}$ , and $S (f) = {s : f (e_{s}) = 0}$ (The dual action, chambers, faces, and root hyperplanes).

[F5]

The dual action is an action by linear maps, and every face of $C$ is nonempty; in particular $C^{\circ} \neq \emptyset$ (The dual action, the faces, and the rank-two chamber tiling).

[F6]

$ρ$ is injective, the dual action $W \to G L (V^{*})$ is injective, and if $w \neq 1$ there is $s \in S$ with $ρ (w) e_{s}$ negative in the root decomposition (The root-length criterion and faithfulness of the canonical reflection representation (3)).

[F7]

If $f, g \in C$ , $w \in W$ and $w \cdot f = g$ , then $f = g$ and $w \in W_{S (f)}$ ; moreover ${Stab}_{W} (f) = W_{S (f)}$ for $f \in C$ (Chamber collisions, point stabilizers, and the intersection rule (3),(4)).

[F8]

$B$ is positive definite when $B (v, v) > 0$ for every $v \neq 0$ (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F9]

$V^{*}$ is the vector space of linear functionals $V \to R$ ; for every bilinear form the map $v \mapsto B (v, \cdot)$ is linear $V \to V^{*}$ , rank-nullity implies that an injective linear map between spaces of equal finite dimension is an isomorphism, and the functionals $f_{s}$ dual to a basis $(e_{s})$ form a basis of $V^{*}$ , so that $\dim V^{*} = \dim V$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ , Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V \to V^{*}$ , Invertible linear maps, linear isomorphisms, and inverse linear maps, Linear map between vector spaces over the same field, The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication, The dual family $(b^{*})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{*} (c) = δ_{b c}$ , The dual family of a finite basis is a basis of the dual space, with the same dimension, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Rank-nullity: $\dim_{F} V = nullity T + rank T$ , If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

[F10]

On $R^{N}$ the open sets are the metric-topology open sets, and a map is continuous exactly when preimages of open sets are open; finite unions and intersections of open sets are open; sums and products of continuous real maps are continuous, and composites of continuous maps are continuous because $(g \circ f)^{- 1} (U) = f^{- 1} (g^{- 1} (U))$ ; a subset of $R^{N}$ is compact exactly when it is closed and bounded; $R^{n}$ carries the metrics $d_{1}, d_{2}, d_{\infty}$ of $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it and any two norms on it are equivalent (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, Metric continuity characterisations, with countable choice for the sequential converse, Open cover, subcover, compact metric space, and compact subset of a metric space, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space, Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed, A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, Equivalent norms, and the dictionary with equivalent metrics, For $n \geq 1$ all norms on $R^{n}$ are equivalent, Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases, The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset).

[F11]

In an orthonormal basis of a finite-dimensional inner product space every vector is $\sum_{j} ⟨ v, φ_{j} ⟩ φ_{j}$ , the inner-product norm is a norm, and $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ with equality exactly for linearly dependent vectors; the matrix $A$ of an isometry satisfies $A^{T} A = I$ , and the adjugate formula gives $A^{- 1} = \det (A)^{- 1} adj (A)$ for invertible matrices (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product, The inner-product norm is definite, homogeneous, and satisfies the triangle inequality, Every finite-dimensional real or complex inner product space has an orthonormal basis, Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors, If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ ). Determinants and cofactors are polynomials in the entries by induction using Laplace expansion computes the determinant along every row and every column over a commutative ring.

[F12]

$W_{J} = ⟨ J ⟩$ for $J \subseteq S$ , the subgroup generated by the empty set is ${1}$ , and $m (s, t)$ is the order of $s t$ in $W$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups, Group and abelian group, Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

Proof

technique · direct; positive definiteness makes $B$ an inner product, and finiteness is read off a compact orthogonal group and an isolated identity

1.1F2F3F4F8F9algebra

(The dual form.) The case $S = \emptyset$ is trivial: then $V = {0}$ , $W = {1}$ is finite by [F12] and $B$ is positive definite vacuously, so assume $S \neq \emptyset$ and put $n : = ∣ S ∣ \geq 1$ . Assume $B$ positive definite. If $B (v, x) = 0$ for all $x$ then $B (v, v) = 0$ , so $v = 0$ by [F8]; hence the linear map $b : V \to V^{*}$ , $b (v) : = B (v, \cdot)$ [F9], has kernel ${0}$ and, since $\dim V^{*} = \dim V = n$ is finite, it is a linear isomorphism [F9]. The form $B^{*} (f, g) : = B (b^{- 1} f, b^{- 1} g)$ is symmetric and bilinear, and positive definite because $f \neq 0$ gives $b^{- 1} f \neq 0$ and $B^{*} (f, f) = B (b^{- 1} f, b^{- 1} f) > 0$ [F8]. For $w \in W$ and $v \in V$ the two functionals $b (ρ (w) v)$ and $w \cdot b (v)$ agree: at $x$ both take the value $B (ρ (w) v, x) = B (v, ρ (w)^{- 1} x)$ , by [F3] applied to the pair $ρ (w)^{- 1} x$ and by the dual-action formula [F4]; hence $b^{- 1} (w \cdot f) = ρ (w) b^{- 1} f$ and $B^{*} (w \cdot f, w \cdot g) = B (ρ (w) b^{- 1} f, ρ (w) b^{- 1} g) = B (b^{- 1} f, b^{- 1} g) = B^{*} (f, g)$ for all $f, g$ , so the dual action preserves $B^{*}$ [F3]. This is clause (2).

1.2F4F5F7F9F10F12algebra

(Isolation of the identity.) Let $f \in C^{\circ}$ , nonempty by [F5], and put $Ω_{f} : = {h \in G L (V^{*}) : h (f) \in C^{\circ}}$ and $Ω_{f}^{'} : = {g \in G L (V) : f \circ g \in C^{\circ}}$ . $C^{\circ}$ is open in $V^{*}$ : it is the finite intersection of the sets ${f^{'} : f^{'} (e_{s}) > 0}$ [F4], each the preimage of the open interval $(0, \infty) \subseteq R$ under the coordinate functional $f^{'} \mapsto f^{'} (e_{s})$ , which is continuous since its difference is bounded by the maximum coordinate difference [F10]; the maps $h \mapsto h (f)$ and $g \mapsto f \circ g$ are linear on the finite-dimensional spaces $E n d (V^{*})$ and $E n d (V)$ and hence continuous (each coordinate is a finite sum of matrix entries multiplied by fixed coordinates) [F9, F10], so $Ω_{f}$ and $Ω_{f}^{'}$ are open neighbourhoods of the identities, because $f \in C^{\circ}$ . If $ρ^{*} (w) \in Ω_{f}$ , that is $w \cdot f = ρ^{*} (w) f \in C^{\circ}$ , then $f \in C$ and $w \cdot f \in C$ , and the collision theorem [F7] gives $w \in W_{S (f)}$ with $S (f) = \emptyset$ because $f \in C^{\circ}$ ; as $W_{\emptyset} = {1}$ [F12], $w = 1$ and $Ω_{f} \cap ρ^{*} (W) = {i d}$ . If instead $ρ (w) \in Ω_{f}^{'}$ , then $w^{- 1} \cdot f = f \circ ρ (w) \in C^{\circ}$ by [F4], so the same argument with $w^{- 1}$ gives $w^{- 1} \in W_{S (f)} = {1}$ and $ρ (w) = {i d}_{V}$ ; hence $Ω_{f}^{'} \cap ρ (W) = {{i d}_{V}}$ . No positive definiteness is used in this step.

1.3F1

(Finite groups have positive definite form.) If $W$ is finite then $B$ is positive definite by [F1]; this proves the finite- $W$ to positive-definite- $B$ direction of (1), and no other argument is needed for it.

2.1F9F10F11step 1.1algebra

(Closedness and boundedness of $O (B^{*})$ .) Assume $B$ positive definite and let $B^{*}$ be the form of step 1.1; put $O (B^{*}) : = {h \in G L (V^{*}) : B^{*} (h φ, h ψ) = B^{*} (φ, ψ) \forall φ, ψ \in V^{*}}$ . Identify $E n d (V^{*})$ with $R^{n^{2}}$ by the dual basis $(f_{s})$ of [F9] [F10]. For each pair $i, j$ the function $h \mapsto B^{*} (h f_{i}, h f_{j})$ is a finite sum of products $\sum_{p, q} h_{p i} h_{q j} B^{*} (f_{p}, f_{q})$ of entries of $h$ with constants, hence continuous [F10]; Any endomorphism preserving $B^{*}$ is injective: $h φ = 0$ implies $B^{*} (φ, φ) = 0$ and hence $φ = 0$ ; it is invertible by rank-nullity in finite dimension [F9]. Thus $O (B^{*})$ is exactly the preimage of the single point $(B^{*} (f_{i}, f_{j}))_{i j}$ under a continuous map $E n d (V^{*}) \to R^{n^{2}}$ , hence closed [F10]. For boundedness fix an orthonormal basis $(φ_{1}, \dots, φ_{n})$ of $V^{*}$ for $B^{*}$ [F11]; if $h \in O (B^{*})$ and $h φ_{i} = \sum_{j} a_{j i} φ_{j}$ , then $a_{j i} = B^{*} (h φ_{i}, φ_{j})$ by the orthonormal expansion [F11], so $∣ a_{j i} ∣ \leq ∥ h φ_{i} ∥ ∥ φ_{j} ∥ = 1$ by Cauchy-Schwarz because $B^{*} (h φ_{i}, h φ_{i}) = B^{*} (φ_{i}, φ_{i}) = 1$ [F11]; hence all matrix entries in this orthonormal basis are bounded by $1$ . A change to the fixed dual basis expresses each new entry as a finite linear combination of these entries with fixed coefficients; its absolute value is bounded by the sum of the absolute values of those coefficients. Therefore $O (B^{*})$ is bounded in the original $R^{n^{2}}$ coordinates [F10].

2.2F10step 1.2algebra

(Discreteness of both images.) Let $γ \in ρ^{*} (W)$ . The set $γ Ω_{f} = {h \in G L (V^{*}) : γ^{- 1} h \in Ω_{f}}$ is open in $G L (V^{*})$ as the preimage of the open set $Ω_{f}$ under the continuous map $h \mapsto γ^{- 1} h$ [F10], and it contains $γ$ ; if $w = γ h \in ρ^{*} (W)$ with $h \in Ω_{f}$ , then $h = γ^{- 1} w \in ρ^{*} (W)$ (a subgroup) and step 1.2 forces $h = i d$ , so $w = γ$ . Hence $γ Ω_{f} \cap ρ^{*} (W) = {γ}$ for every $γ$ , so each point of $ρ^{*} (W)$ is open in the subspace topology and $ρ^{*} (W)$ is discrete [F10]. The same translation argument with $Ω_{f}^{'}$ shows that each point of $ρ (W)$ is isolated, so $ρ (W)$ is discrete as well. This proves clause (3), and no positive definiteness was used.

2.3F10F11step 1.2algebra

(Continuity of the operations, and a symmetric neighbourhood squaring into $Ω_{f}$ .) With $E n d (V^{*}) ≅ R^{n^{2}}$ and $E n d (V) ≅ R^{n^{2}}$ as in [F10], matrix multiplication has entries that are finite sums of products of entries of the factors, hence is continuous [F10], and the entries of the inverse are given by the adjugate formula $h^{- 1} = \det (h)^{- 1} adj (h)$ [F11], a polynomial in the entries divided by the continuous function $\det$ , which is nonzero on $G L$ [F10]; hence multiplication and inversion are continuous on the general linear groups [F10], and $G L (V^{*})$ is open in $E n d (V^{*})$ as the preimage of $R ∖ {0}$ under $\det$ [F10, F11]. Since multiplication sends $(i d, i d)$ to $i d \in Ω_{f}$ with $Ω_{f}$ open [step 1.2], there are open neighbourhoods $V_{1}, V_{2} \subseteq G L (V^{*})$ of $i d$ with $m (V_{1} \times V_{2}) \subseteq Ω_{f}$ : the open preimage $m^{- 1} (Ω_{f})$ contains a maximum-coordinate ball around $(i d, i d)$ in the paired matrix coordinates, and such a ball is a product of two balls about $i d$ [F10]; then $U : = V_{1} \cap V_{2} \cap V_{1}^{- 1} \cap V_{2}^{- 1}$ (where $W^{- 1} : = {h^{- 1} : h \in W}$ ) is an open symmetric neighbourhood of $i d$ in $G L (V^{*})$ , since $U = U^{- 1}$ , with $U \cdot U \subseteq V_{1} \cdot V_{2} \subseteq Ω_{f}$ [F10].

3.1F10step 2.1

(Compactness of $O (B^{*})$ .) By step 2.1, $O (B^{*})$ is a closed and bounded subset of $E n d (V^{*}) ≅ R^{n^{2}}$ ; by the Heine-Borel theorem a subset of $R^{n^{2}}$ is compact if and only if it is closed and bounded [F10], so $O (B^{*})$ is compact.

4.1F6F10F11step 1.1step 1.2step 1.3step 2.2step 2.3step 3.1algebra∎

(Positive definite $B$ gives finite $W$ ; conclusion.) Assume $B$ positive definite and $S \neq \emptyset$ as in step 1.1; put $Γ : = ρ^{*} (W)$ , a subgroup of $G L (V^{*})$ that is contained in $O (B^{*})$ by the invariance proved in step 1.1 [step 1.1]. Let $f \in C^{\circ}$ and let $U$ be the open symmetric neighbourhood of $i d$ with $U \cdot U \subseteq Ω_{f}$ from step 2.3 [step 2.3]. Each $h U$ ( $h \in O (B^{*})$ ) is open in $E n d (V^{*})$ , being the image of the open set $U$ under the linear isomorphism $u \mapsto h u$ with inverse $k \mapsto h^{- 1} k$ [F10, F11], so the family ${h U \cap O (B^{*}) : h \in O (B^{*})}$ is an open cover of the compact space $O (B^{*})$ from step 3.1 [step 3.1]; choose a finite subcover, say $O (B^{*}) = ⋃_{i = 1}^{N} (h_{i} U \cap O (B^{*}))$ [F10]. Each $h_{i} U$ contains at most one element of $Γ$ : if $w_{1} = h_{i} u_{1}$ and $w_{2} = h_{i} u_{2}$ with $u_{1}, u_{2} \in U$ and $w_{1}, w_{2} \in Γ$ , then $w_{2}^{- 1} w_{1} = u_{2}^{- 1} u_{1} \in U \cdot U \subseteq Ω_{f}$ because $U = U^{- 1}$ , and $w_{2}^{- 1} w_{1} \in Γ$ , so step 1.2 gives $w_{2}^{- 1} w_{1} = i d$ and $w_{1} = w_{2}$ [step 1.2]. Hence $Γ$ has at most $N$ elements, and faithfulness of the dual action [F6] gives $∣ W ∣ = ∣ Γ ∣ \leq N < \infty$ . Therefore $W$ finite if and only if $B$ is positive definite, which is (1); clause (2) is step 1.1, clause (3) is steps 1.2 and 2.2, and the finite- $W$ direction of (1) is step 1.3.

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

Classification of finite Coxeter systems, including the H and dihedral families

Statement

Let $S$ be a finite set with Coxeter matrix $m$ and diagram $Γ$ (Coxeter diagrams: edges, labels, components and finite type), $W$ the presented group with length $ℓ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), $V = R^{S}$ with Coxeter form $B$ and cosine matrix $C = (B (e_{s}, e_{t}))_{s, t \in S} = (- \cos (π / m (s, t)))_{s, t}$ (The real Coxeter form, its radical, reflections, and form-preserving maps), with $\cos (π / \infty) : = 1$ in this matrix notation.

(1) Irreducible case. If $Γ$ is connected, then $W$ is finite if and only if $Γ$ is isomorphic as a labelled graph (Graph isomorphisms, automorphisms and graph complements) to one of the following standard diagrams:

$A_{n}$ , $n \geq 1$ : a path on $n$ vertices, all edges labelled $3$ ;
$B_{n}$ , $n \geq 2$ : a path on $n$ vertices whose labels are $3, \dots, 3, 4$ (one edge of label $4$ , at an end);
$D_{n}$ , $n \geq 4$ : the star with one degree- $3$ vertex and three arms of $1, 1, n - 3$ vertices, all edges labelled $3$ ;
$E_{6}$ , $E_{7}$ , $E_{8}$ : the stars with arms of $1, 2, 2$ ; $1, 2, 3$ ; $1, 2, 4$ vertices, all edges labelled $3$ ;
$F_{4}$ : the path on four vertices with labels $3, 4, 3$ ;
$H_{3}$ : the path on three vertices with labels $3, 5$ , and $H_{4}$ : the path on four vertices with labels $3, 3, 5$ ;
$I_{2} (m)$ , $3 \leq m < \infty$ : two vertices joined by a single edge labelled $m$ .

(2) Reducible case. For an arbitrary finite $S$ , with components of $Γ$ on the vertex sets $S_{1}, \dots, S_{k}$ , the group $W$ is finite if and only if every component $Γ_{S_{i}}$ is one of the diagrams of (1); in that case $W ≅ W_{S_{1}} \times \dots \times W_{S_{k}}$ (Disconnected diagrams, direct products, and comparison of invariant forms (1), The external direct product $G \times H$ with componentwise multiplication). Thus the finite Coxeter systems are exactly the direct products of the irreducible types listed in (1).

(3) Positivity of the listed diagrams. For every diagram $Γ$ of the list (1) the form $B$ is positive definite. In more detail: every proper principal submatrix of the matrix of $B$ is a block diagonal matrix whose blocks are matrices of listed diagrams of smaller rank, and the determinant of the full $n \times n$ cosine matrix of each listed diagram, multiplied by $2^{n}$ (i.e. $\det (2 C)$ ), is $A_{n} : n + 1, B_{n} : 2, D_{n} : 4, E_{6} : 3, E_{7} : 2, E_{8} : 1, F_{4} : 1, H_{3} : 3 - \sqrt{5}, H_{4} : \frac{7 - 3 \sqrt{5}}{2}, I_{2} (m) : 4 \sin^{2} (π / m),$ all of which are positive. Hence $B$ is positive definite for each listed diagram (Sylvester's criterion; 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), and by (1) of Finiteness criterion: W is finite exactly when the Coxeter form is positive definite each such diagram defines a finite Coxeter group.

(4) Coincidences. As Coxeter systems one has $A_{2} = I_{2} (3)$ , $B_{2} = C_{2} = I_{2} (4)$ and $G_{2} = I_{2} (6)$ ; more generally with the usual naming conventions $A_{1} = B_{1}$ (the one-vertex diagram), $A_{3} = D_{3}$ (the path on three vertices), $H_{2} = I_{2} (5)$ , and $A_{1} \times A_{1} = I_{2} (2)$ if the notation $I_{2} (m)$ is extended to $m = 2$ , the disconnected two-vertex diagram (no edge, since $m = 2$ draws no edge). Apart from these identifications the diagrams of (1) are pairwise non-isomorphic, and the classification list is therefore the duplicate-free list $A_{n}$ ( $n \geq 1$ ), $B_{n}$ ( $n \geq 2$ ), $D_{n}$ ( $n \geq 4$ ), $E_{6}, E_{7}, E_{8}, F_{4}, H_{3}, H_{4}$ and $I_{2} (m)$ with $m \geq 3$ , $m \notin {3, 4}$ , where $I_{2} (3), I_{2} (4)$ are also written $A_{2}, B_{2}$ and $I_{2} (5), I_{2} (6)$ are also written $H_{2}, G_{2}$ .

Facts & Assumptions

Given: A finite set $S$ with Coxeter matrix $m$ , its diagram $Γ$ , the presented group $W$ with length $ℓ$ , the space $V = R^{S}$ with Coxeter form $B$ and cosine matrix $C = (B (e_{s}, e_{t}))$ .

[F1]

$W$ is finite if and only if $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)).

[F2]

Every connected positive definite diagram is isomorphic as a labelled graph to one of $A_{n}$ ( $n \geq 1$ ), $B_{n}$ ( $n \geq 2$ ), $D_{n}$ ( $n \geq 4$ ), $E_{6}, E_{7}, E_{8}$ , $F_{4}$ , $H_{3}$ , $H_{4}$ , $I_{2} (m)$ ( $m \geq 3$ ); moreover for a path with labels $m_{1}, \dots, m_{n - 1}$ the determinants $d_{k}$ of the leading $k \times k$ principal submatrices of $C$ satisfy $d_{0} = 1$ , $d_{1} = 1$ and $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ , with $d_{k} = (k + 1) / 2^{k}$ and $\det (2 C) = n + 1$ when all labels are $3$ (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (5)(i),(7)).

[F3]

Distinct vertices are joined exactly when $m (s, t) \geq 3$ and carry the label $m (s, t)$ , a label $3$ edge being drawn unlabelled; the subdiagram $Γ_{T}$ is the induced labelled graph, so deleting a vertex removes exactly its incident edges; an isomorphism of labelled graphs is a bijection of vertex sets preserving edges and labels (Coxeter diagrams: edges, labels, components and finite type, Graph isomorphisms, automorphisms and graph complements).

[F4]

$B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ and $B (e_{s}, e_{t}) = - 1$ for $m (s, t) = \infty$ , with $B (e_{s}, e_{s}) = 1$ (The real Coxeter form, its radical, reflections, and form-preserving maps), with $\cos (π / \infty) : = 1$ in this matrix notation.

[F5]

A symmetric real matrix $A \in M_{n} (R)$ , $n \geq 1$ , is positive definite if and only if the determinants of all its leading $k \times k$ principal submatrices, $1 \leq k \leq n$ , are positive (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).

[F6]

Determinants are multilinear in the columns, a determinant scales by $λ^{n}$ when the $n \times n$ matrix is multiplied by $λ$ , the determinant of a block diagonal matrix is the product of the determinants of its blocks, and each minor $M_{i j} (A)$ is the determinant of the matrix with row $i$ and column $j$ deleted (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, Laplace expansion computes the determinant along every row and every column over a commutative ring).

[F7]

For every real $θ$ one has $T_{5} (\cos θ) = \cos (5 θ)$ , and the Chebyshev polynomials of the first kind satisfy $T_{0} = 1$ , $T_{1} = t$ and $T_{n + 2} = 2 t T_{n + 1} - T_{n}$ ; $\cos (x + π) = - \cos x$ , $\cos (π / 2) = 0$ , $\cos π = - 1$ ; $\cos (2 x) = 2 \cos^{2} x - 1$ ; $\cos (π / 4) = \sqrt{2} / 2$ ; $\sin^{2} x + \cos^{2} x = 1$ 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, The finite Viete cosine product and its positive nested-radical factors, Pi is the first positive zero of sine, 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 ${x^{2} : x \neq 0}$ , Signs, monotonicity intervals, and ranges of sine and cosine).

[F8]

A finite product of groups is finite if and only if every factor is finite: products of finite sets are finite by successive enumeration, and each factor embeds by putting identities in the other coordinates. The external direct product $W_{1} \times \dots \times W_{k}$ is a group, and for diagram components its multiplication map is an isomorphism by Disconnected diagrams, direct products, and comparison of invariant forms (1) (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, Internal direct products of finitely many normal subgroups).

[F9]

A direct sum of symmetric bilinear forms is positive definite if and only if every summand is; in a direct sum the off-diagonal blocks are zero and a principal submatrix of a block diagonal matrix is block diagonal with the corresponding principal submatrices as blocks; the $e_{s}$ form a basis and $\dim V = ∣ S ∣$ (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form, Internal direct sum $V = ⨁_{i < n} U_{i}$ : the sum is everything and each summand meets the sum of the others only in $0_{V}$ , Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Linear combination of a finite list, and the span $span (S)$ as the smallest linear subspace containing $S$ , Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, The cardinality $∣ A ∣$ of a finite set).

[F10]

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

Proof

technique · induction on the rank, together with the finiteness criterion and the exclusion tree

1.1F2F4F6F7algebra

(The special cosine values and the determinant recurrences.) First the cosine values used below. For $c : = \cos (π / 3)$ the double-angle and shift identities of [F7] give $2 c^{2} - 1 = \cos (2 π / 3) = - \cos (π / 3) = - c$ , 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 $c = 1 / 2$ ; also $\cos (π / 4) = \sqrt{2} / 2$ by [F7]. For $c_{5} : = \cos (π / 5)$ , iterating the recurrence of [F7] 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; since $0 < π / 5 < π / 2$ gives $0 < c_{5} < 1$ [F7], one has $c_{5} \neq - 1$ , hence $4 c_{5}^{2} - 2 c_{5} - 1 = 0$ 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 $c_{5} > 0$ excludes $\frac{1 - \sqrt{5}}{4} < 0$ (as $2 < \sqrt{5}$ [F7]), so $c_{5} = (1 + \sqrt{5}) / 4$ . Now for a path on $k$ vertices with labels $m_{1}, \dots, m_{k - 1}$ put $D_{k} : = \det (2 C_{k})$ , where $C_{k}$ is the leading $k \times k$ principal submatrix of $C$ ; since multiplying a $k \times k$ matrix by $2$ multiplies its determinant by $2^{k}$ [F6] and $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ with $2^{k} d_{k} = D_{k}$ [F2], expanding along the last row gives $D_{0} = 1$ , $D_{1} = 2$ , $D_{2} = 4 \sin^{2} (π / m_{1})$ and $D_{k} = 2 D_{k - 1} - 4 \cos^{2} (π / m_{k - 1}) D_{k - 2} (k \geq 2) .$ In particular $D_{2} (A_{2}) = 4 - 4 \cos^{2} (π / 3) = 4 - 1 = 3$ , $D_{2} (I_{2} (m)) = 4 \sin^{2} (π / m)$ , and for a disjoint union the matrix is block diagonal, so $D_{2} (A_{1} ⊔ A_{1}) = 2 \cdot 2 = 4$ and $D_{3} (A_{3}) = 2 D_{2} (A_{2}) - D_{1} (A_{1}) = 6 - 2 = 4$ [F6, F7].

1.2F10ih

(Induction hypothesis, recorded for use in the proof of (3).) Assume $n \geq 3$ and assume that for every disjoint union $Δ$ of listed diagrams with total rank $< n$ both statements hold: every principal minor of $C (Δ)$ is positive, and $B_{Δ}$ is positive definite.

1.3F1F2F3algebra

(The irreducible case: both directions of (1).) Let $Γ$ be connected. If $W$ is finite then $B$ is positive definite by [F1], so clause (7) of the exclusion lemma in [F2] gives that $Γ$ is isomorphic as a labelled graph to one of $A_{n}$ ( $n \geq 1$ ), $B_{n}$ ( $n \geq 2$ ), $D_{n}$ ( $n \geq 4$ ), $E_{6}, E_{7}, E_{8}$ , $F_{4}$ , $H_{3}$ , $H_{4}$ , $I_{2} (m)$ ( $m \geq 3$ ) with the labels displayed in (1). Conversely, if $Γ$ is one of those labelled graphs then $B$ is positive definite by the positivity clause (3), proved below, and then $W$ is finite by [F1]. Hence $W$ is finite if and only if $Γ$ is one of those labelled graphs, which is (1).

1.4F3F8algebra

(Coincidences and the duplicate-free list (4).) Reading the labelled graphs of (1) [F3]: $I_{2} (3)$ is one edge labelled $3$ , which is $A_{2}$ ; $I_{2} (4)$ is one edge labelled $4$ , which is $B_{2}$ , also called $C_{2}$ in the root-system naming, and for $n \geq 3$ the paths with labels $3, \dots, 3, 4$ are the $B_{n}$ ; $I_{2} (6)$ is one edge labelled $6$ , conventionally called $G_{2}$ ; $A_{1}$ is the one-vertex diagram, written $B_{1}$ in the root-system naming; $A_{3}$ and $D_{3}$ are both the path on three vertices, since the third arm of $D_{3}$ has length $n - 3 = 0$ ; $H_{2}$ is $I_{2} (5)$ by the rank-two naming convention; and $I_{2} (2)$ denotes the two-vertex diagram with no edge, which by Disconnected diagrams, direct products, and comparison of invariant forms (1) is the direct product $A_{1} \times A_{1}$ of two one-vertex systems [F8]. For the non-isomorphism claim, the type is recovered from invariants of the labelled graph: the number of vertices, the degree sequence, the multiset of edge labels, the position of the unique label $\geq 4$ or of the unique vertex of degree $3$ , and, for a diagram with a vertex of degree $3$ , the multiset of the lengths of the three arms (the numbers of vertices in the components obtained by deleting that vertex); these separate every pair of the list with exactly the exceptions displayed above, and distinct $m$ give non-isomorphic $I_{2} (m)$ because the label is read off the single edge. Removing the duplicates gives the displayed duplicate-free list, with $I_{2} (3), I_{2} (4), I_{2} (6)$ named alternatively as $A_{2}, B_{2}, G_{2}$ .

2.1F2F4F6F7step 1.1algebra

(The determinant table for the listed diagrams.) Using the recurrence of 1.1 [step 1.1] with the last edge label and deleting the last one or two vertices: $A_{n}$ has all labels $3$ , so $D_{n} = 2 D_{n - 1} - D_{n - 2}$ with $D_{1} = 2$ , $D_{2} = 3$ , giving $D_{n} = n + 1$ by induction on $n$ [F10]; $B_{n}$ (labels $3, \dots, 3, 4$ , the $4$ at the end) has $D_{n} = 2 D_{n - 1} - 2 D_{n - 2}$ with $D_{n - 1} = n$ (the all- $3$ path $A_{n - 1}$ ) and $D_{n - 2} = n - 1$ , giving $D_{n} = 2$ ; $B_{3}$ has $D_{3} = 2 D_{2} (A_{2}) - 2 D_{1} (A_{1}) = 6 - 4 = 2$ and $F_{4}$ (labels $3, 4, 3$ ) has $D_{4} = 2 D_{3} (B_{3}) - D_{2} (A_{2}) = 4 - 3 = 1$ ; $H_{3}$ (labels $3, 5$ ) has $D_{3} = 2 D_{2} (A_{2}) - 4 \cos^{2} (π / 5) D_{1} = 6 - 8 \cos^{2} (π / 5) = 3 - \sqrt{5}$ and $H_{4}$ (labels $3, 3, 5$ ) has $D_{4} = 2 D_{3} (A_{3}) - 4 \cos^{2} (π / 5) D_{2} (A_{2}) = 8 - \frac{9 + 3 \sqrt{5}}{2} = \frac{7 - 3 \sqrt{5}}{2}$ ; $I_{2} (m)$ has $D_{2} = 4 \sin^{2} (π / m)$ . For any leaf $a$ with sole neighbour $b$ and off-diagonal entry $- t$ in $M = 2 C$ , order $a$ last and $b$ next to last. Expansion along the last row gives $\det M = 2 \det M_{\hat{a}} - t^{2} \det M_{\hat{a}, \hat{b}}$ : the off-diagonal cofactor has its last column zero except for $- t$ , whose expansion gives the second term with its negative sign [F6]. For label $3$ , $t = 1$ . For $D_{4}$ , deletion of $a$ gives $A_{3}$ , and deletion of $a, b$ gives $A_{1} ⊔ A_{1}$ ; for $D_{n}$ with $n \geq 5$ , use the end of the long arm, giving $D_{n - 1}$ and $D_{n - 2}$ (with $D_{3} = A_{3}$ ). Thus $D_{n} (D) = 2 D_{n - 1} - D_{n - 2}$ with $D_{3} (A_{3}) = 4$ , $D_{2} (A_{1} ⊔ A_{1}) = 4$ , whence $D_{n} (D) = 4$ for $n \geq 4$ ; and deleting the end vertex of the long arm of $E_{6}, E_{7}, E_{8}$ respectively one and two vertices of it gives $D (E_{6}) = 2 D_{5} (D) - D_{4} (A) = 8 - 5 = 3$ , $D (E_{7}) = 2 D (E_{6}) - D_{5} (D) = 6 - 4 = 2$ , $D (E_{8}) = 2 D (E_{7}) - D (E_{6}) = 4 - 3 = 1$ . This is the table of (3).

2.2F8F9step 1.3algebra

(The reducible case (2).) Let the connected components of $Γ$ be on $S_{1}, \dots, S_{k}$ . By Disconnected diagrams, direct products, and comparison of invariant forms (1) the multiplication map $W_{S_{1}} \times \dots \times W_{S_{k}} \to W$ is an isomorphism, and by its (2) the form is the orthogonal direct sum $B = B_{S_{1}} \oplus \dots \oplus B_{S_{k}}$ [F9]. A direct product of groups is finite exactly when each factor is finite [F8], and a direct sum of forms is positive definite exactly when each summand is [F9]; by 1.3 [step 1.3] applied to each component this happens exactly when every $Γ_{S_{i}}$ is one of the diagrams of (1). This is (2).

2.3F5F7step 1.1algebra

(Base of the induction: rank $\leq 2$ .) The empty diagram has rank $0$ and no principal minors, and its form is positive definite vacuously; $A_{1}$ has $D_{1} = 2 > 0$ , and $I_{2} (m)$ ( $m \geq 3$ ) has $D_{2} = 4 \sin^{2} (π / m) > 0$ together with the $1 \times 1$ minors $2 > 0$ , by 1.1 [step 1.1] and $\sin (π / m) > 0$ for $0 < π / m < π$ [F7]; by Sylvester's criterion [F5] the forms of $A_{1}$ , $A_{2} = I_{2} (3)$ , $B_{2} = I_{2} (4)$ and all $I_{2} (m)$ are positive definite. The remaining disjoint union of rank $2$ is $A_{1} ⊔ A_{1}$ , with $2 C = diag (2, 2)$ , positive principal minors $2, 2, 4$ and positive definite form. This completes the base of the induction. [base],

3.1F3F5F6F9step 1.2step 2.1algebra

(Inductive step: every listed diagram is positive definite; clause (3).) Let $Δ$ be a disjoint union of listed diagrams of total rank $n \geq 3$ , and assume the induction hypothesis of 1.2 [step 1.2]. If $Δ$ is disconnected, its matrix is block diagonal with blocks the matrices of its components, and every principal submatrix of a block diagonal matrix is block diagonal with blocks the corresponding principal submatrices of the components, so multiplicativity of determinants [F6] and the induction hypothesis give positivity of every principal minor and positive definiteness of $B_{Δ}$ [F9]. If $Δ$ is connected, it is one of the listed diagrams: deleting vertices from an all- $3$ path leaves $A$ paths; a subpath retaining the label- $4$ endpoint edge of $B_{n}$ is a smaller $B$ path; proper subpaths of $F_{4}$ are $A$ paths, $B_{2}$ or $B_{3}$ ; and proper subpaths of $H_{3}, H_{4}$ are $A$ paths, $I_{2} (5)$ or $H_{3}$ . Deleting a vertex of $I_{2} (m)$ leaves $A_{1}$ . Deleting vertices from a star with arm lengths $(a, b, c)$ leaves stars with arm lengths $a^{'} \leq a$ , $b^{'} \leq b$ , $c^{'} \leq c$ (or paths, when an arm disappears); the triples $(1, 1, n - 3)$ of $D_{n}$ and $(1, 2, 2), (1, 2, 3), (1, 2, 4)$ of $E_{6}, E_{7}, E_{8}$ dominate componentwise every smaller triple, and the result is again a $D$ or $E$ diagram of smaller rank, or a path $A$ [F3]. Hence every proper principal submatrix of a listed diagram is block diagonal with blocks listed diagrams of smaller rank, and every principal minor is positive by the induction hypothesis; the full determinant is the positive table value of 2.1 [step 2.1]; since the leading principal minors are principal minors, Sylvester's criterion [F5] gives positive definiteness of the form of every listed diagram, which is (3).

4.1step 1.3step 1.4step 2.2step 2.3step 3.1∎

(Discharge of the induction; assembly.) The base case is 2.3 [step 2.3] and the inductive step is 3.1 [step 3.1], so by the induction principle on the rank, every disjoint union of listed diagrams, and in particular every diagram of the list (1), has positive definite form [discharge-induction: step 3.1]. Clause (1) is 1.3 [step 1.3], clause (2) is 2.2 [step 2.2], clause (3) is 2.1 and 3.1 [step 2.1, step 3.1], and clause (4) is 1.4 [step 1.4]; hence the connected finite diagrams are exactly the standard list, the finite Coxeter systems are exactly the direct products of the irreducible types, every listed diagram is positive definite and therefore finite, and the list is duplicate-free after the stated identifications.

5 · Examples, counterexamples and false statements

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Finite Coxeter Diagrams and Complete Classification

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

Coxeter diagrams: edges, labels, components and finite type

Definition

Remarks

Exclusions for positive definite diagrams: trees, valency, labels, chains and arms

Statement

Facts & Assumptions

Proof

Disconnected diagrams, direct products, and comparison of invariant forms

Statement

Facts & Assumptions

Proof

Finiteness criterion: W is finite exactly when the Coxeter form is positive definite

Statement

Facts & Assumptions

Proof

Classification of finite Coxeter systems, including the H and dihedral families

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources