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Affine Coxeter Diagrams and Semidefinite Classification

1 · Prerequisites

2 · Summary

This page defines affine form type by the canonical Coxeter form: the diagram is connected and the form is positive semidefinite of corank one. It proves the positive-radical and proper-submatrix properties, classifies the standard affine diagrams, and constructs the associated Euclidean simplex reflection action. The crystallographic alcove model is matched by a facet-preserving similarity. An infinite Coxeter group is not automatically of affine form type; twisted Lie-theoretic diagrams and extended affine Weyl groups are separate conventions.

The items below are in current dependency order. The simplex-similarity result uses only published prerequisites and supplies the geometric comparison needed later in the classification.

Items

Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet proves that bounded Euclidean simplices with the same inward-unit-normal Gram matrix are similar with labelled facets matched, so their facet-reflection groups are conjugate.
Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice defines affine form type and the radical quotient and affine slice.
Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions proves the positive radical ray, corank one, positive definiteness of proper principal submatrices, finiteness of proper standard parabolics, and the local domination exclusions.
The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde records the standard affine list and its low-rank naming conventions.
The affine slice: faithful isometric action, the alcove simplex, and its facet reflections constructs the faithful Euclidean slice action, simplex, facet reflections, and closed-face intersection rule.
Crystallographic alcove diagrams: the affine list realized by Weyl types A–G computes the crystallographic highest-root data, affine facet Gram matrices, and standard affine realizations.
Enumeration of the connected positive semidefinite corank-one diagrams completes the connected semidefinite corank-one diagram enumeration.
Classification of affine Coxeter diagrams and their Euclidean simplex realization assembles the classification, Euclidean realization, fundamental-domain properties, and conventions for coroot and coweight translation groups.

The page requires the finite Coxeter classification and the affine reflection/coroot-translation theory. Its companion page tests the low-rank, reducible, and indefinite cases. No Kac–Moody classification or twisted Lie-theoretic diagram classification is asserted here.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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

Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet

Statement

Let $I$ be a finite set with $∣ I ∣ = n + 1$ , and let $E, E^{'}$ be Euclidean affine spaces of dimension $n$ (Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis) with direction spaces $V, V^{'}$ (Affine subspaces as translates $x + U$ of linear subspaces). Choose origins in $E, E^{'}$ so points are written in $V, V^{'}$ . Let $σ \subseteq E$ and $σ^{'} \subseteq E^{'}$ be nonempty bounded $n$ -simplices (The geometric simplex spanned by affinely independent vertices) presented as $σ = {φ \in V : ⟨ u_{i}, φ ⟩ \geq c_{i} for all i \in I}, σ^{'} = {φ^{'} \in V^{'} : ⟨ u_{i}^{'}, φ^{'} ⟩ \geq c_{i}^{'} for all i \in I},$ where each half-space defines one of the distinct facets, $u_{i} \in V$ and $u_{i}^{'} \in V^{'}$ are unit inward normals, and $c_{i}, c_{i}^{'} \in R$ . Suppose $⟨ u_{i}, u_{j} ⟩ = ⟨ u_{i}^{'}, u_{j}^{'} ⟩$ for all $i, j \in I$ . Then there are a linear isometry $T : V \to V^{'}$ , a point $y_{0} \in E^{'}$ and a number $t > 0$ such that the similarity $f (φ) = y_{0} + t T (φ)$ satisfies $f (σ) = σ^{'}$ and carries the facet of $σ$ with normal $u_{i}$ onto the facet of $σ^{'}$ with normal $u_{i}^{'}$ for every $i \in I$ . In particular the groups generated by the reflections of $E$ in the facets of $σ$ and of $E^{'}$ in the facets of $σ^{'}$ are conjugate by the similarity $f$ , hence isomorphic as reflection groups acting on Euclidean spaces.

Facts & Assumptions

Given: A finite set $I$ with $∣ I ∣ = n + 1$ ; Euclidean affine spaces $E, E^{'}$ of dimension $n$ with direction spaces $V, V^{'}$ , each identified with its direction space by a fixed origin, so that the points of $E$ and $E^{'}$ are written as vectors of $V$ and $V^{'}$ ; nonempty bounded $n$ -simplices $σ \subseteq E$ , $σ^{'} \subseteq E^{'}$ presented by the half-spaces of the statement, with unit normals $u_{i} \in V$ , $u_{i}^{'} \in V^{'}$ , offsets $c_{i}, c_{i}^{'}$ and facet hyperplanes $H_{i} = {φ : ⟨ u_{i}, φ ⟩ = c_{i}}$ , $H_{i}^{'} = {φ^{'} : ⟨ u_{i}^{'}, φ^{'} ⟩ = c_{i}^{'}}$ ; and the common Gram matrix, $⟨ u_{i}, u_{j} ⟩ = ⟨ u_{i}^{'}, u_{j}^{'} ⟩$ for all $i, j \in I$ .

[F1]

A real inner product space satisfies $⟨ v, v ⟩ \geq 0$ for every $v$ , with $⟨ v, v ⟩ = 0$ only for $v = 0$ , and the induced norm is $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ (Real and complex inner-product spaces and their induced length).

[F2]

A geometric simplex is the convex hull of finitely many affinely independent points, its vertices, and its points are exactly the convex combinations of those vertices (The geometric simplex spanned by affinely independent vertices).

[F3]

A subset $U$ of a Euclidean space is convex when it contains the segment between any two of its points (A convex subset of $R^{m}$ contains every line segment between two of its points).

[F4]

A face of a convex set $K$ is a nonempty convex subset $F \subseteq K$ such that $(1 - t) y + t z \in F$ with $y, z \in K$ and $0 < t < 1$ implies $y, z \in F$ ; so a face is closed under the operation of splitting off vertices of convex combinations (Extreme point and face).

[F5]

The orthogonal complement of a subspace $W$ of an inner product space is $W^{⊥} = {v : ⟨ v, w ⟩ = 0 for every w \in W}$ , a linear subspace (The orthogonal complement $W^{⊥} = {v : ⟨ v, w ⟩ = 0 for all w \in W}$ ).

[F6]

For every subspace $W$ of a finite-dimensional real inner product space $V$ one has $V = W \oplus W^{⊥}$ : every $v \in V$ is uniquely $w + z$ with $w \in W$ and $z \in W^{⊥}$ (For a subspace $W$ of a finite-dimensional inner product space, $V = W \oplus W^{⊥}$ ).

[F7]

A linear map $T : V \to V^{'}$ of inner product spaces is a linear isometry when $∥ T v ∥ = ∥ v ∥$ for all $v$ ; if $T$ preserves inner products then it is a linear isometry by [F1] (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces).

[F8]

Let $A : V \to W$ be linear with $V$ finite-dimensional. There are a basis $K$ of $\ker A$ and a basis $B$ of $V$ with $K \subseteq B$ , and for $C = B ∖ K$ the restriction $A ∣_{C}$ is a bijection onto a basis $A [C]$ of $im A$ ; hence $\dim V = \dim \ker A + \dim im A$ (Extending a basis of the kernel to a basis of the domain gives a basis of the image).

[F9]

If $U$ is a linear subspace of a finite-dimensional space $V$ with $\dim U = \dim V$ , then $U = V$ (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]

A function $T : V \to W$ is linear when $T (a u + b v) = a T (u) + b T (v)$ for all scalars $a, b$ and vectors $u, v$ (Linear map between vector spaces over the same field).

[F11]

A linear map between inner-product spaces is a linear isometry when it preserves the induced norm (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces).

[F12]

A map between metric spaces is an isometry when it preserves distances and is bijective (Isometry, isometric embedding, and the subspace metric on a subset).

Proof

technique · direct; the vertex equations of the two simplices force all coefficients of the unique relation among the normals to have one sign, and the relation then reconstructs the translation

1.1F5F6givenalgebra

(Complements of one normal span.) Fix $k \in I$ and put $W_{k} : = span {u_{j} : j \neq k}$ . If $v \in W_{k}^{⊥}$ , then $⟨ u_{j}, v ⟩ = 0$ for all $j \neq k$ . For $φ \in σ$ and $λ \in R$ , the point $φ + λ v$ satisfies those other inequalities for every $λ$ ; the $k$ -th slack $⟨ u_{k}, φ + λ v ⟩ - c_{k}$ is affine in $λ$ and is nonnegative at $0$ , so if $⟨ u_{k}, v ⟩ \neq 0$ its admissible values contain an unbounded half-line, while if $⟨ u_{k}, v ⟩ = 0$ they are all of $R$ . This contradicts boundedness of $σ$ , so $W_{k}^{⊥} = {0}$ [F5]. By [F6], every $x \in V$ is $w + z$ with $w \in W_{k}$ and $z \in W_{k}^{⊥} = {0}$ ; hence $W_{k} = V$ . Thus every family obtained by omitting one $u_{k}$ spans $V$ , and in particular all the $u_{i}$ span $V$ . The same argument applied to $σ^{'}$ shows that the $u_{i}^{'}$ span $V^{'}$ .

1.2F2F3F4givenalgebra

(Vertices opposite facets.) Write $σ = conv {v_{0}, \dots, v_{n}}$ with affinely independent vertices [F2]. Every face $F$ of $σ$ is the convex hull of the vertices it contains: if $z = \sum_{k} λ_{k} v_{k} \in F$ is a convex combination, a single positive coefficient gives $z = v_{k} \in F$ ; otherwise split off any term with $0 < λ_{k} < 1$ as $z = λ_{k} v_{k} + (1 - λ_{k}) y$ . The face property [F4] puts $v_{k}, y$ in $F$ , and induction on the number of positive coefficients puts every vertex used in $z$ in $F$ . Thus $z \in conv (F \cap {v_{0}, \dots, v_{n}})$ ; the reverse inclusion follows from convexity [F3]. For the facet $F_{i} = σ \cap H_{i}$ , let $S_{i} : = F_{i} \cap {v_{0}, \dots, v_{n}}$ . Then $F_{i} = conv (S_{i})$ and $S_{i}$ is nonempty; its points are affinely independent, so its affine span has dimension $∣ S_{i} ∣ - 1$ (subtracting one point gives $∣ S_{i} ∣ - 1$ linearly independent vectors spanning the direction space). Since this affine span is $H_{i}$ , of dimension $n - 1$ , $∣ S_{i} ∣ = n$ . Hence there is a unique vertex $w_{i}$ outside $F_{i}$ , and $⟨ u_{i}, w_{i} ⟩ > c_{i}$ . If $j \neq i$ and $w_{i} \notin S_{j}$ , then $S_{j} \subseteq S_{i}$ and both have size $n$ , so $S_{j} = S_{i}$ and $F_{j} = F_{i}$ , contradicting that the indexed facets are distinct. Thus $w_{i} \in S_{j}$ , so $⟨ u_{j}, w_{i} ⟩ = c_{j}$ for every $j \neq i$ . Applying the same argument to $σ^{'}$ gives opposite vertices $w_{i}^{'}$ with $⟨ u_{j}^{'}, w_{i}^{'} ⟩ = c_{j}^{'}$ for $j \neq i$ and $⟨ u_{i}^{'}, w_{i}^{'} ⟩ > c_{i}^{'}$ .

2.1F1step 1.1algebra

(A nonzero relation valid for both normal families.) Fix $k \in I$ . By step 1.1, $u_{k} = \sum_{j \neq k} a_{j} u_{j}$ for some real $a_{j}$ ; put $p_{k} : = 1$ and $p_{j} : = - a_{j}$ for $j \neq k$ . Then $p \in R^{I}$ is nonzero and $\sum_{i} p_{i} u_{i} = 0$ , so $p$ is a relation among the normals. The same vector is a relation among the primed normals: $∥ \sum_{i} p_{i} u_{i}^{'} ∥^{2} = \sum_{i, j} p_{i} p_{j} ⟨ u_{i}^{'}, u_{j}^{'} ⟩ = \sum_{i, j} p_{i} p_{j} ⟨ u_{i}, u_{j} ⟩ = ∥ \sum_{i} p_{i} u_{i} ∥^{2} = 0$ , hence $\sum_{i} p_{i} u_{i}^{'} = 0$ by [F1].

2.2F1F7F10step 1.1algebra

(The linear isometry.) Define $T : V \to V^{'}$ on a combination of the $u_{i}$ by $T (\sum_{i} a_{i} u_{i}) : = \sum_{i} a_{i} u_{i}^{'}$ . This is well defined: if $\sum_{i} a_{i} u_{i} = 0$ then $∥ \sum_{i} a_{i} u_{i}^{'} ∥^{2} = \sum_{i, j} a_{i} a_{j} ⟨ u_{i}^{'}, u_{j}^{'} ⟩ = \sum_{i, j} a_{i} a_{j} ⟨ u_{i}, u_{j} ⟩ = ∥ \sum_{i} a_{i} u_{i} ∥^{2} = 0$ , so $\sum_{i} a_{i} u_{i}^{'} = 0$ by [F1]. By construction $T$ is linear in the sense of [F10], satisfies $T (u_{i}) = u_{i}^{'}$ , and preserves inner products: $⟨ T (\sum_{i} a_{i} u_{i}), T (\sum_{j} b_{j} u_{j}) ⟩ = \sum_{i, j} a_{i} b_{j} ⟨ u_{i}^{'}, u_{j}^{'} ⟩ = \sum_{i, j} a_{i} b_{j} ⟨ u_{i}, u_{j} ⟩ = ⟨ \sum_{i} a_{i} u_{i}, \sum_{j} b_{j} u_{j} ⟩$ ; so $T$ is a linear isometry [F1, F7, F10]. Since the $u_{i}^{'}$ span $V^{'}$ [step 1.1], the image of $T$ is $V^{'}$ , and $T$ is bijective.

3.1step 2.1step 1.2algebra

(Sign consistency and the two offset sums.) Put $R : = \sum_{i} p_{i} c_{i}$ and $R^{'} : = \sum_{i} p_{i} c_{i}^{'}$ . For each $i$ , using $⟨ u_{j}, w_{i} ⟩ = c_{j}$ for $j \neq i$ from step 1.2 and $\sum_{k} p_{k} u_{k} = 0$ from step 2.1, $0 = ⟨ \sum_{k} p_{k} u_{k}, w_{i} ⟩ = p_{i} ⟨ u_{i}, w_{i} ⟩ + \sum_{k \neq i} p_{k} c_{k} = p_{i} (⟨ u_{i}, w_{i} ⟩ - c_{i}) + R$ . Since $⟨ u_{i}, w_{i} ⟩ - c_{i} > 0$ [step 1.2], the number $R$ is nonzero and $p_{i}$ has the sign of $- R$ , for every $i$ : all coefficients $p_{i}$ are nonzero and of one sign. Applying the same computation to the primed simplex, using the opposite vertices from step 1.2 and the primed relation from step 2.1, gives $p_{i} (⟨ u_{i}^{'}, w_{i}^{'} ⟩ - c_{i}^{'}) = - R^{'}$ for every $i$ , so $R^{'} \neq 0$ and $p_{i}$ has the sign of $- R^{'}$ . Thus $R^{'}$ and $R$ have the same sign, and $t : = R^{'} / R > 0$ .

4.1F5F8F9step 1.1step 2.1step 3.1algebra

(The translation solving the offset equations.) Put $Δ : = (c_{i}^{'} - t c_{i})_{i \in I} \in R^{I}$ . By step 3.1, $\sum_{i} p_{i} Δ_{i} = R^{'} - t R = 0$ , i.e. $Δ ⊥ p$ [F5]. Let $A : V^{'} \to R^{I}$ be the linear map $A (y) : = (⟨ u_{i}^{'}, y ⟩)_{i \in I}$ . Its kernel is trivial: if $A (y) = 0$ then $⟨ u_{i}^{'}, y ⟩ = 0$ for all $i$ , so $y ⊥ span {u_{i}^{'}} = V^{'}$ by step 1.1 and $y = 0$ by [F1]. By [F8] (applied to the coordinate basis of $R^{I}$ , of $∣ I ∣ = n + 1$ elements) $\dim im A = \dim V^{'} = n$ ; the kernel of the nonzero functional $R^{I} \to R$ , $x \mapsto \sum_{i} p_{i} x_{i}$ , is $p^{⊥}$ . It is nonzero because $p \neq 0$ [step 2.1], so its image is the one-dimensional space $R$ ; [F8] therefore gives $\dim p^{⊥} = (n + 1) - 1 = n$ . Moreover $im A \subseteq p^{⊥}$ , because for $x = A (y)$ one has $\sum_{i} p_{i} x_{i} = ⟨ \sum_{i} p_{i} u_{i}^{'}, y ⟩ = 0$ by step 2.1. As $im A$ and $p^{⊥}$ are subspaces of $p^{⊥}$ of the same dimension $n$ , [F9] gives $im A = p^{⊥}$ ; since $Δ ⊥ p$ , there is $y_{0} \in V^{'}$ with $⟨ u_{i}^{'}, y_{0} ⟩ = c_{i}^{'} - t c_{i}$ for every $i \in I$ .

5.1step 2.2step 4.1givenalgebra

(The similarity and the facet correspondence.) For $φ \in V$ and $i \in I$ , using $T (u_{i}) = u_{i}^{'}$ and inner-product preservation of step 2.2 and the point $y_{0}$ of step 4.1, $⟨ u_{i}^{'}, y_{0} + t T (φ) ⟩ = (c_{i}^{'} - t c_{i}) + t ⟨ u_{i}, φ ⟩$ . Hence $y_{0} + t T (φ) \in σ^{'}$ if and only if $⟨ u_{i}, φ ⟩ \geq c_{i}$ for all $i$ , that is, if and only if $φ \in σ$ ; so $f (φ) : = y_{0} + t T (φ)$ maps $σ$ onto $σ^{'}$ , and $⟨ u_{i}^{'}, f (φ) ⟩ = c_{i}^{'}$ if and only if $⟨ u_{i}, φ ⟩ = c_{i}$ , so $f$ carries the facet $F_{i}$ of $σ$ onto the facet $F_{i}^{'}$ of $σ^{'}$ . Since $t > 0$ and $T$ is a linear isometry, $f$ is a similarity.

6.1F11F12step 2.2step 4.1step 5.1algebra∎

(Conjugating the facet reflections.) For $i \in I$ define $s_{i} : V \to V$ by $s_{i} (φ) : = φ + 2 (c_{i} - ⟨ u_{i}, φ ⟩) u_{i}$ . Its linear part $r_{i} (φ) : = φ - 2 ⟨ u_{i}, φ ⟩ u_{i}$ preserves the norm: using $∥ u_{i} ∥ = 1$ and bilinearity, $∥ r_{i} (φ) ∥^{2} = ∥ φ ∥^{2} - 4 ⟨ u_{i}, φ ⟩^{2} + 4 ⟨ u_{i}, φ ⟩^{2} ∥ u_{i} ∥^{2} = ∥ φ ∥^{2}$ . It fixes $u_{i}^{⊥}$ and sends $u_{i}$ to $- u_{i}$ , so it is the reflection across $u_{i}^{⊥}$ and is a linear isometry [F11]. Since $s_{i} = r_{i} + 2 c_{i} u_{i}$ , it preserves distances; direct substitution gives $s_{i}^{2} = i d$ , and $s_{i}$ fixes $H_{i}$ pointwise. Thus $s_{i}$ is the affine reflection in $H_{i}$ and is an isometry [F12]. Let $s_{i}^{'}$ be the corresponding reflection of $E^{'}$ defined by the primed data. Then for every $ψ \in V^{'}$ , writing $f^{- 1} (ψ) = T^{- 1} ((ψ - y_{0}) / t)$ and using $⟨ u_{i}, f^{- 1} (ψ) ⟩ = (⟨ u_{i}^{'}, ψ ⟩ - c_{i}^{'} + t c_{i}) / t$ , $f (s_{i} (f^{- 1} (ψ))) = ψ + 2 t (c_{i} - ⟨ u_{i}, f^{- 1} (ψ) ⟩) u_{i}^{'} = ψ + 2 (c_{i}^{'} - ⟨ u_{i}^{'}, ψ ⟩) u_{i}^{'} = s_{i}^{'} (ψ)$ . Thus $f \circ s_{i} \circ f^{- 1} = s_{i}^{'}$ for every $i$ , and the map $C_{f} : g \mapsto f \circ g \circ f^{- 1}$ carries the group generated by the $s_{i}$ onto the group generated by the $s_{i}^{'}$ . It preserves composition since $C_{f} (g \circ h) = C_{f} (g) \circ C_{f} (h)$ , and its inverse is $h \mapsto f^{- 1} \circ h \circ f$ ; hence it is an isomorphism of the two reflection groups.

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

Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice

Definition

Let $S$ be finite, let $m$ be a Coxeter matrix on $S$ , let $W$ be the presented Coxeter group (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), let $Γ$ be its Coxeter diagram (Coxeter diagrams: edges, labels, components and finite type), and let $V = R^{S}$ carry the real Coxeter form $B$ (The real Coxeter form, its radical, reflections, and form-preserving maps): $B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite labels, and $B (e_{s}, e_{t}) = - 1$ when $m (s, t) = \infty$ . Put $rad (B) : = {v \in V : B (v, w) = 0 for every w \in V}$ (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space).

(1) Affine form type. The system $(W, S)$ , and with it $m$ , $Γ$ and $B$ , is of affine form type when $Γ$ is connected and $B$ is positive semidefinite (meaning $B (v, v) \geq 0$ for every $v \in V$ ) of corank one, that is, $\dim_{R} rad (B) = 1$ (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis). By Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions ↗ (1), this is equivalent to requiring that $Γ$ be connected, $B$ positive semidefinite, and $B$ not positive definite. This is the only notion called affine on this page. It is a condition on the Coxeter form, not a synonym for an infinite abstract Coxeter group. Disconnected systems are outside this definition; the companion examples page treats reducible forms separately.

(2) Positive radical vector. A positive radical vector for affine form type is a vector $δ = \sum_{s \in S} δ_{s} e_{s} \in rad (B)$ with $δ_{s} > 0$ for every $s \in S$ . For affine form type such a vector exists and $rad (B) = R δ$ , by Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions ↗ (1); this clause names the object and records the property supplied by that lemma.

(3) Radical quotient. Let $U : = V / rad (B)$ (The quotient vector space $V / W$ and its canonical projection), and define $b (\overset{ˉ}{v}, \overset{ˉ}{w}) : = B (v, w)$ . The form $b$ is well-defined, symmetric and positive definite. Thus $U$ is a Euclidean vector space (Real and complex inner-product spaces and their induced length) of dimension $∣ S ∣ - 1$ (Linear subspace of a vector space, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

(4) Affine slice and walls. For a positive radical vector $δ$ , put $E_{δ} : = {φ \in V^{*} : φ (δ) = 1},$ where $V^{*}$ is the algebraic dual (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ). This is an affine subspace (Affine subspaces as translates $x + U$ of linear subspaces) with direction space $K_{δ} : = {φ \in V^{*} : φ (δ) = 0} = Ann (rad (B)) .$ Precomposition with the quotient projection identifies $K_{δ}$ with $U^{*}$ . Let $b^{♭} : U \to U^{*}$ be $b^{♭} (u) : = b (u, \cdot)$ (Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V \to V^{*}$ ); it is an isomorphism, as verified in the remarks, and the dual form is $b^{*} (α, β) : = b ((b^{♭})^{- 1} α, (b^{♭})^{- 1} β) .$ Transporting $b^{*}$ to the direction space $K_{δ}$ makes $E_{δ}$ a Euclidean affine space of dimension $∣ S ∣ - 1$ . For $s \in S$ , its wall is $H_{s} : = {φ \in E_{δ} : φ (e_{s}) = 0}$ , its open alcove is $A : = {φ \in E_{δ} : φ (e_{s}) > 0 for every s \in S}$ , and $\overset{ˉ}{A}$ denotes the closure of $A$ in $E_{δ}$ .

(5) Abstentions and conventions. Beyond the constructions and properties above, this definition does not assert that the $H_{s}$ are affine hyperplanes, that $A$ is nonempty, bounded or a simplex, that $W$ acts on $E_{δ}$ as a group generated by affine reflections, or that the translates $w \overset{ˉ}{A}$ cover $E_{δ}$ . The phrase affine form type refers only to the Coxeter-matrix condition in (1); extended affine Weyl groups and the directed affine Dynkin diagrams used in Lie theory are separate conventions and are not defined here. No choice principle is used in these finite-dimensional constructions.

Remarks

The quotient-form assertions in (3) follow directly from positive semidefiniteness. If $r \in rad (B)$ , then replacing either representative by one differing by a radical vector does not change $B (v, w)$ , so $b$ is well-defined; symmetry is inherited from $B$ . If $b (\overset{ˉ}{v}, \overset{ˉ}{v}) = 0$ , then for every $w \in V$ and every $t \in R$ , $0 \leq B (v + t w, v + t w) = 2 t B (v, w) + t^{2} B (w, w) .$ Both signs of arbitrarily small $t$ force $B (v, w) = 0$ . This holds for every $w$ , so $v \in rad (B)$ and $\overset{ˉ}{v} = 0$ .

The dimension claim in (3) is also explicit. Since the radical has dimension one, choose a nonzero $r = \sum_{s} r_{s} e_{s}$ in it and an index $s_{0}$ with $r_{s_{0}} \neq 0$ . The classes ${\overset{ˉ}{e}}_{s}$ for $s \neq s_{0}$ span $U$ , because ${\overset{ˉ}{e}}_{s_{0}} = - \sum_{s \neq s_{0}} (r_{s} / r_{s_{0}}) {\overset{ˉ}{e}}_{s}$ . They are independent: if $\sum_{s \neq s_{0}} a_{s} e_{s}$ lies in $R r$ , its $s_{0}$ coordinate forces that multiple of $r$ to be zero, and then every $a_{s} = 0$ . They are therefore a basis of size $∣ S ∣ - 1$ .

For (4), $E_{δ}$ is nonempty: choose $s_{0}$ with $δ_{s_{0}} > 0$ and use the coordinate functional $φ_{0} (v) : = v_{s_{0}} / δ_{s_{0}}$ . Its direction is $K_{δ}$ , which equals the annihilator of $rad (B)$ because that radical is the line $R δ$ . The map $U^{*} \to K_{δ}$ , $λ \mapsto λ \circ π$ , with $π : V \to U$ the quotient projection, is injective since $π$ is onto; it is surjective because each functional in $K_{δ}$ vanishes on the radical and therefore factors through $π$ . A basis of $U$ gives dual coordinate functionals: each functional is determined by its values on that basis, and arbitrary values extend linearly. Thus they form a basis of $U^{*}$ and $\dim U^{*} = \dim U = ∣ S ∣ - 1$ . Finally, $b^{♭}$ is injective: if $b^{♭} (u) = 0$ , then $b (u, u) = 0$ , hence $u = 0$ by positive definiteness. The images under $b^{♭}$ of a basis of $U$ are therefore $∣ S ∣ - 1$ independent vectors in the $∣ S ∣ - 1$ dimensional space $U^{*}$ , so they form a basis and $b^{♭}$ is an isomorphism. The displayed formula therefore defines a positive-definite inner product on $U^{*}$ and hence on $K_{δ}$ . All selections above are single finite-dimensional constructions; no arbitrary choice principle is used.

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

Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions

Statement

Let $S$ be a finite set with Coxeter matrix $m$ , let $W$ be the presented group (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), let $Γ$ be its diagram (Coxeter diagrams: edges, labels, components and finite type), and let $V = R^{S}$ carry the Coxeter form $B$ (The real Coxeter form, its radical, reflections, and form-preserving maps). Assume that $Γ$ is connected, that $B (e_{s}, e_{t}) \leq 0$ for distinct $s, t$ , and that $B$ is positive semidefinite (Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form) with $rad (B) \neq {0}$ . These are the raw hypotheses; no corank-one condition is assumed.

(1) Positive radical and corank one. If $x = \sum_{s} x_{s} e_{s} \in rad (B)$ and $x \neq 0$ , then $x_{s} \neq 0$ for every $s \in S$ . The set of $x \in rad (B)$ with $x_{s} > 0$ for all $s$ is nonempty and consists of the positive multiples of one vector $δ$ ; in particular $\dim rad (B) = 1$ and $rad (B) = R δ$ . [No Perron-Frobenius theorem is used.]

(2) Proper principal submatrices and parabolics. For every proper subset $T ⊊ S$ , the principal submatrix $(B (e_{s}, e_{t}))_{s, t \in T}$ is positive definite, with the $T = \emptyset$ case vacuous. For nonempty $T$ , the standard parabolic $W_{T} : = ⟨ s : s \in T ⟩$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups) has the Coxeter presentation with restricted matrix $m ∣_{T \times T}$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2)); its Coxeter form is the displayed principal submatrix, so $W_{T}$ is finite by Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1). For $T = \emptyset$ , $W_{T} = {1}$ is finite by definition.

(3) Domination. Let $T \subseteq S$ and let $Γ^{'}$ be a Coxeter diagram on $T$ , with edge labels in ${3, 4, \dots} \cup {\infty}$ , whose underlying graph is a subgraph of the induced subdiagram $Γ_{T}$ and whose retained edge labels are no larger than the corresponding labels of $Γ_{T}$ . For a nonedge use label $2$ ; put $m^{'} (s, s) = 1$ , and let $B_{Γ^{'}}$ be the symmetric form on $R^{T}$ with $B_{Γ^{'}} (e_{s}, e_{s}) = 1$ , $B_{Γ^{'}} (e_{s}, e_{t}) = - \cos (π / m^{'} (s, t))$ for finite $m^{'} (s, t)$ , and $B_{Γ^{'}} (e_{s}, e_{t}) = - 1$ when $m^{'} (s, t) = \infty$ . Thus nonedges have entry $0$ . If $Γ^{'} \neq Γ_{T}$ (a strict instance of the usual label/subgraph domination relation), then its cosine matrix is positive definite. Here the relation is applied to $Γ_{T}$ even when it is disconnected; when $T = S$ , it is the relation of [Davis, Appendix C.3]. Consequently, if the cosine matrix of some such $Γ^{'}$ is not positive definite - for instance if it has a nonzero vector $v$ with $B_{Γ^{'}} (v, v) \leq 0$ , or, when $T \neq \emptyset$ , if its determinant is $\leq 0$ while some proper principal submatrix is positive definite (Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix) - then $Γ_{T}$ cannot strictly dominate $Γ^{'}$ .

(4) Use. Clauses (1)-(3) supply the positive-radical structure and domination exclusion used to enumerate connected diagrams of the affine form type defined in Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (1). This reference supplies terminology only: the hypotheses above are raw and the proof does not assume corank one.

Facts & Assumptions

Given: A finite set $S$ with Coxeter matrix $m$ , presented group $W$ , diagram $Γ$ (connected) and the form $B$ on $V = R^{S}$ , with $B (e_{s}, e_{s}) = 1$ , $B (e_{s}, e_{t}) \leq 0$ for $s \neq t$ and $B (e_{s}, e_{t}) = - 1$ exactly when $m (s, t) = \infty$ ; $B$ is positive semidefinite, $B (v, v) \geq 0$ for all $v$ , and $rad (B) \neq {0}$ .

[F1]

$B$ is symmetric and bilinear, with $B (e_{s}, e_{s}) = 1$ , the stated cosine entries, and the given inequalities $B (e_{s}, e_{t}) \leq 0$ for $s \neq t$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms). The coordinate functions $e_{s}$ form a basis of $V = R^{S}$ : each $u$ is $\sum_{s} u (s) e_{s}$ , and evaluation at each index gives uniqueness.

[F2]

The radical $rad (B) = {v : B (v, w) = 0 for all w \in V}$ is closed under linear combinations by bilinearity, hence is a linear subspace; $B$ vanishes on $rad (B) \times V$ (The real Coxeter form, its radical, reflections, and form-preserving maps, The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space, Linear subspace of a vector space).

[F3]

By the hypothesis of positive semidefiniteness, $B (v, v) \geq 0$ for every $v$ ; positive definiteness means $B (v, v) > 0$ 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).

[F4]

Two vertices $s \neq t$ of $Γ$ are adjacent exactly when $m (s, t) \geq 3$ (Coxeter diagrams: edges, labels, components and finite type).

[F5]

For a real symmetric $n \times n$ matrix with $n \geq 1$ , positive definiteness is equivalent to positivity of all leading principal minors; in particular, a nonpositive determinant rules out positive definiteness (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, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[F7]

Cosine is strictly decreasing on $[0, π]$ (Signs, monotonicity intervals, and ranges of sine and cosine).

[F8]

The singleton ${δ}$ is a basis of the line $R δ$ , so that this line has dimension $1$ (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 $S$ , Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

[F9]

The standard parabolic is $W_{T} = ⟨ s : s \in T ⟩$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups); $⟨ \emptyset ⟩ = {1}$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups); and the group presented by the restricted matrix maps isomorphically to $W_{T}$ , making $(W_{T}, T)$ a Coxeter system (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (2)).

[F10]

For a Coxeter system, its group is finite if and only if its Coxeter form is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)).

[F11]

Connectedness of $Γ$ means its underlying graph is connected (Coxeter diagrams: edges, labels, components and finite type).

[F12]

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

Proof

technique · direct; the $|x|$ trick replaces Perron-Frobenius, and the domination clause is the equality case of a three-term comparison. All sums and choices are finite; no choice principle is used

1.1F1algebra

(The absolute-value inequality.) For $x = \sum_{s} x_{s} e_{s} \in V$ put $q (x) : = B (x, x)$ and $y : = \sum_{s} ∣ x_{s} ∣ e_{s}$ . Expanding in the basis $(e_{s})$ , $q (x) - q (y) = 2 \sum_{{s, t} \in (\binom{S}{2})} B (e_{s}, e_{t}) (x_{s} x_{t} - ∣ x_{s} ∣ ∣ x_{t} ∣)$ , where the sum runs over unordered pairs of distinct indices. Each summand is $\geq 0$ because $B (e_{s}, e_{t}) \leq 0$ and $x_{s} x_{t} \leq ∣ x_{s} x_{t} ∣ = ∣ x_{s} ∣ ∣ x_{t} ∣$ ; hence $q (y) \leq q (x)$ [F1].

1.2F2F3F1algebra

(The radical of a positive semidefinite form.) If $q (v) = 0$ then $v \in rad (B)$ : for every $w \in V$ and every $t \in R$ one has $0 \leq q (v + t w) = 2 t B (v, w) + t^{2} q (w)$ . If $q (w) > 0$ then choosing $t = - B (v, w) / q (w)$ gives $- B (v, w)^{2} / q (w) \geq 0$ , so $B (v, w) = 0$ ; if $q (w) = 0$ then $2 t B (v, w) \geq 0$ for every $t$ , so again $B (v, w) = 0$ .

1.3F1F2F4F7F11F12algebra

(Propagation of zeros along the diagram.) Let $y = \sum_{s} y_{s} e_{s} \in rad (B)$ with $y_{s} \geq 0$ for all $s$ , and suppose $y_{i} = 0$ for some $i \in S$ . Since $B$ is symmetric and $y$ is radical, $0 = B (e_{i}, y) = \sum_{t} B (e_{i}, e_{t}) y_{t}$ [F1, F2]. Each summand is $\leq 0$ : this follows from $y_{t} \geq 0$ and $B (e_{i}, e_{t}) \leq 0$ for $t \neq i$ , while the $t = i$ term is $1 \cdot 0 = 0$ [F1]. Hence every summand is $0$ ; in particular $y_{j} = 0$ for every neighbour $j$ of $i$ , because neighbours have $B (e_{i}, e_{j}) < 0$ by [F1, F4, F7, F12]. Iterating along the connected diagram $Γ$ [F11] gives $y_{t} = 0$ for all $t \in S$ .

2.1F3step 1.1step 1.2step 1.3algebra

(Full support of nonzero radical vectors.) Let $x = \sum_{s} x_{s} e_{s} \in rad (B)$ , $x \neq 0$ , and put $y : = \sum_{s} ∣ x_{s} ∣ e_{s}$ . By step 1.1, $q (y) \leq q (x) = 0$ , and by positive semidefiniteness $q (y) \geq 0$ , so $q (y) = 0$ [F3]; by step 1.2, $y \in rad (B)$ , and $y \neq 0$ because $x \neq 0$ . If some $x_{i}$ were $0$ , then $y_{i} = 0$ and step 1.3 would give $y = 0$ , a contradiction. Hence $x_{s} \neq 0$ for every $s \in S$ .

3.1F1F2step 2.1step 1.3algebra

(Existence, uniqueness and full span of the positive radical vector.) Replacing any nonzero $x \in rad (B)$ by $y = \sum_{s} ∣ x_{s} ∣ e_{s}$ gives a nonzero radical vector with nonnegative coordinates; step 1.3 shows every coordinate is positive. Thus the positive radical set is nonempty. If $x, x^{'}$ both belong to it and were linearly independent, let $t_{*} : = \min_{s} x_{s} / x_{s}^{'} > 0$ and $w : = x - t_{*} x^{'}$ . Bilinearity shows $w \in rad (B)$ ; some coordinate of $w$ is $0$ , and $w \neq 0$ , contradicting step 2.1. Hence any two positive radical vectors are positive scalar multiples. Fix one such vector $δ$ . For an arbitrary $z = \sum_{s} z_{s} e_{s} \in rad (B)$ , if some $z_{s} < 0$ put $M : = \sum_{z_{s} < 0} (- z_{s} / δ_{s}) > 0$ and $ε : = 1 / (2 M)$ ; otherwise put $ε : = 1$ . In either case every coordinate of $δ + ε z$ is positive. Bilinearity puts this vector in $rad (B)$ , so the uniqueness just proved gives $δ + ε z = t δ$ for some $t > 0$ . Therefore $z = ((t - 1) / ε) δ$ . This proves $rad (B) = R δ$ and $\dim rad (B) = 1$ by [F8].

3.2F1F3step 1.2step 2.1algebra

(Proper principal submatrices are positive definite.) Let $T ⊊ S$ and let $u \in R^{T}$ be nonzero. Pad its coordinates by zero to obtain $\tilde{u} \in V$ . Since $B$ is positive semidefinite, $B (\tilde{u}, \tilde{u}) \geq 0$ . If equality held, step 1.2 would put $\tilde{u}$ in $rad (B)$ ; it is nonzero and has a zero coordinate outside $T$ , contradicting step 2.1. Hence $B (\tilde{u}, \tilde{u}) > 0$ for every nonzero $u$ , exactly positive definiteness of the principal submatrix. If $T = \emptyset$ , this condition is vacuous and the zero-dimensional form is positive definite by definition.

3.3F1F2F3F4F7step 1.2step 2.1algebra

(Domination.) Let $Γ^{'}$ , $T$ and $B_{Γ^{'}}$ be as in (3), and write $A^{'}$ for its cosine matrix. Let $A$ be the matrix of $B$ ; after reordering the vertices so that $T$ comes first, $A^{'}$ is indexed by the same first $∣ T ∣$ vertices. For an edge of $Γ^{'}$ with label $m^{'} \leq m$ , the entry is $- \cos (π / m^{'}) \geq - \cos (π / m)$ by [F7]; if the labels differ, the inequality is strict, including the convention $π / \infty = 0$ . For a pair not joined in $Γ^{'}$ , its entry is $0 \geq A_{s t}$ . Thus $A_{s t} \leq A_{s t}^{'} \leq 0$ for all distinct $s, t \in T$ , while both diagonals are $1$ [F1, F4]. Suppose $A^{'}$ is not positive definite. By [F3], some nonzero $x \in R^{T}$ satisfies $x^{T} A^{'} x \leq 0$ . Pad $z : = (∣ x_{s} ∣)_{s \in T}$ by zero outside $T$ . Then $0 \leq z^{T} A z \leq \sum_{s, t \in T} A_{s t}^{'} ∣ x_{s} ∣ ∣ x_{t} ∣ \leq x^{T} A^{'} x \leq 0$ . The first inequality is positive semidefiniteness; the second follows termwise from $A_{s t} \leq A_{s t}^{'}$ and nonnegative coordinate products; the third follows termwise because $A_{s t}^{'} \leq 0$ off the diagonal and $x_{s} x_{t} \leq ∣ x_{s} ∣ ∣ x_{t} ∣$ . Equality throughout gives $B (z, z) = 0$ , so $z \in rad (B)$ by step 1.2. Since $z \neq 0$ , step 2.1 forces every coordinate of $z$ to be nonzero, hence $T = S$ and every $x_{s} \neq 0$ . Equality in the second inequality then forces $A_{s t} = A_{s t}^{'}$ for every distinct pair, so the strict monotonicity in [F7] gives the same edges and labels: $Γ^{'} = Γ_{T}$ , contrary to the hypothesis. Therefore $A^{'}$ is positive definite.

4.1F1F9F10step 3.2algebra

(The proper standard parabolics are finite.) Let $T ⊊ S$ . If $T \neq \emptyset$ , step 3.2 makes its restricted Coxeter form positive definite, and [F9] identifies $(W_{T}, T)$ as a Coxeter system with that restricted form; [F10] then gives that $W_{T}$ is finite. If $T = \emptyset$ , [F9] gives $W_{T} = {1}$ , also finite.

5.1F3F5step 3.3algebra∎

(The exclusion consequence.) If $A^{'}$ has a nonzero vector $v$ with $B_{Γ^{'}} (v, v) \leq 0$ , then it is not positive definite by definition [F3]. If $T \neq \emptyset$ and $\det A^{'} \leq 0$ , then its last leading principal minor is nonpositive, so $A^{'}$ is not positive definite by Sylvester's criterion [F5]; this also covers the statement's example that additionally assumes a proper principal submatrix is positive definite. By step 3.3, neither obstruction is compatible with strict domination.

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

The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde

Definition

With the Coxeter-diagram conventions of Coxeter diagrams: edges, labels, components and finite type (a finite simple graph with edge labels in ${3, 4, 5, \dots} \cup {\infty}$ , label $3$ omitted), the standard affine diagrams are the following labelled graphs. Within each clause, distinct vertex symbols denote distinct vertices; pairs not listed as edges are nonedges and have Coxeter label $2$ .

(1) The A family. ${\tilde{A}}_{1}$ is the graph with two vertices joined by one edge labelled $\infty$ . For $n \geq 2$ , ${\tilde{A}}_{n}$ is the cycle $v_{0}, v_{1}, \dots, v_{n}$ on $n + 1$ vertices, with edges ${v_{i}, v_{i + 1}}$ for $0 \leq i < n$ and ${v_{n}, v_{0}}$ , all labelled $3$ (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges). Thus the family " ${\tilde{A}}_{n}$ , $n \geq 1$ " is ${\tilde{A}}_{1}$ for $n = 1$ and the $(n + 1)$ -cycle for $n \geq 2$ .

(2) The B family. For $n \geq 3$ , ${\tilde{B}}_{n}$ has vertices $v_{0}, v_{1}, \dots, v_{n}$ and edges ${v_{0}, v_{2}}$ and ${v_{1}, v_{2}}$ with label $3$ , together with the path edges ${v_{i}, v_{i + 1}}$ for $2 \leq i \leq n - 1$ , where ${v_{n - 1}, v_{n}}$ carries the label $4$ and every other edge the label $3$ . Thus ${\tilde{B}}_{n}$ is the path $v_{1} - \dots - v_{n}$ with the $B_{n}$ labels $3, \dots, 3, 4$ , with one extra vertex $v_{0}$ attached to $v_{2}$ by a $3$ -edge. (The recipe is stated for $n \geq 3$ ; for $n = 2$ the family is defined by the convention ${\tilde{B}}_{2} : = {\tilde{C}}_{2}$ of (7), the three-vertex path with both edges labelled $4$ .)

(3) The C family. For $n \geq 2$ , ${\tilde{C}}_{n}$ is the path on $n + 1$ vertices $v_{0} - v_{1} - \dots - v_{n}$ with label $4$ on the two end edges ${v_{0}, v_{1}}$ and ${v_{n - 1}, v_{n}}$ and label $3$ on all other edges.

(4) The D family. ${\tilde{D}}_{4}$ is the star with one centre and four leaves (four arms of length $1$ ), all labels $3$ . For $n \geq 5$ , ${\tilde{D}}_{n}$ has two branch vertices $a, b$ joined by a chain $a = w_{0}, w_{1}, \dots, w_{n - 4} = b$ with exactly $n - 4$ edges, with two leaves attached to $a$ and two leaves attached to $b$ ; all labels $3$ . (The chain has $n - 3$ vertices, so the total number of vertices is $(n - 3) + 4 = n + 1$ .)

(5) The E family. A star with arms $(p, q, r)$ is a tree with one vertex of degree $3$ and three paths (arms) of $p$ , $q$ , $r$ edges from it (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree, Connected graphs and connected components defined by the existence of vertex paths). Then ${\tilde{E}}_{6}$ is the star with arms $(2, 2, 2)$ (seven vertices), ${\tilde{E}}_{7}$ the star with arms $(1, 3, 3)$ (eight vertices), and ${\tilde{E}}_{8}$ the star with arms $(1, 2, 5)$ (nine vertices); all labels are $3$ .

(6) The F and G families. ${\tilde{F}}_{4}$ is the path on five vertices with labels $(3, 3, 4, 3)$ ; ${\tilde{G}}_{2}$ is the path on three vertices with labels $(3, 6)$ .

(7) Coincidences. The convention ${\tilde{B}}_{2} : = {\tilde{C}}_{2}$ gives the three-vertex path with both edges labelled $4$ . The finite diagrams $B_{2}$ , $C_{2}$ and $I_{2} (4)$ are each the two-vertex graph with one edge labelled $4$ ; their coincidence as finite Coxeter systems is Classification of finite Coxeter systems, including the H and dihedral families (4). With the naming conventions ${\tilde{C}}_{1} : = {\tilde{A}}_{1}$ , $D_{3} : = A_{3}$ (the finite coincidence $A_{3} = D_{3}$ of (4) of that theorem), $E_{4} : = A_{4}$ and $E_{5} : = D_{5}$ , one has ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ and ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ . Apart from these identifications the diagrams (1)-(6) are pairwise non-isomorphic as labelled graphs (Graph isomorphisms, automorphisms and graph complements); the verification is clause (7) of Crystallographic alcove diagrams: the affine list realized by Weyl types A–G ↗. ${\tilde{A}}_{1}$ is the only diagram of the list with a label $\infty$ , and every other label in the list lies in ${3, 4, 6}$ .

Remarks

Each recipe gives a finite labelled simple graph. The bounds $n \geq 3$ in the ${\tilde{B}}_{n}$ recipe and $n \geq 2$ in the ${\tilde{C}}_{n}$ recipe make the terminal $4$ -edge distinct from the branch edges and make the two ${\tilde{C}}_{n}$ end edges distinct, respectively. The low-rank aliases in (7) complete the family naming. Every vertex and edge is explicitly specified, so no choice principle is used.

Remarks

The list is presented once, here, so that every later item, the classification theorem and the companion examples page use the same names and the same low-rank conventions. The identifications of (7) are conventions about which family member a diagram belongs to; they are not claims that the corresponding Coxeter systems are isomorphic as abstract Coxeter groups, and no such claim is used on this page.

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The affine slice: faithful isometric action, the alcove simplex, and its facet reflections

Statement

Let $(W, S, m, V, B, ρ, Γ)$ be of affine form type, and let $δ = \sum_{s \in S} δ_{s} e_{s}$ be positive with $rad (B) = R δ$ (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (1)-(4), Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (1)). Put $U = V / rad (B)$ with its quotient Euclidean form $b$ , and let $E = E_{δ}$ , $H_{s}$ , $A$ , and $\overset{ˉ}{A}$ be as in Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (3)-(4). Let $ρ^{*} (w) φ = φ \circ ρ (w)^{- 1}$ be the dual action (The dual action, chambers, faces, and root hyperplanes (1)); give the direction space $K_{δ}$ of $E$ its dual Euclidean form $b^{*}$ from (4). For each $s \in S$ let ${\overset{ˉ}{e}}_{s}$ be the image of $e_{s}$ in $U$ .

(1) The action. Every $ρ^{*} (w)$ preserves $E$ and acts there by an affine isometry. It sends $H_{s}$ to ${φ \in E : φ (ρ (w) e_{s}) = 0}$ and permutes the family of alcove translates ${ρ^{*} (w) A : w \in W}$ . The induced homomorphism $W \to Isom (E)$ is faithful.

(2) The affine relation. For every $φ \in E$ , $\sum_{s \in S} δ_{s} φ (e_{s}) = φ (δ) = 1.$

(3) The alcove is a simplex. For each $s \in S$ there is exactly one point $v_{s} \in E$ with $v_{s} (e_{t}) = 0$ for $t \neq s$ , and it satisfies $v_{s} (e_{s}) = 1 / δ_{s}$ and $v_{s} \in \overset{ˉ}{A}$ . The points $(v_{s})_{s \in S}$ are affinely independent and $\overset{ˉ}{A} = conv {v_{s} : s \in S}$ is a Euclidean simplex of dimension $∣ S ∣ - 1$ (The geometric simplex spanned by affinely independent vertices). Its facets are $\overset{ˉ}{A} \cap H_{s}$ . The inward unit normal to the facet $\overset{ˉ}{A} \cap H_{s}$ is $b^{♭} ({\overset{ˉ}{e}}_{s}) \in K_{δ}$ , equivalently ${\overset{ˉ}{e}}_{s}$ under the identification $K_{δ} ≅ U$ ; its Gram matrix is $b^{*} (b^{♭} ({\overset{ˉ}{e}}_{s}), b^{♭} ({\overset{ˉ}{e}}_{t})) = b ({\overset{ˉ}{e}}_{s}, {\overset{ˉ}{e}}_{t}) = B (e_{s}, e_{t}) .$ In particular, $A$ is nonempty and $\overset{ˉ}{A}$ is compact.

(4) Facet reflections. For every $s \in S$ , $ρ^{*} (s) ∣_{E}$ is the reflection in $H_{s}$ ; thus the action on $E$ is generated by the $∣ S ∣$ facet reflections of $\overset{ˉ}{A}$ . Every point of $A$ has trivial stabilizer.

(5) Alcove intersections. If $w \neq u$ , then $ρ^{*} (w) A \cap ρ^{*} (u) A = \emptyset$ . For every $w \in W$ , $ρ^{*} (w) \overset{ˉ}{A} \cap \overset{ˉ}{A} = {φ \in \overset{ˉ}{A} : φ (e_{s}) = 0 for all s \in S (w)},$ the closed face of $\overset{ˉ}{A}$ indexed by $S (w)$ (allowing the empty face when $S (w) = S$ ), where $S (w)$ is the support of $w$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1)). This intersection has nonempty interior in $E$ only when $w = 1$ .

(6) Abstention. This lemma does not assert that $E = ⋃_{w \in W} ρ^{*} (w) \overset{ˉ}{A}$ , or that the action is properly discontinuous or cocompact. No choice principle is used.

Facts & Assumptions

Given: A finite Coxeter system of affine form type, its real Coxeter space $(V, B)$ , a positive radical vector $δ$ , the radical quotient $(U, b)$ , the affine slice $E$ , walls $H_{s}$ , strict alcove $A$ , and its closure $\overset{ˉ}{A}$ as in the cited definition.

[F1]

$rad (B) = R δ$ and every $δ_{s} > 0$ (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (2)-(4), Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (1)).

[F2]

The quotient form $b$ is well-defined and positive definite, so $U$ is a Euclidean vector space of dimension $∣ S ∣ - 1$ (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (3)).

[F3]

The direction space of $E$ is $K_{δ} ≅ U^{*}$ ; $b^{♭} : U \to U^{*}$ is an isomorphism, and $b^{*}$ is transported to $K_{δ}$ through it. Also $A = {φ \in E : φ (e_{s}) > 0 \forall s}$ while $\overset{ˉ}{A}$ is its closure (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (4)).

[F4]

$V = R^{S}$ has basis $(e_{s})$ , $B$ is symmetric, $B (e_{s}, e_{s}) = 1$ , and $r_{e_{s}} (v) = v - 2 B (v, e_{s}) e_{s}$ (The real Coxeter form, its radical, reflections, and form-preserving maps (1)-(3)).

[F5]

The homomorphism $ρ$ satisfies $ρ (s) = r_{e_{s}}$ and preserves $B$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (1)-(2)).

[F6]

The dual action $w \cdot f = f \circ ρ (w)^{- 1}$ is a left action by linear bijections; $C = {f \in V^{*} : f (e_{s}) \geq 0 \forall s}$ and $C^{\circ} = {f : f (e_{s}) > 0 \forall s}$ (The dual action, chambers, faces, and root hyperplanes (1)-(2)).

[F7]

Both $ρ$ and its dual action are injective (The root-length criterion and faithfulness of the canonical reflection representation (3)).

[F8]

For $f \in C$ , ${Stab}_{W} (f) = W_{S (f)}$ ; $w C \cap C = {f \in C : w \in W_{S (f)}}$ ; and the open chambers $w C^{\circ}$ are pairwise disjoint (Chamber collisions, point stabilizers, and the intersection rule (4)-(6)).

[F9]

$S (w)$ is independent of the reduced expression, and $w \in W_{J}$ iff $S (w) \subseteq J$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1)); $W_{\emptyset} = ⟨ \emptyset ⟩ = {1}$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[F10]

Cauchy-Schwarz holds for the real inner product $b$ : $∣ b (x, y) ∣ \leq ∥ x ∥_{b} ∥ y ∥_{b}$ (The induced length is a norm, Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[F11]

A finite convex hull of points in a real topological vector space is compact (Convex closures and hulls of finitely many compact convex sets).

[F12]

The topology on $E$ is that of the metric induced by the norm from $b^{*}$ , and an isometry preserves that metric (The induced length is a norm, 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, Isometry, isometric embedding, and the subspace metric on a subset).

[F13]

A geometric simplex is the convex hull of affinely independent vertices (The geometric simplex spanned by affinely independent vertices).

Proof

technique · describe the slice using its coordinate evaluations, then identify its vertices, faces, and Euclidean normals

1.1F1F2F3F4F5F6F7F12algebra

(The action on the slice.) Since every generator fixes $rad (B)$ pointwise by the reflection formula [F4], $ρ (w) δ = δ$ for all $w$ ; hence $(ρ^{*} (w) φ) (δ) = φ (δ)$ and $E$ is invariant. The induced map $T$ on $U$ preserves $b$ by [F5]. For $α = b^{♭} (z)$ and $β = b^{♭} (z^{'})$ in $K_{δ}$ , the contragredient action sends them to $b^{♭} (T z)$ and $b^{♭} (T z^{'})$ , so their $b^{*}$ -pairing remains $b (T z, T z^{'}) = b (z, z^{'})$ ; hence the restriction to $E$ is an affine isometry. For each $s$ , $φ (e_{s}) = 0$ iff $(ρ^{*} (w) φ) (ρ (w) e_{s}) = 0$ , so walls are sent to the stated walls and the family of translates of $A$ is permuted. Since affine form type has corank one, $S$ is nonempty. Put $D = \sum_{t} δ_{t} > 0$ and define $φ_{0}$ by $φ_{0} (e_{s}) = 1 / D$ for every $s$ ; then $φ_{0} \in E$ . If $ρ^{*} (w)$ fixes $E$ pointwise, it fixes $φ_{0}$ and every direction vector (a difference of two points of $E$ ). Every $f \in V^{*}$ decomposes as $f = (f - f (δ) φ_{0}) + f (δ) φ_{0}$ , where the first term is a direction vector, so $ρ^{*} (w)$ fixes all of $V^{*}$ . Injectivity in [F7] gives $w = 1$ .

1.2F1F3F4algebra

(The affine relation.) For $φ \in E$ , linearity and $δ = \sum_{s} δ_{s} e_{s}$ give $\sum_{s} δ_{s} φ (e_{s}) = φ (δ) = 1$ . Since $δ_{s} > 0$ , the point $φ_{0}$ with $φ_{0} (e_{s}) = 1 / (\sum_{t} δ_{t})$ for all $s$ is in $E$ and in $A$ . Also $∣ S ∣ \geq 2$ : the nonzero radical rules out $S = \emptyset$ , and for $∣ S ∣ = 1$ the matrix $B = (1)$ has zero radical.

1.3F1F4algebra

(The coordinate vertices.) For fixed $s$ , a functional vanishing on every $e_{t}$ with $t \neq s$ is a multiple of the coordinate functional $q_{s}$ defined by $q_{s} (e_{s}) = 1$ , $q_{s} (e_{t}) = 0$ for $t \neq s$ ; since $q_{s} (δ) = δ_{s} \neq 0$ , at most one such functional lies in $E$ . The functional $v_{s}$ with these zero values and $v_{s} (e_{s}) = 1 / δ_{s}$ is well-defined on the basis $(e_{t})$ and satisfies $v_{s} (δ) = 1$ , so it is that unique point of $E$ . If $\sum_{s} μ_{s} v_{s} = 0$ and $\sum_{s} μ_{s} = 0$ , evaluation at $e_{t}$ gives $μ_{t} / δ_{t} = 0$ ; hence the vertices are affinely independent. For every $φ \in E$ , both sides of $φ = \sum_{s} δ_{s} φ (e_{s}) v_{s}$ agree on each basis vector $e_{t}$ , so this barycentric identity holds.

2.1F3F4F10F12step 1.2algebra

(The closure is the coordinate simplex.) For $ψ \in K_{δ}$ , let $z \in U$ be the vector corresponding to $ψ$ under $b^{♭}$ ; then $ψ (e_{s}) = b (z, {\overset{ˉ}{e}}_{s})$ and $∥ {\overset{ˉ}{e}}_{s} ∥_{b}^{2} = B (e_{s}, e_{s}) = 1$ . By [F10], $∣ ψ (e_{s}) ∣ \leq ∥ z ∥_{b} = ∥ ψ ∥_{b^{*}}$ , so each affine coordinate map $φ \mapsto φ (e_{s})$ is continuous in the metric topology [F12]. It follows that $\overset{ˉ}{A} \subseteq {φ \in E : φ (e_{s}) \geq 0 \forall s}$ . Conversely, if all these coordinates are nonnegative, then for $0 < ϵ < 1$ the point $φ_{ϵ} = (1 - ϵ) φ + ϵ φ_{0}$ lies in $E$ and has every coordinate strictly positive; as $ϵ \to 0$ , $∥ φ_{ϵ} - φ ∥_{b^{*}} = ϵ ∥ φ_{0} - φ ∥_{b^{*}} \to 0$ . Thus $\overset{ˉ}{A} = E \cap C = {φ \in E : φ (e_{s}) \geq 0 \forall s} .$ In particular each $v_{s}$ belongs to $\overset{ˉ}{A}$ .

3.1F2F4F11F13step 1.3step 2.1algebra

(Simplex, facets, normals, and compactness.) By the barycentric identity in step 1.3 and the closure description in step 2.1, each $φ \in \overset{ˉ}{A}$ has nonnegative coefficients $λ_{s} = δ_{s} φ (e_{s})$ summing to $1$ , and conversely, for any coefficients $λ_{s} \geq 0$ with $\sum_{s} λ_{s} = 1$ , the combination $\sum_{s} λ_{s} v_{s}$ lies in $E$ and has coordinate $λ_{t} / δ_{t} \geq 0$ at each $e_{t}$ , so it lies in $\overset{ˉ}{A} = E \cap C$ . Hence $\overset{ˉ}{A} = conv {v_{s} : s \in S}$ is a simplex of dimension $∣ S ∣ - 1$ . The face where $φ (e_{s}) = 0$ is exactly the convex hull of the vertices $v_{t}$ with $t \neq s$ , so these are its facets; in the barycentric model ${(λ_{s}) : λ_{s} \geq 0, \sum_{s} λ_{s} = 1}$ , the relative interior is exactly where every $λ_{s} > 0$ , so ${int}_{E} (\overset{ˉ}{A}) = A$ . The linear part of $φ \mapsto φ (e_{s})$ on $K_{δ}$ is represented under $b^{*}$ by $b^{♭} ({\overset{ˉ}{e}}_{s})$ , whose squared norm is $b ({\overset{ˉ}{e}}_{s}, {\overset{ˉ}{e}}_{s}) = B (e_{s}, e_{s}) = 1$ ; it points inward because the coordinate is nonnegative on $\overset{ˉ}{A}$ and positive on $A$ . Its pairings with the other normals are $b ({\overset{ˉ}{e}}_{s}, {\overset{ˉ}{e}}_{t}) = B (e_{s}, e_{t})$ . A finite convex hull of points is compact by [F11], hence $\overset{ˉ}{A}$ is compact.

4.1F4F5F8F9step 3.1algebra

(Facet reflections and interior stabilizers.) Put $n_{s} : = B (e_{s}, -) \in K_{δ}$ ; under the quotient identification this is $b^{♭} ({\overset{ˉ}{e}}_{s})$ , the unit normal from step 3.1. For $v \in V$ , [F4] gives $B (r_{e_{s}} v, e_{s}) = B (v, e_{s}) - 2 B (v, e_{s}) B (e_{s}, e_{s}) = - B (v, e_{s})$ , hence $r_{e_{s}}^{2} (v) = v$ and $ρ (s)^{- 1} = r_{e_{s}}$ by [F5]. Thus for $φ \in E$ , using symmetry of $B$ , $(ρ^{*} (s) φ) (v) = φ (v - 2 B (v, e_{s}) e_{s}) = φ (v) - 2 φ (e_{s}) B (e_{s}, v),$ so $ρ^{*} (s) φ = φ - 2 φ (e_{s}) n_{s}$ . Since $n_{s}$ is unit and $n_{s} (e_{s}) = 1$ , the affine coordinate $φ (e_{s})$ is signed distance from $H_{s}$ , and this is the Euclidean reflection in $H_{s}$ . If $φ \in A$ is fixed by $w$ , then $φ \in C$ and $S (φ) = \emptyset$ ; [F8] gives $w \in W_{\emptyset} = {1}$ by [F9].

4.2F3F6F8F9step 2.1step 3.1algebra

(Alcove intersections.) Since $E$ is invariant and $\overset{ˉ}{A} = E \cap C$ , one has $ρ^{*} (w) \overset{ˉ}{A} = E \cap w C$ and $ρ^{*} (w) A = E \cap w C^{\circ}$ . For $φ \in \overset{ˉ}{A}$ , the chamber intersection rule [F8] and the support property [F9] give $φ \in ρ^{*} (w) \overset{ˉ}{A} \cap \overset{ˉ}{A} ⟺ w \in W_{S (φ)} ⟺ S (w) \subseteq S (φ),$ which is equivalent to $φ (e_{s}) = 0$ for all $s \in S (w)$ . This coordinate condition is the face opposite the vertices indexed by $S (w)$ , interpreted as empty when $S (w) = S$ . If $w \neq u$ , then $w^{- 1} u \neq 1$ , so $w C^{\circ} \cap u C^{\circ} = \emptyset$ by [F8], and therefore $ρ^{*} (w) A \cap ρ^{*} (u) A = \emptyset$ . Finally, for $w \neq 1$ , $S (w) \neq \emptyset$ by [F9], so the displayed face is proper and has empty interior in $E$ ; for $w = 1$ it is $\overset{ˉ}{A}$ , whose interior is the nonempty set $A$ .

5.1given∎

(Scope and Choice.) The proof establishes only the action, simplex, wall-reflection, stabilizer, and intersection claims stated above; it does not establish that these translates cover $E$ , or that the action is properly discontinuous or cocompact. All constructions use finite coordinates and explicit convex interpolation, so no choice principle is used.

LemmaStatement: Literature-sourcedProof: AI-adaptedaudited 2026-10-08Open item page →

Crystallographic alcove diagrams: the affine list realized by Weyl types A–G

Statement

Let $Φ \neq \emptyset$ be an irreducible reduced crystallographic Euclidean root system with positive system and base $Δ = {α_{1}, \dots, α_{n}}$ . Its finite Weyl group and chosen simple-root lengths give a crystallographic scaling of the finite Coxeter form; by Crystallographic finite type: the Weyl types, reduced realizations and lattice stability the based finite type is one of $A_{n}$ ( $n \geq 1$ ), $B_{n}$ or $C_{n}$ ( $n \geq 2$ ), $D_{n}$ ( $n \geq 4$ ), $E_{6}, E_{7}, E_{8}, F_{4}$ , or $G_{2}$ . The root lengths of $Φ$ are part of the chosen root system: in particular the two rank-two systems denoted $B_{2}$ and $C_{2}$ have the same finite Coxeter diagram but the two dual root-length assignments. Use the standard simple-root numbering: the classical coordinate bases of Classical root systems in coordinates, Bourbaki numbering for $E_{6}, E_{7}, E_{8}, F_{4}$ , and $α_{1}$ short and $α_{2}$ long for $G_{2}$ . Let $θ$ be the highest root and let $A_{Φ} : = {x \in E : (x, α_{i}) > 0 (1 \leq i \leq n), (x, θ) < 1}$ be the fundamental alcove of Highest-root dominance and the fundamental alcove. Put $s_{0} = r_{θ, 1}$ and $s_{i} = r_{α_{i}, 0}$ for $1 \leq i \leq n$ , and define $m_{i j}^{Φ} : = ord (s_{i} s_{j})$ , allowing $m_{i j}^{Φ} = \infty$ .

(1) Highest-root table and affine labels. The highest roots and the new labels from the affine facet $H_{θ, 1}$ are as follows; all unlisted $m_{0 i}^{Φ}$ equal $2$ , and the entries $m_{i j}^{Φ}$ for $i, j \geq 1$ are the finite-type labels.

$A_{n}$ : $θ = α_{1} + \dots + α_{n}$ . If $n = 1$ , $m_{01}^{Φ} = \infty$ . If $n \geq 2$ , $m_{01}^{Φ} = m_{0 n}^{Φ} = 3$ .
$B_{n}$ : $θ = α_{1} + 2 α_{2} + \dots + 2 α_{n}$ . For $n \geq 3$ , $m_{02}^{Φ} = 3$ ; for $n = 2$ , $m_{02}^{Φ} = 4$ .
$C_{n}$ : $θ = 2 α_{1} + \dots + 2 α_{n - 1} + α_{n}$ and $m_{01}^{Φ} = 4$ .
$D_{n}$ : $θ = α_{1} + 2 α_{2} + \dots + 2 α_{n - 2} + α_{n - 1} + α_{n}$ and $m_{02}^{Φ} = 3$ .
$E_{6}$ : $θ = α_{1} + 2 α_{2} + 2 α_{3} + 3 α_{4} + 2 α_{5} + α_{6}$ and $m_{02}^{Φ} = 3$ .
$E_{7}$ : $θ = 2 α_{1} + 2 α_{2} + 3 α_{3} + 4 α_{4} + 3 α_{5} + 2 α_{6} + α_{7}$ and $m_{01}^{Φ} = 3$ .
$E_{8}$ : $θ = 2 α_{1} + 3 α_{2} + 4 α_{3} + 6 α_{4} + 5 α_{5} + 4 α_{6} + 3 α_{7} + 2 α_{8}$ and $m_{08}^{Φ} = 3$ .
$F_{4}$ : $θ = 2 α_{1} + 3 α_{2} + 4 α_{3} + 2 α_{4}$ and $m_{01}^{Φ} = 3$ .
$G_{2}$ : $θ = 3 α_{1} + 2 α_{2}$ , with $m_{02}^{Φ} = 3$ , $m_{01}^{Φ} = 2$ , and $m_{12}^{Φ} = 6$ .

Thus $m^{Φ}$ is the standard affine diagram of the corresponding line of The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde; for $B_{2}$ and $C_{2}$ it is the common path with labels $(4, 4)$ .

(2) Facet-normal Gram matrix. The inward unit normals of $A_{Φ}$ are $u_{0} = - θ / ∥ θ ∥$ and $u_{i} = α_{i} / ∥ α_{i} ∥$ for $i \geq 1$ . Their Gram matrix is the cosine matrix of $m^{Φ}$ : $(u_{i}, u_{j}) = - \cos (π / m_{i j}^{Φ}),$ where $\cos (π / \infty) = 1$ . Hence the facet-reflection matrix of $A_{Φ}$ equals the cosine matrix of the displayed affine diagram.

(3) Semidefinite consequences. For every standard affine diagram $D$ , its cosine matrix $C_{D}$ is positive semidefinite of corank one and has a kernel vector with every coordinate positive. Every proper principal submatrix of $C_{D}$ is positive definite; the empty principal submatrix case is vacuous.

(4) Surjectivity. Every standard affine diagram occurs in (1): ${\tilde{A}}_{1}$ is obtained from $A_{1}$ ; ${\tilde{A}}_{n}$ from $A_{n}$ for $n \geq 2$ ; ${\tilde{B}}_{n}$ from $B_{n}$ for $n \geq 3$ ; ${\tilde{B}}_{2} = {\tilde{C}}_{2}$ from either $B_{2}$ or $C_{2}$ ; and ${\tilde{C}}_{n}$ from $C_{n}$ for $n \geq 2$ , ${\tilde{D}}_{n}$ from $D_{n}$ for $n \geq 4$ , and ${\tilde{E}}_{6}, {\tilde{E}}_{7}, {\tilde{E}}_{8}, {\tilde{F}}_{4}, {\tilde{G}}_{2}$ from their matching finite types. The aliases ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ , and ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ are therefore realized by $A_{3}, A_{4}, D_{5}$ , respectively. The convention ${\tilde{C}}_{1} : = {\tilde{A}}_{1}$ is covered by $A_{1}$ .

(5) Affine Weyl group and simplex reflection presentation. The assignment from the standard generators of the abstract Coxeter group $W (m^{Φ})$ to $s_{0}, \dots, s_{n}$ is an isomorphism onto $W_{a} (Φ)$ , and $W_{a} (Φ) = Q^{\lor} ⋊ W (Φ) .$ Thus each standard affine diagram is the Coxeter diagram of a Euclidean simplex reflection group with fundamental alcove $A_{Φ}$ . In rank one the two endpoint hyperplanes are disjoint and their product has infinite order.

(6) Verification data. The coefficient vectors in (1) are those of the highest roots in the stated standard numbering. For simply laced types $A, D, E$ , the values $2 c_{i} - \sum_{j \sim i} c_{j}$ for $θ = \sum_{i} c_{i} α_{i}$ give the displayed attachment node (and for $A_{1}$ the single value is $2$ ). For $B, C, F, G$ , the coordinate root models and the root-length ratios give exactly the normalized pairings recorded in the proof below. The $B_{2} / C_{2}$ coincidence is only a coincidence of the finite Coxeter diagram; the two root-length assignments are both included.

(7) Non-isomorphism. Apart from the naming conventions ${\tilde{C}}_{1} = {\tilde{A}}_{1}$ , ${\tilde{B}}_{2} = {\tilde{C}}_{2}$ , ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ , and ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ in The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde, the diagrams in clauses (1)–(6) of that definition are pairwise non-isomorphic as labelled graphs. No twisted affine diagrams or extended affine Weyl group $P^{\lor} ⋊ W$ are asserted here. No choice principle is used.

Facts & Assumptions

Given: The finite irreducible reduced crystallographic root system $Φ$ , its positive system and base $Δ$ , highest root $θ$ , Weyl group $W (Φ)$ , root and coroot lattices, and the affine hyperplanes and reflections of Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group.

[F1]

The system is finite and reduced; the positive system has base $Δ$ ; the simple roots form a basis; positive roots have nonnegative integral simple-root coordinates; the highest root exists, is unique, dominates every positive root, and is dominant against every positive root (Reduced crystallographic Euclidean root system, Positive systems and simple roots, Height and highest root, Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis, Simple roots form a signed integral basis, Existence and uniqueness of the highest root).

[F2]

The standard coordinate root systems of types $A_{n}, B_{n}, C_{n}, D_{n}$ and their simple-root bases are as in Classical root systems in coordinates. In particular these include $B_{2}$ and $C_{2}$ ; the latter has $α_{1} = e_{1} - e_{2}$ , $α_{2} = 2 e_{2}$ .

[F3]

The exceptional based diagrams use Bourbaki numbering: the $E$ chain is $1 - 3 - 4 - 5 - 6$ (continued through $7, 8$ when present), with node $2$ attached to $4$ . For $F_{4}$ , in order $1, 2, 3, 4$ the squared simple lengths are proportional to $(2, 2, 1, 1)$ and its Cartan matrix has rows $(2, - 1, 0, 0), (- 1, 2, - 1, 0), (0, - 2, 2, - 1), (0, 0, - 1, 2)$ . For $G_{2}$ , $α_{1}$ is short, the squared lengths are proportional to $(2, 6)$ and $(α_{1}, α_{2}) = - 3$ in that normalization. These are the based type and length data of [F7], not assertions of membership or maximality of a table vector. A linear map between based root systems with the same Cartan matrix carries their simple roots and roots correspondingly (The Cartan matrix determines a based root system).

[F4]

The coroot is $α^{\lor} = 2 α / (α, α)$ ; the affine reflections satisfy $r_{α, k} = t_{k α^{\lor}} s_{α}$ , $r_{α, 0} = s_{α}$ , and $r_{α, 1} r_{α, 0} = t_{α^{\lor}}$ ; by definition $Q^{\lor}$ is generated by the coroots of all roots, and $W$ preserves both the root system and the inner product (Coroot and dual root system, Affine reflections: translation form, involutivity, local finiteness, and $W_{a} = Q^{\lor} ⋊ W$ , Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group, Root, coroot, weight, and coweight lattices, Weyl group).

[F5]

The closed fundamental alcove is a bounded geometric simplex with exactly the walls $H_{α_{i}, 0}$ and $H_{θ, 1}$ as its facets; its open interior is an alcove (Highest-root dominance and the fundamental alcove (2)).

[F6]

For the finite simple roots, the orders of products of their reflections are the finite Coxeter labels. For an affine facet paired with a finite facet, the rank-two mirror angle is $π / m$ for $m \in {2, 3, 4, 6}$ when the walls meet; the rank-one endpoint case has disjoint walls and $m = \infty$ (Weyl group, Point stabilizers, vertex residues, and rank-two boundary words (2), Rank-two root-system classification).

[F7]

The Weyl group of $Φ$ is finite. The normalized simple roots form a basis; their pairwise inner products are the Coxeter-form entries $- \cos (π / m_{i j})$ by the rank-two root-angle classification, so identifying $e_{i}$ with $α_{i} / ∥ α_{i} ∥$ identifies the Coxeter form with their positive-definite Gram form. Set $c_{i} = ∥ α_{i} ∥$ in the Coxeter scaling definition. Its scaled Cartan entry is $a_{i j} = 2 (α_{i}, α_{j}) / ∥ α_{j} ∥^{2} = C_{j i}$ , the transpose of the usual based Cartan matrix $C$ of $Φ$ (Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices, Cartan matrix of a based root system); hence the scaling is crystallographic. The scaled-root theorem Crystallographic finite type: the Weyl types, reduced realizations and lattice stability (1)–(2) gives exactly the finite crystallographic types $A, B, C, D, E, F, G$ with their standard ranks and the $B_{n} / C_{n}$ dual length assignments. Its root system $Φ_{c}$ has based Cartan matrix $A^{T} = C$ , so The Cartan matrix determines a based root system identifies $Φ_{c}$ with $Φ$ . Since $Φ_{c} = ⋃_{i} ρ (W) a_{i}$ by the scaling definition and reflections preserve the form, every root has one of the simple-root lengths; [F15] transports these lengths up to a common positive factor. Thus every root has squared length at most $M : = \max_{i} ∥ α_{i} ∥^{2}$ , including the short-root orbits. The finiteness and angle inputs are The Weyl group is finite and faithful and Rank-two root-system classification.

[F8]

Across a label- $3$ edge the squared simple-root lengths are equal; across label $4$ their ratio is $2$ or $1 / 2$ , and across label $6$ it is $3$ or $1 / 3$ . The Cartan products are $0, 1, 2, 3$ (Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices, Cartan-number products, allowed edge labels, tree scalings and reflection stability (1)–(2)).

[F9]

A connected positive-semidefinite cosine matrix with nonzero radical has a positive radical vector, radical of dimension one, and positive-definite proper principal submatrices (Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (1)–(2)).

[F10]

The standard affine diagrams have the explicit graph recipes in The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (1)–(6); every graph in those clauses is connected. Their low-rank aliases are recorded separately in [F16].

[F11]

The local theorem Alcove transitivity, the affine Coxeter presentation, and the length function (1) proves that the fundamental facet reflections generate $W_{a}$ ; (2) proves that the homomorphism from their actual Coxeter presentation to $W_{a}$ is an isomorphism, including rank one. These are the precise generation and presentation inputs.

[F12]

For an $n$ -simplex with $n \geq 2$ , two distinct facets share the hull of the $n - 1$ vertices omitted by neither facet, a codimension-two face by affine independence (The geometric simplex spanned by affinely independent vertices). In its two-dimensional normal section, the inward normals make an angle supplementary to the interior wedge angle: the two boundary rays are perpendicular to the respective inward normals. Thus their pairing is the negative cosine of the interior dihedral angle. For $n = 1$ the two facets are distinct endpoints.

[F13]

A graph isomorphism preserves vertex count, degrees, and adjacency; for the Coxeter diagrams defined in The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (1)–(6), the labels attached to corresponding edges are part of the labelled-graph isomorphism data (Graph isomorphisms, automorphisms and graph complements).

[F14]

The inner product on $E$ is symmetric and positive definite; for vectors $u_{i}$ and scalars $x_{i}$ , the Gram quadratic form is $\sum_{i, j} x_{i} x_{j} (u_{i}, u_{j}) = ∥ \sum_{i} x_{i} u_{i} ∥^{2} \geq 0$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form).

[F15]

If two based root systems on a connected diagram have the same Cartan matrix $C$ , then their simple-root Gram matrices $G, G^{'}$ are positive scalar multiples. Indeed, from $C_{i j} = 2 G_{i j} / G_{i i}$ , an edge $i \sim j$ gives $C_{i j} G_{i i} = 2 G_{i j} = 2 G_{j i} = C_{j i} G_{j j}$ , so the Cartan entries determine the ratio of adjacent squared lengths; connectedness fixes all diagonal entries up to one common scalar, and the same formula fixes all off-diagonal entries. Consequently normalized pairings of corresponding linear combinations of simple roots agree (Cartan matrix of a based root system).

[F16]

The low-rank naming conventions are ${\tilde{C}}_{1} = {\tilde{A}}_{1}$ , ${\tilde{B}}_{2} = {\tilde{C}}_{2}$ , ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ , and ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (7)).

[F17]

Two unit normals with pairing $- \cos (π / m)$ for finite $m \geq 2$ give a positive-definite rank-two Gram form, and the product of their linear reflections has exact order $m$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (3)(i),(iii)–(iv)). To apply this to two intersecting affine walls, translate an intersection point to the origin and identify their normal span with that rank-two plane; both reflections fix its orthogonal complement.

Proof

technique · establish candidate membership and maximality locally, compute the alcove normals, and apply the positive-radical lemma and the proved local affine presentation. All computations are finite and explicit; no Choice is used

1.1F2F3F7F8F15algebra

By [F7] the finite crystallographic type is among the rows in (1). Define a candidate $t$ by the coefficient vector in its row. In the classical coordinate systems [F2], $t$ is respectively $e_{1} - e_{n + 1}$ , $e_{1} + e_{2}$ , $2 e_{1}$ , and $e_{1} + e_{2}$ for $A, B, C, D$ , so it is a root; expansion gives the displayed coefficients, including $(1, 2)$ for $B_{2}$ and $(2, 1)$ for $C_{2}$ . For simply laced $A, D, E$ , put $L^{2} = ∥ α_{i} ∥^{2}$ . Their Gram matrix gives $(t, α_{i}) = L^{2} (2 c_{i} - \sum_{j \sim i} c_{j}) / 2$ . Substituting the stated coefficients in the graphs gives bracket $1$ at the two $A_{n}$ endpoints ( $n \geq 2$ ), $2$ at the sole $A_{1}$ node, $1$ at node $2$ of $D_{n}$ , and $1$ at node $2, 1, 8$ of $E_{6}, E_{7}, E_{8}$ respectively, and zero elsewhere. Therefore $(t, α_{i}) \geq 0$ and $∥ t ∥^{2} = \sum_{i} c_{i} (t, α_{i}) = L^{2}$ in every simply laced row. For $B, C$ , the same coordinate calculations give nonnegative simple pairings and squared norm $M$ ; the $F_{4}, G_{2}$ Gram data [F3] give respectively $∥ t ∥^{2} = 2$ , $(t, α_{i}) = (1, 0, 0, 0)$ and $∥ t ∥^{2} = 6$ , $(t, α_{i}) = (0, 3)$ in the stated normalizations. All candidates have nonnegative integral coefficients, squared norm $M$ , and nonnegative simple pairings.

2.1F1F3F4step 1.1algebra

Membership for the exceptional simply laced candidates follows locally by descent, without a highest-root table. Let $z = \sum_{i} d_{i} α_{i}$ have nonnegative integral coefficients and $∥ z ∥^{2} = L^{2}$ . If it is not a simple root, $L^{2} = \sum_{i} d_{i} (z, α_{i}) > 0$ supplies a positive pairing. For that $i$ , $k = 2 (z, α_{i}) / L^{2} = 2 d_{i} - \sum_{j \sim i} d_{j}$ is a positive integer. Since $∥ z - α_{i} ∥^{2} = L^{2} (2 - k) \geq 0$ , one has $k \leq 2$ ; equality forces $z = α_{i}$ . Thus for a nonsimple $z$ , $k = 1$ and $d_{i} \geq 1$ (otherwise its pairing is nonpositive). The reflection $s_{i} z = z - α_{i}$ keeps all coefficients nonnegative and the norm unchanged while decreasing their sum by one. Repetition ends at a simple root; reversing this finite reflection word proves that $z$ is a root by reflection invariance. Apply this to the $E$ candidates from Step 1.1. For $F_{4}$ , the coefficient vector $(2, 3, 4, 2)$ is carried by successive reflections at nodes $1, 2, 3, 2, 1, 4, 3$ through $(1, 3, 4, 2), (1, 2, 4, 2), (1, 2, 2, 2), (1, 1, 2, 2), (0, 1, 2, 2), (0, 1, 2, 0), (0, 1, 0, 0)$ ; the reflection coefficients are the row pairings with [F3]'s Cartan matrix. Reversing the word from $α_{2}$ proves membership. For $G_{2}$ , reflections at nodes $2, 1$ carry $(3, 2)$ to $(3, 1)$ and $(0, 1)$ , again a simple root. This proves membership of every candidate, preserving its exact coefficients.

3.1F1F7step 1.1step 2.1algebra

Each candidate $t$ is a positive root of squared norm $M$ and has $(t, α_{i}) \geq 0$ . If a root $t + η$ lay strictly above it in the root order, $η$ would be a nonzero nonnegative integral combination of simple roots. Positive definiteness then gives $∥ t + η ∥^{2} = M + 2 (t, η) + ∥ η ∥^{2} > M$ , contrary to the all-root length bound of [F7]. Thus $t$ is maximal. The unique-highest-root supplier [F1] identifies it with $θ$ . Every positive root lies below a maximal root by finiteness (extend an upward chain until it stops), and uniqueness makes that maximal root this candidate. This proves membership and the full highest-root assertion in (1), including the short-root orbits of $F_{4}, G_{2}$ , without importing table maximality.

4.1F1F3F7F8F15step 1.1step 3.1algebra

For simply laced $A, D, E$ , the simple roots have one length $L$ by [F8], and all roots have that length by [F7]. Adjacent simple roots have angle $2 π / 3$ , so their pairing is $- L^{2} / 2$ . Thus $L^{2} = (α_{i}, α_{i})$ is independent of $i$ , and for the coefficients $c_{i}$ just listed, $(θ, α_{i}) = \frac{L^{2}}{2} (2 c_{i} - \sum_{j \sim i} c_{j})$ . The bracket is $1$ at both endpoints and $0$ elsewhere in $A_{n}$ for $n \geq 2$ , is $2$ for the single node of $A_{1}$ , is $1$ at node $2$ and $0$ elsewhere in $D_{n}$ , and is $1$ at node $2, 1, 8$ respectively and $0$ elsewhere in $E_{6}, E_{7}, E_{8}$ . The highest root $θ$ also has length $L$ , since it is a root in the same simply laced system. Hence the normalized pairing $(θ, α_{i}) / (∥ θ ∥ ∥ α_{i} ∥)$ is $1 / 2$ at each displayed attachment and $0$ elsewhere, including value $1$ for the sole $A_{1}$ node.

4.2F2F3F6F8F15step 1.1step 3.1algebra

In the standard $B_{n}$ coordinates, $(θ, α_{i})$ vanishes except at $i = 2$ , where it is $1$ ; $∥ θ ∥^{2} = 2$ and $∥ α_{2} ∥^{2} = 2$ for $n \geq 3$ , while $∥ α_{2} ∥^{2} = 1$ for $B_{2}$ . Thus the normalized pairing is $1 / 2$ for $n \geq 3$ and $1 / \sqrt{2}$ for $B_{2}$ . In $C_{n}$ , only $(θ, α_{1})$ is nonzero and equals $2$ ; $∥ θ ∥^{2} = 4$ and $∥ α_{1} ∥^{2} = 2$ , so the normalized pairing is $1 / \sqrt{2}$ . In the Bourbaki $F_{4}$ basis, $∥ θ ∥^{2} = 2$ and the only nonzero simple-root pairing is $(θ, α_{1}) = 1$ , so its normalized value is $1 / 2$ . For $G_{2}$ , normalize $∥ α_{1} ∥^{2} = 2$ , $∥ α_{2} ∥^{2} = 6$ , and $(α_{1}, α_{2}) = - 3$ ; then $θ = 3 α_{1} + 2 α_{2}$ has squared length $6$ , pairs to $0$ with $α_{1}$ and to $3$ with $α_{2}$ , so the normalized values are $0$ and $1 / 2$ . These model calculations give the same normalized pairings for $Φ$ by [F15], hence exactly the finite values in (1).

5.1F5F6F12F17step 4.1step 4.2algebra

By [F5], the walls $H_{α_{i}, 0}$ and $H_{θ, 1}$ are precisely the facets of the bounded simplex $A_{Φ}$ , with inward unit normals $u_{i} = α_{i} / ∥ α_{i} ∥$ and $u_{0} = - θ / ∥ θ ∥$ . For $i, j \geq 1$ , their Gram entries are $- \cos (π / m_{i j}^{Φ})$ by the finite-type root angles. For $0, i$ , one has $(u_{0}, u_{i}) = - (θ, α_{i}) / (∥ θ ∥ ∥ α_{i} ∥)$ ; Steps 4.1–4.2 give $- 1 / 2, - 1 / \sqrt{2},$ or $0$ in rank at least two. The corresponding walls intersect by [F12], so [F17] gives exact product orders $3, 4,$ or $2$ , respectively. In type $A_{1}$ , $u_{0} = - u_{1}$ , giving the entry $- 1$ ; in the coordinate $a = (x, α_{1})$ the endpoint reflections are $a \mapsto - a$ and $a \mapsto 2 - a$ , whose product is translation by two and has infinite order. Hence the full Gram matrix is the cosine matrix of $m^{Φ}$ .

5.2F2F3F10F16step 1.1step 4.1step 4.2algebra

Reading the types in Step 1.1 against the graph recipes in [F10] gives the listed affine diagram for each finite type. The bounds are explicit: $A_{n}$ covers $n = 1$ and $n \geq 2$ separately; $B_{2}$ gives the path $(4, 4)$ and $B_{n}$ for $n \geq 3$ gives the branch diagram; $C_{n}$ covers every $n \geq 2$ ; and $D_{n}$ covers every $n \geq 4$ . The three exceptional simply laced vectors attach at the nodes giving arms $(2, 2, 2), (1, 3, 3), (1, 2, 5)$ , while the $F_{4}, G_{2}$ pairings give the displayed labelled paths. The low-rank conventions [F16] realize ${\tilde{C}}_{1}, {\tilde{B}}_{2}, {\tilde{D}}_{3}, {\tilde{E}}_{4}, {\tilde{E}}_{5}$ via $A_{1}, B_{2}, A_{3}, A_{4}, D_{5}$ , respectively.

6.1F1F9F10F14F16step 5.1step 5.2algebra

Take any standard affine diagram $D$ . By Step 5.2, including the aliases [F16], it is the affine diagram of a finite root system $Φ$ of rank $n$ . Step 5.1 identifies its cosine matrix with the Gram matrix of the $n + 1$ inward unit normals of $A_{Φ}$ , so the matrix is positive semidefinite by [F14]. The finite simple-root normals $u_{1}, \dots, u_{n}$ form a basis of $E$ . For any coefficient vector $x$ , $x$ lies in the Gram kernel exactly when $\sum_{i} x_{i} u_{i} = 0$ , since $x^{T} G x = ∥ \sum_{i} x_{i} u_{i} ∥^{2}$ ; hence the Gram matrix has rank $n$ and a nonzero radical. The matrix graph is connected with nonpositive off-diagonal entries and diagonal entries $1$ by [F10, F16]. Apply [F9] to obtain a one-dimensional radical generated by a vector with every coordinate positive; the same clause gives positive definiteness of every nonempty proper principal submatrix, while the empty case is vacuous.

6.2F4F5F11step 5.1algebra

Let $G = ⟨ s_{0}, \dots, s_{n} ⟩$ . By F11, $G = W_{a}$ , and by F11 the abstract Coxeter group on the actual reflection-product matrix $m^{Φ}$ maps isomorphically onto it. This uses the proved local generation and presentation statements in every rank, including the endpoint reflections in rank one, whose product is a nonzero coroot translation. The identities [F4] give $W_{a} = Q^{\lor} ⋊ W$ directly: every affine generator $r_{α, k} = t_{k α^{\lor}} s_{α}$ lies in that semidirect product, while $s_{α} = r_{α, 0}$ and $t_{α^{\lor}} = r_{α, 1} r_{α, 0}$ lie in $W_{a}$ for every root. The coroots generate $Q^{\lor}$ , and $w (α^{\lor}) = (w α)^{\lor}$ shows that $W$ normalizes its translations. A translation and an element of $W$ agree only at the identity, since $W$ fixes the origin. Together with the alcove simplex and matrix computation, this proves (5).

7.1step 1.1step 4.1step 4.2step 5.1step 6.1algebra

The kernel and proper-minor claims are those established in Step 6.1 for the cosine matrix identified in Step 5.1. The coefficient and normalized-pairing computations of Steps 1.1–4.2 give the stated verification data, including the $A_{1}$ parallel-wall case and the distinct $B_{2} / C_{2}$ root-length assignments.

8.1F10F13F16step 1.1step 4.1step 4.2algebra∎

To distinguish the labelled graphs without using the coincidence assertion in clause (7) of the definition, first note that ${\tilde{A}}_{1}$ is the only listed graph with an $\infty$ -edge, and the cycles ${\tilde{A}}_{n}$ for $n \geq 2$ have all degrees $2$ and are distinguished by vertex count. Among the remaining trees, ${\tilde{B}}_{n}$ has one degree- $3$ vertex and exactly one $4$ -edge; ${\tilde{C}}_{n}$ is a path with exactly two $4$ -edges; ${\tilde{F}}_{4}$ is a path with one interior $4$ -edge; and ${\tilde{G}}_{2}$ is the three-vertex path with a $6$ -edge. The all- $3$ diagrams are distinguished by degree data: ${\tilde{D}}_{4}$ has a degree- $4$ vertex, ${\tilde{D}}_{n}$ for $n \geq 5$ has two degree- $3$ vertices, and each $\tilde{E}$ diagram has one degree- $3$ vertex with its stated arm lengths, which distinguish ${\tilde{E}}_{6}, {\tilde{E}}_{7}, {\tilde{E}}_{8}$ . Vertex counts distinguish successive members within each family. Thus only the explicit naming conventions in [F16] identify two family names. No Choice is used: all root-coordinate checks and graph invariants are finite and explicit.

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

Enumeration of the connected positive semidefinite corank-one diagrams

Statement

Let $(W, S, m, V, B, ρ, Γ)$ be of affine form type (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (1)). Then $Γ$ is isomorphic as a labelled graph (Coxeter diagrams: edges, labels, components and finite type) to one of the standard affine diagrams: ${\tilde{A}}_{1}$ , ${\tilde{A}}_{n}$ for $n \geq 2$ , ${\tilde{B}}_{n}$ for $n \geq 3$ , ${\tilde{C}}_{n}$ for $n \geq 2$ , ${\tilde{D}}_{n}$ for $n \geq 4$ , ${\tilde{E}}_{6}$ , ${\tilde{E}}_{7}$ , ${\tilde{E}}_{8}$ , ${\tilde{F}}_{4}$ , or ${\tilde{G}}_{2}$ (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde). The low-rank names ${\tilde{C}}_{1} = {\tilde{A}}_{1}$ , ${\tilde{B}}_{2} = {\tilde{C}}_{2}$ , ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ , and ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ are represented by the corresponding listed diagrams.

In particular, if $∣ S ∣ \geq 3$ , no edge is labelled $\infty$ , every edge label is in ${3, 4, 6}$ , and $Γ$ is either an all- $3$ cycle ${\tilde{A}}_{n}$ ( $n \geq 2$ ) or a tree. There are at most two edges labelled $\geq 4$ . Thus the cyclic case in the list is precisely the ${\tilde{A}}_{n}$ family; the ${\tilde{B}}_{2}$ , ${\tilde{C}}_{1}$ , ${\tilde{D}}_{3}$ , ${\tilde{E}}_{4}$ , and ${\tilde{E}}_{5}$ aliases are represented by ${\tilde{C}}_{2}$ , ${\tilde{A}}_{1}$ , ${\tilde{A}}_{3}$ , ${\tilde{A}}_{4}$ , and ${\tilde{D}}_{5}$ , respectively.

Facts & Assumptions

Given: A Coxeter matrix $m$ on a finite set $S$ , its diagram $Γ$ , the vector space $V = R^{S}$ , and the cosine matrix $C = (B (e_{s}, e_{t}))$ of the Coxeter form.

[F1]

Affine form type means that $Γ$ is connected and $C$ is positive semidefinite of corank one (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (1)).

[F2]

Every proper principal submatrix of this positive-semidefinite corank-one form is positive definite (Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (2)).

[F3]

${\tilde{A}}_{1}$ is the two-vertex graph with an edge labelled $\infty$ , and for $n \geq 2$ , ${\tilde{A}}_{n}$ is the all- $3$ cycle on $n + 1$ vertices (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (1)).

[F4]

Every standard affine diagram has a positive semidefinite cosine matrix of corank one (Crystallographic alcove diagrams: the affine list realized by Weyl types A–G (3)); hence its cosine matrix is not positive definite. The consumer uses this clause only, not the crystallographic or group-presentation claims of that lemma.

[F5]

In a Coxeter diagram, distinct vertices are joined exactly when their label is at least $3$ , an omitted edge has label $2$ , and the subdiagram on $T \subseteq S$ is induced (Coxeter diagrams: edges, labels, components and finite type (1)-(2)).

[F6]

A Coxeter system is finite if and only if its cosine form is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)).

[F7]

The connected finite Coxeter diagrams are exactly the listed $A, B, D, E, F, H$ and $I_{2} (m)$ diagrams (Classification of finite Coxeter systems, including the H and dihedral families (1)).

[F8]

For a path, the leading cosine determinants satisfy $d_{k} = d_{k - 1} - \cos^{2} (π / m_{k - 1}) d_{k - 2}$ ; an all- $3$ path has $d_{k} = (k + 1) / 2^{k} > 0$ (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (5)(i)).

[F9]

A real symmetric matrix is positive definite exactly when all its leading principal minors 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).

[F10]

The determinant is the Leibniz signed sum (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix); deleted-row-and-column minors and their signed cofactors are defined in Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring. Grouping the Leibniz terms by the row entry in the last row gives the cofactor expansion along that row.

[F11]

The form $B$ is symmetric and bilinear, so its quadratic value in the basis $(e_{s})$ is the sum of diagonal terms and twice the unordered off-diagonal terms (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms).

[F12]

Sine and cosine are defined by their real power series; in particular cosine is even, sine is odd, and $\cos 0 = 1$ (Sine and cosine defined by their real power series).

[F13]

The roots-of-unity theorem lists the fifth roots $e^{2 π i k / 5}$ , $0 \leq k < 5$ , and Euler's formula identifies $e^{i θ} = \cos θ + i \sin θ$ (The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ , Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ).

[F14]

If a connected affine diagram strictly dominates another Coxeter diagram, the latter's cosine matrix is positive definite (Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (3)).

[F15]

The low-rank naming conventions include ${\tilde{C}}_{1} : = {\tilde{A}}_{1}$ , ${\tilde{B}}_{2} : = {\tilde{C}}_{2}$ , ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ , and ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (7)).

[F16]

In the library's real-number construction, $R$ is a complete ordered field, so every nonnegative real has a unique nonnegative square root (The real numbers, The reals form a totally ordered field, The Cauchy-sequence reals have the least-upper-bound property, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

[F17]

Squaring is strictly increasing on nonnegative reals (Squaring is monotone on the nonnegatives).

[F18]

For $n \geq 3$ , ${\tilde{B}}_{n}$ is the path-and-branch graph with one terminal label $4$ specified in the standard recipe (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (2)).

[F19]

For $n \geq 2$ , ${\tilde{C}}_{n}$ is the path on $n + 1$ vertices with label $4$ on both end edges and $3$ on the others (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (3)).

[F20]

The standard $\tilde{D}$ diagrams are the all- $3$ star with four leaves and the two-branch trees specified in the standard recipe (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (4)).

[F21]

The standard ${\tilde{E}}_{6}, {\tilde{E}}_{7}, {\tilde{E}}_{8}$ diagrams are the all- $3$ stars with arms $(2, 2, 2)$ , $(1, 3, 3)$ , and $(1, 2, 5)$ (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (5)).

[F22]

The standard ${\tilde{F}}_{4}$ and ${\tilde{G}}_{2}$ diagrams are the paths with labels $(3, 3, 4, 3)$ and $(3, 6)$ (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde (6)).

[F23]

For a positive-definite path with one edge labelled $m \geq 4$ , the split sizes $i \leq j$ satisfy the strict inequality $(i + 1) (j + 1) > 4 i j \cos^{2} (π / m)$ (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (5)(ii)); this hypothesis is not available for the semidefinite form here.

[F24]

A positive-definite all- $3$ diagram with one degree- $3$ vertex and arm sizes $p, q, r$ satisfies $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} > 1$ (Exclusions for positive definite diagrams: trees, valency, labels, chains and arms (6)).

[F25]

Cycles, paths, and acyclicity are defined in the underlying simple graph (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

[F26]

A vertex degree is the number of its neighbours (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[F27]

$π > 0$ , $\cos (π / 2) = 0$ , $\cos π = - 1$ , and $\cos (x + π) = - \cos x$ (Pi as twice the smallest positive zero of cosine, Quarter-turn values and shifts by pi/2 and pi).

[F28]

The double-angle and power-reduction identities hold, including $\cos^{2} x = (1 + \cos 2 x) / 2$ (Double-angle and quadratic power-reduction identities).

[F29]

Cosine is strictly decreasing on $[0, π]$ (Signs, monotonicity intervals, and ranges of sine and cosine).

Proof

technique · use the corank-one proper-minor property and strict domination to force standard affine subdiagrams to be the whole diagram; in the remaining path and arm cases, give the semidefinite inequalities and finite determinant calculations explicitly. No choice principle is used

1.1F1F3F5F9F12algebra

Let $n = ∣ S ∣$ . If $n = 0$ , the diagram is not connected, and if $n = 1$ its cosine matrix is $[1]$ , so it is positive definite rather than corank one [F1]. For $n = 2$ , connectedness gives one edge labelled $m \geq 3$ or $m = \infty$ : if $m < \infty$ , then $0 < \cos (π / m) < 1$ because $0 < π / m \leq π / 3 < π / 2$ , $\cos (π / 2) = 0$ , $\cos 0 = 1$ by its power series, and cosine is strictly decreasing; hence the leading minors $1$ and $1 - \cos^{2} (π / m)$ are positive, so Sylvester's criterion makes the matrix positive definite [F9,F12,F27,F29]. If $m = \infty$ , the matrix is $(\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array})$ , the standard ${\tilde{A}}_{1}$ . Hence assume $n \geq 3$ .

1.2F12F13F16F27F28F29algebra

The cosine values needed below are $\cos (π / 3) = 1 / 2$ , $\cos (π / 4) = \sqrt{2} / 2$ , $\cos (π / 6) = \sqrt{3} / 2$ , and $c_{5}^{2} : = \cos^{2} (π / 5) = (3 + \sqrt{5}) / 8$ . For the first, if $c = \cos (π / 3)$ then $2 c^{2} - 1 = \cos (2 π / 3) = - c$ , so $(2 c - 1) (c + 1) = 0$ ; strict decrease of cosine and $π / 3 < π$ give $c > - 1$ , hence $c = 1 / 2$ . The other two follow from $\cos^{2} x = (1 + \cos 2 x) / 2$ , $\cos (π / 2) = 0$ , positivity on $(0, π / 2)$ , and the nonnegative square root [F16]. For $c_{5} = \cos (π / 5)$ , put $ζ = e^{2 π i / 5}$ . The five distinct fifth roots of unity include $1$ and $ζ \neq 1$ , and since $ζ^{5} = 1$ the geometric-series identity gives $1 + ζ + ζ^{2} + ζ^{3} + ζ^{4} = 0$ . Set $y = ζ + ζ^{- 1}$ ; dividing by $ζ^{2}$ and using $y^{2} = ζ^{2} + 2 + ζ^{- 2}$ gives $y^{2} + y - 1 = 0$ . Since $ζ^{4} = ζ^{- 1}$ , Euler's formula and the even/odd parity of cosine and sine give $y = 2 \cos (2 π / 5) > 0$ ; positivity follows from $0 < 2 π / 5 < π / 2$ and strict decrease to $\cos (π / 2) = 0$ . Thus $(2 y + 1)^{2} = 5$ and $2 y + 1 > 0$ , so uniqueness of the nonnegative square root [F16] gives $y = (\sqrt{5} - 1) / 2$ . The double-angle identity then gives $c_{5}^{2} = (1 + \cos (2 π / 5)) / 2 = (3 + \sqrt{5}) / 8$ .

1.3F2F3F5F11F14F25algebra

Suppose the underlying graph contains a cycle on a vertex set $T$ , with $r \geq 3$ vertices. On $T$ put the all- $3$ cycle $Δ = {\tilde{A}}_{r - 1}$ and let $u$ be the sum of its $r$ basis vectors. Its quadratic value is $r - 2 r (1 / 2) = 0$ , so its cosine matrix is not positive definite. The induced graph $Γ_{T}$ contains $Δ$ and has labels at least those of $Δ$ ; if it strictly dominates $Δ$ , the full affine diagram $Γ$ also strictly dominates $Δ$ on $T$ , so [F14] would make $Δ$ positive definite, a contradiction. If $Γ_{T} = Δ$ but $T ⊊ S$ , its principal matrix is not positive definite, again contradicting [F2]. Therefore $T = S$ and $Γ = Δ$ , which is a listed all- $3$ cycle.

2.1F2F5step 1.1algebra

If an edge has label $\infty$ , its two-vertex principal matrix is $(\begin{array}{cc} 1 & - 1 \\ - 1 & 1 \end{array})$ , which is not positive definite because $(1, 1)$ has quadratic value $0$ . Since this is a proper principal submatrix when $n \geq 3$ , it contradicts [F2]; thus no edge has label $\infty$ .

2.2F2F19F4F5F14F25step 1.3algebra

Suppose two distinct edges have labels at least $4$ . Since the graph is acyclic by Step 1.3, the minimal connected subdiagram containing these edges is a path whose first and last edges have label at least $4$ . Replace those two labels by $4$ and every internal edge by $3$ ; the resulting diagram is ${\tilde{C}}_{k}$ for some $k \geq 2$ by [F19], and is not positive definite by [F4]. If it is a strict subdiagram or any retained label is larger, [F14] would make it positive definite. If it is the whole diagram with equal labels, $Γ = {\tilde{C}}_{k}$ . Hence the only possibility with two or more such edges is exactly a ${\tilde{C}}_{k}$ diagram; in particular there cannot be three such edges.

2.3F5F8F9F10F25step 1.2algebra

For a path with edge labels $m_{1}, \dots, m_{k - 1}$ , let $d_{k}$ be the determinant of its leading $k \times k$ cosine matrix and put $d_{0} = d_{1} = 1$ . The last row has only the entries $- c$ and $1$ , where $c = \cos (π / m_{k - 1})$ ; the diagonal cofactor is $d_{k - 1}$ , while the minor for the $- c$ entry is $- c d_{k - 2}$ and its cofactor sign is $(- 1)^{k + (k - 1)} = - 1$ , so that cofactor is $c d_{k - 2}$ . The expansion from [F10] therefore gives $d_{k} = d_{k - 1} - c^{2} d_{k - 2}$ . This is also the recurrence in the positive-definite path result F8(i). When every label is $3$ , Step 1.2 gives $c = 1 / 2$ , and induction yields $d_{k} = (k + 1) / 2^{k} > 0$ . Thus an all- $3$ path is positive definite by [F9] and cannot have corank one.

2.4F1F5F23F11F25step 1.2algebra

For later use, suppose a path has exactly one edge labelled $m \geq 4$ . Let that edge split the path into $i \leq j$ vertices, both at least $1$ , and put $c = \cos (π / m)$ . Weight the vertices on the two sides from their remote ends toward the large edge by $1, 2, \dots, i$ and $1, 2, \dots, j$ , giving nonzero vectors $u, v$ . Expanding along the all- $3$ parts gives $B (u, u) = \sum_{h = 1}^{i} h^{2} - \sum_{h = 1}^{i - 1} h (h + 1) = i (i + 1) / 2$ , $B (v, v) = j (j + 1) / 2$ , and the only cross term is $B (u, v) = - i j c$ . Since $B$ is positive semidefinite, $B (t u + v, t u + v) \geq 0$ for every real $t$ ; choosing $t = - B (u, v) / B (u, u)$ yields $B (u, v)^{2} \leq B (u, u) B (v, v)$ and hence $(i + 1) (j + 1) \geq 4 i j c^{2}$ . The strict inequality in F23(ii) assumes positive definiteness and cannot be used for the present semidefinite form; the derived non-strict inequality includes the affine equality cases.

3.1F2F4F5F14F18F20F25F26step 2.2algebra

Suppose exactly one edge has label at least $4$ and $Γ$ has a vertex of degree at least $3$ . A vertex of degree at least $4$ , together with four of its neighbours, gives a subdiagram dominating ${\tilde{D}}_{4}$ ; two distinct degree- $3$ vertices, the path between them, and two additional neighbours at each end give a subdiagram dominating some ${\tilde{D}}_{k}$ . These are standard affine diagrams and are not positive definite by [F20,F4]. A strict domination contradicts [F14]; an equal proper subdiagram contradicts [F2]. Equality on all vertices would make $Γ$ a $\tilde{D}$ diagram with all labels $3$ , contrary to the assumed large edge. Thus there is exactly one degree- $3$ vertex $v$ . The unique large edge lies on one of its three arms. Retain the path from $v$ through that edge to its endpoint farther from $v$ , and retain just the first edge on each of the other two arms. Lower the retained large label to $4$ (and all other retained edges already have label $3$ ). The resulting comparison diagram is ${\tilde{B}}_{k}$ with its terminal label $4$ ; it is not positive definite by [F4]. The same strict-domination and proper-principal arguments force it to be all of $Γ$ with the large label exactly $4$ . Therefore the branched case is precisely ${\tilde{B}}_{k}$ . If there is no vertex of degree at least $3$ , the connected acyclic graph is a path.

3.2F5F6F7F24F11F25step 2.3algebra

For an all- $3$ star with arms of $p, q, r$ vertices, each arm block is the positive definite all- $3$ path matrix $A_{j}$ from Step 2.3. Solving $A_{j} z = e_{1}$ gives $z_{h} = 2 (j + 1 - h) / (j + 1)$ : the entries form an arithmetic progression, satisfy the endpoint and interior tridiagonal equations, and the solution is unique since $A_{j}$ is positive definite. Thus $(A_{j}^{- 1})_{11} = 2 j / (j + 1)$ . If $x$ is the central coordinate and $y_{a}$ are the arm vectors, the quadratic form is $x^{2} + \sum_{a} (y_{a}^{T} A_{j_{a}} y_{a} - x e_{1}^{T} y_{a})$ . For each arm, $y_{a}^{T} A_{j_{a}} y_{a} - x e_{1}^{T} y_{a} = (y_{a} - \frac{x}{2} A_{j_{a}}^{- 1} e_{1})^{T} A_{j_{a}} (y_{a} - \frac{x}{2} A_{j_{a}}^{- 1} e_{1}) - \frac{x^{2}}{4} e_{1}^{T} A_{j_{a}}^{- 1} e_{1}$ . Thus the remaining central coefficient is $1 - \frac{p}{2 (p + 1)} - \frac{q}{2 (q + 1)} - \frac{r}{2 (r + 1)} = \frac{1}{2} (\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r + 1} - 1)$ . Hence the star is positive definite when that reciprocal sum exceeds $1$ . In particular the finite stars $(1, 1, r)$ for $r \geq 1$ and $(1, 2, 2), (1, 2, 3), (1, 2, 4)$ have positive definite forms, with central coefficients respectively $1 / (2 (r + 1)), 1 / 12, 1 / 24, 1 / 60$ . By [F6] they are finite Coxeter systems, and F7,(3) identifies these as the finite $D$ and $E$ diagrams. The same reciprocal sum is the necessary three-arm bound supplied for positive definite diagrams by [F24].

3.3F22F5F6F7F9F16F17F25F29step 1.1step 1.2step 2.3step 2.4algebra

The integer cases from Step 2.4 are as follows. If $m = 4$ , then $(i - 1) (j - 1) \leq 2$ , so $i = 1$ with arbitrary $j$ , or $(i, j) = (2, 2)$ or $(2, 3)$ ; the first paths are finite $B$ diagrams, $(2, 2)$ is finite $F_{4}$ , and $(2, 3)$ is ${\tilde{F}}_{4}$ up to reversal. If $m = 5$ , then $4 c^{2} = (3 + \sqrt{5}) / 2 > 9 / 4$ because $2 \sqrt{5} > 3$ (indeed $\sqrt{5} > 2$ by [F16,F17]); if $i \geq 2$ , then $j \geq i$ and $(i + 1) (j + 1) / (i j) = (1 + 1 / i) (1 + 1 / j) \leq 9 / 4$ , contradicting Step 2.4. Thus $i = 1$ , and $2 (j + 1) \geq 4 c^{2} j$ forces $j \leq 4 / (\sqrt{5} - 1) = \sqrt{5} + 1 < 4$ since $\sqrt{5} < 3$ by [F16,F17]; the formal cases $j = 1, 2, 3$ are $I_{2} (5)$ , $H_{3}$ , and $H_{4}$ , with $j = 1$ already treated in rank $2$ . If $m \geq 6$ , then $4 c^{2} \geq 3$ ; the same ratio bound excludes $i \geq 2$ , so $i = 1$ , and $2 (j + 1) \geq 3 j$ gives $j \leq 2$ . For $j = 1$ rank $2$ was treated in Step 1.1, while $j = 2$ requires $m = 6$ (for $m > 6$ , strict monotonicity gives $4 c^{2} > 3$ ) and gives the path ${\tilde{G}}_{2} = (3, 6)$ . For completeness, the finite path claims just used follow from Step 2.3 and Sylvester: a terminal $4$ -edge has final determinant $2^{- j}$ after an all- $3$ prefix, the $(3, 4, 3)$ path has leading determinants $1, 3 / 4, 1 / 4, 1 / 16$ , and the $(5, 3)$ and $(5, 3, 3)$ paths have leading determinants $1, (5 - \sqrt{5}) / 8, (3 - \sqrt{5}) / 8$ and $1, (5 - \sqrt{5}) / 8, (3 - \sqrt{5}) / 8, (7 - 3 \sqrt{5}) / 32$ , respectively. The roots obey $2 < \sqrt{5} < 3$ by [F16,F17], and $3 \sqrt{5} < 7$ since both sides are positive and their squares satisfy $45 < 49$ ; hence all these determinants are positive. Thus the finite cases are positive definite by [F9], their groups are finite by [F6], and the finite classification [F7] gives their names; the equality paths are exactly the listed affine diagrams.

4.1F2F20F4F5F24F25F26step 3.1step 3.2algebra

Now suppose there is no edge labelled at least $4$ , so every edge has label $3$ , and the tree has a vertex of degree at least $3$ . A degree- $4$ vertex produces ${\tilde{D}}_{4}$ as in Step 3.1; if there are two degree- $3$ vertices, the same construction there produces a ${\tilde{D}}_{k}$ subdiagram. The subdiagram has the exact standard $\tilde{D}$ labels, is not positive definite by [F4], and therefore cannot be a proper principal submatrix by [F2]; it must be all of $Γ$ . If there is one degree- $3$ vertex, let $1 \leq p \leq q \leq r$ be the numbers of vertices on its arms. When $r \geq 2$ , deleting a terminal vertex of the longest arm leaves a proper connected positive definite subdiagram by [F2]; applying the three-arm inequality [F24] to that subdiagram gives $\frac{1}{p + 1} + \frac{1}{q + 1} + \frac{1}{r} > 1$ . If $r = 1$ , the arms are $(1, 1, 1)$ and Step 3.2 shows the star is finite $D_{4}$ and positive definite, so it cannot be affine.

5.1F21F25step 3.2step 4.1algebra

The integer solutions to the inequality in Step 4.1, with $p \leq q \leq r$ , are $(1, 1, r)$ for $r \geq 2$ , $(1, 2, r)$ for $2 \leq r \leq 5$ , $(1, 3, 3)$ , and $(2, 2, 2)$ . Indeed, $p \geq 3$ makes the sum at most $1 / 4 + 1 / 4 + 1 / 3 < 1$ . If $p = 2$ and $q \geq 3$ , the sum is at most $1 / 3 + 1 / 4 + 1 / 3 < 1$ , so $q = 2$ and then $1 / 3 + 1 / 3 + 1 / r > 1$ forces $r = 2$ . If $p = 1$ and $q \geq 4$ , the sum is at most $1 / 2 + 1 / 5 + 1 / 4 < 1$ , so $q \leq 3$ ; with $q = 2$ the inequality gives $r \leq 5$ , with $q = 3$ it gives $r = 3$ , and $q = 1$ allows every $r \geq 2$ . The $(1, 1, r)$ stars and $(1, 2, 2), (1, 2, 3), (1, 2, 4)$ are the positive definite finite stars of Step 3.2 and so are excluded. The remaining cases $(2, 2, 2)$ , $(1, 3, 3)$ , and $(1, 2, 5)$ are precisely ${\tilde{E}}_{6}, {\tilde{E}}_{7}, {\tilde{E}}_{8}$ by [F21], as required.

6.1F1F3F5F15F18F19F20F21F22F25step 1.1step 2.1step 1.3step 2.2step 2.3step 3.1step 4.1step 5.1step 2.4step 3.3algebra∎

The cases above exhaust connected affine form type diagrams: rank at most $2$ gives only ${\tilde{A}}_{1}$ ; in rank at least $3$ , Step 1.3 gives the all- $3$ cycle case or a tree, Step 2.2 handles two or more labels at least $4$ , Step 3.1 handles a single large label with a branch, Step 2.3 excludes an all- $3$ path, Steps 4.1-5.1 handle the remaining all- $3$ trees, and Steps 2.4-3.3 handle the remaining paths. Reading the resulting graph shapes against [F3,F18,F19,F20,F21,F22] gives exactly the list in the statement. Its families have no infinite labels above rank $2$ , all finite labels among $3, 4, 6$ , and at most two edges labelled at least $4$ ; the cyclic case is exactly ${\tilde{A}}_{n}$ ( $n \geq 2$ ). The low-rank aliases in the statement follow from [F15]. All witnesses and constructions use finitely many vertices and explicit formulas, so no Choice is used.

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

Classification of affine Coxeter diagrams and their Euclidean simplex realization

Statement

Let $S$ be finite, $m$ a Coxeter matrix, $W$ the presented Coxeter group with length $ℓ$ , $V = R^{S}$ , the Coxeter form $B$ , the canonical reflection homomorphism $ρ$ and the diagram $Γ$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Coxeter diagrams: edges, labels, components and finite type).

(1) Classification. Assume $Γ$ is connected. Then $B$ is positive semidefinite of corank one (Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice (1)) if and only if $Γ$ is isomorphic as a labelled graph to one of the standard affine diagrams ${\tilde{A}}_{1}; {\tilde{A}}_{n} (n \geq 2); {\tilde{B}}_{n} (n \geq 3); {\tilde{C}}_{n} (n \geq 2); {\tilde{D}}_{n} (n \geq 4); {\tilde{E}}_{6}, {\tilde{E}}_{7}, {\tilde{E}}_{8}; {\tilde{F}}_{4}; {\tilde{G}}_{2}$ (The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde), with the low-rank coincidences ${\tilde{A}}_{1} = {\tilde{C}}_{1}$ , ${\tilde{B}}_{2} = {\tilde{C}}_{2}$ , ${\tilde{D}}_{3} = {\tilde{A}}_{3}$ , ${\tilde{E}}_{4} = {\tilde{A}}_{4}$ , ${\tilde{E}}_{5} = {\tilde{D}}_{5}$ ; modulo these the list is duplicate-free, ${\tilde{A}}_{1}$ is the only diagram with a label $\infty$ , and every other label lies in ${3, 4, 6}$ . Equivalently: $Γ$ is connected, $B$ is positive semidefinite and $B$ is not positive definite.

(2) The radical and the affine slice. For a connected affine $Γ$ : $rad (B) = R δ$ for a positive radical vector $δ$ (Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (1)); with the radical quotient $U$ , the Euclidean form $b$ , the affine slice $E = E_{δ}$ , the walls $H_{s}$ and the alcove $A$ , the conclusions of The affine slice: faithful isometric action, the alcove simplex, and its facet reflections hold: the dual action of $W$ on $E$ is faithful and by affine isometries, $\overset{ˉ}{A} = conv {v_{s}}$ is a Euclidean simplex of dimension $∣ S ∣ - 1$ whose facets are the walls $H_{s}$ , $\sum_{s} δ_{s} φ (e_{s}) = 1$ on $E$ , the facet normals have Gram matrix $C = (B (e_{s}, e_{t}))$ , distinct alcove interiors are disjoint and $w \overset{ˉ}{A} \cap \overset{ˉ}{A}$ is the closed face of type $S (w)$ .

(3) Realization and matching with the crystallographic alcoves. If $Γ$ is one of the standard diagrams $D$ , let $Φ$ be an irreducible finite crystallographic root system of the matching Weyl type supplied by Crystallographic alcove diagrams: the affine list realized by Weyl types A–G (4), with affine Weyl group $W_{a} (Φ) = Q^{\lor} ⋊ W (Φ)$ , fundamental alcove $A_{Φ}$ and facet reflections $s_{0}, s_{1}, \dots$ (Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group, Highest-root dominance and the fundamental alcove). Then the facet-reflection matrix of $A_{Φ}$ equals the Coxeter matrix of $D$ (Crystallographic alcove diagrams: the affine list realized by Weyl types A–G (1)), so $W ≅ W (m) ≅ W_{a} (Φ)$ (Alcove transitivity, the affine Coxeter presentation, and the length function (2)); $W_{a} (Φ)$ acts on its Euclidean space properly discontinuously and cocompactly, $\overline{A_{Φ}}$ is a strict fundamental domain, and $W_{a} (Φ)$ acts simply transitively on the alcoves, with $ℓ (g)$ the number of walls separating $A_{Φ}$ from $g A_{Φ}$ (clauses (1)-(3) of that theorem). Moreover the inward unit normals of the facets of $A_{Φ}$ have the same Gram matrix $C$ as the facet normals of the slice simplex $\overset{ˉ}{A}$ of (2), with facets matched by the labelling; by Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet the two simplices are similar with facets matched, so the similarity intertwines the two reflection group actions. Consequently, for the slice realization of (2): the alcoves $w \overset{ˉ}{A}$ ( $w \in W$ ) tile $E$ , $W$ acts properly discontinuously and cocompactly on $E$ with $\overset{ˉ}{A}$ a strict fundamental domain, $W$ acts simply transitively on the alcoves, and $ℓ (w)$ is the number of walls separating $\overset{ˉ}{A}$ from $w \overset{ˉ}{A}$ .

(4) Consequences and conventions. For connected $Γ$ : $W$ is finite if and only if $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)); if $Γ$ is affine then $W$ is infinite and every proper standard parabolic $W_{T}$ $(T ⊊ S)$ is finite (Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (2)), and conversely an infinite connected Coxeter system with all proper parabolics finite need not be affine: its form need not be positive semidefinite (the companion page gives an indefinite example). Equivalently, a connected system is of affine form type if and only if it has a faithful Euclidean simplex reflection realization with a bounded fundamental Coxeter chamber, whose interior dihedral angles are $π / m_{s t}$ and whose facet-reflection matrix is its own Coxeter matrix, the realization being the one constructed in (2)-(3). The facet-generated group of this realization is $Q^{\lor} ⋊ W (Φ)$ ; the extended group $P^{\lor} ⋊ W (Φ)$ , with $P^{\lor}$ the coweight lattice and $Q^{\lor}$ the coroot lattice (Root, coroot, weight, and coweight lattices), strictly contains it when $P^{\lor} \neq Q^{\lor}$ , and coincides with it when the two lattices are equal. No assertion is made about twisted Lie-theoretic diagrams. No choice principle is used.

Facts & Assumptions

Given: The finite Coxeter matrix, group, canonical representation, diagram and form of the statement.

[F1]

Connected positive-semidefinite non-positive-definite cosine forms have a positive radical ray, corank one and positive-definite proper principal submatrices; every proper standard parabolic is finite (Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions (1)–(2)).

[F2]

Connected affine form type diagrams are exactly the standard affine list, including the all- $3$ cycles, as proved by Enumeration of the connected positive semidefinite corank-one diagrams.

[F3]

Each standard affine diagram has positive-semidefinite corank-one cosine form, occurs as a crystallographic alcove diagram, and has the listed aliases, labels and facet-normal Gram matrix (Crystallographic alcove diagrams: the affine list realized by Weyl types A–G (1)–(4),(7)).

[F4]

The slice action is faithful and isometric, its closure is a bounded simplex with the stated vertices, facet normals, affine relation, facet-reflection action and intersection formula (The affine slice: faithful isometric action, the alcove simplex, and its facet reflections (1)–(5)).

[F5]

Every standard affine diagram occurs from a finite crystallographic root system of the matching Weyl type (Crystallographic alcove diagrams: the affine list realized by Weyl types A–G (4)).

[F6]

Fundamental facet reflections generate the affine Weyl group, their actual Coxeter presentation is exact, the action on alcoves is simply transitive, and length counts separating walls (Alcove transitivity, the affine Coxeter presentation, and the length function (1)–(3)).

[F7]

Bounded Euclidean simplices with the same inward unit-normal Gram matrix admit a facet-matching similarity conjugating their facet reflections (Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet).

[F8]

Finiteness of a Coxeter group is equivalent to positive definiteness of its form (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)).

[F9]

A finite-dimensional real vector space is not a finite union of proper linear subspaces (A finite-dimensional vector space over an infinite field is not a finite union of proper subspaces).

[F10]

A geometric simplex is the convex hull of finitely many affinely independent vertices (The geometric simplex spanned by affinely independent vertices).

[F11]

The convex hull of finitely many points in a finite-dimensional real topological vector space is compact (Convex closures and hulls of finitely many compact convex sets).

[F12]

The support criterion is $w \in W_{J}$ if and only if $S (w) \subseteq J$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (1)).

[F13]

The fundamental root alcove is a bounded geometric simplex (Highest-root dominance and the fundamental alcove (2)).

[F14]

Finite-dimensional normed spaces are locally compact (A normed space is locally compact if and only if it is finite-dimensional), and in a locally compact metric space every point has arbitrarily small compact closed balls (In a locally compact metric space every point has arbitrarily small compact closed balls, hence a neighbourhood base of compact sets).

[F15]

The affine-wall arrangement is locally finite: every compact set meets only finitely many walls (Affine reflections: translation form, involutivity, local finiteness, and $W_{a} = Q^{\lor} ⋊ W$ (3)).

[F16]

The affine Weyl group has the Euclidean decomposition $W_{a} = Q^{\lor} ⋊ W (Φ)$ (Affine reflections: translation form, involutivity, local finiteness, and $W_{a} = Q^{\lor} ⋊ W$ (4)).

[F17]

The Weyl group of a finite crystallographic root system is finite (The Weyl group is finite and faithful).

[F18]

The simple coroots form a real basis of the root-system space and integrally generate every coroot, so $Q^{\lor} = ⨁_{s} Z α_{s}^{\lor}$ (Affine reflections: translation form, involutivity, local finiteness, and $W_{a} = Q^{\lor} ⋊ W$ , Remark).

[F19]

The real Coxeter form has entries $B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite labels (The real Coxeter form, its radical, reflections, and form-preserving maps (2)).

[F20]

A basis-coordinate map identifies a finite-dimensional normed space with its coordinate space and is continuous (A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space).

[F21]

$Q^{\lor}$ is contained in the coweight lattice $P^{\lor}$ ; both are the lattices defined from the roots and coroots (Root, coroot, weight, and coweight lattices).

[F22]

$π > 0$ (Pi as twice the smallest positive zero of cosine).

[F23]

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

[F24]

$\cos (π - x) = - \cos x$ for every real $x$ (Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions).

[F25]

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

[F26]

Cosine is strictly decreasing on $[0, π]$ (Signs, monotonicity intervals, and ranges of sine and cosine).

[F27]

Every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Proof

technique · direct; all finite coordinate constructions use no choice principle

1.1F1F2F3algebra

For a connected $Γ$ with positive-semidefinite corank-one $B$ , [F2] enumerates the standard affine diagram. Conversely [F3] proves that each standard diagram has positive-semidefinite corank-one cosine form and occurs as a crystallographic alcove diagram; a relabelling permutes matrix rows and columns and preserves these properties. The aliases, nonisomorphism and label assertions are also [F3]. Finally [F1] proves that connected positive-semidefinite non-positive-definite is equivalent to corank one, without assuming it in advance.

1.2F1F4

Under these equivalent conditions, [F1] gives $rad (B) = R δ$ with all $δ_{s} > 0$ . Apply [F4] to obtain exactly the radical quotient, Euclidean slice, faithful action, coordinates, simplex, normals and intersection conclusions in (2), including rank one.

1.3F3F4F5F6F7F13

Choose the matching root system from [F5], using the actual affine type and aliases of [F3]. Its bounded fundamental alcove is supplied by [F13], and the inward unit normals have the same Gram matrix as the slice normals by [F3,F4]. The simplex similarity [F7] therefore matches their labelled facets and conjugates the corresponding reflections. By [F6], the fundamental reflections give the Coxeter group of this matrix and generate $W_{a} (Φ)$ . Thus this similarity intertwines the homomorphisms from $W$ to the two reflection groups and identifies $W$ with $W_{a} (Φ)$ , preserving its simple generators and lengths.

1.4F1F8F19F22F23F24F25F26F27algebra

The finite criterion is [F8]. For affine $Γ$ , its form is not positive definite, so $W$ is infinite, while every proper standard parabolic is finite by [F1]. The converse fails already for the triangle with labels $(3, 3, 4)$ . Put $c_{3} = \cos (π / 3)$ . Since $0 < π / 3 < π / 2$ by [F22], strict decrease [F26] and [F25] give $c_{3} > 0$ ; [F23] and [F24] give $2 c_{3}^{2} - 1 = \cos (2 π / 3) = - c_{3}$ , so $(2 c_{3} - 1) (c_{3} + 1) = 0$ and $c_{3} = 1 / 2$ . Put $c_{4} = \cos (π / 4)$ . Again $0 < π / 4 < π / 2$ gives $c_{4} > 0$ , and [F23,F25] give $2 c_{4}^{2} - 1 = 0$ ; by [F27], $c_{4} = \sqrt{2} / 2$ . Therefore the displayed Coxeter matrix [F19] has quadratic value $3 + 2 (- c_{3} - c_{3} - c_{4}) = 1 - \sqrt{2} < 0$ on $(1, 1, 1)$ , while its value on each basis vector is $1$ ; the strict inequality follows from $(1 - \sqrt{2}) (1 + \sqrt{2}) = - 1$ and $1 + \sqrt{2} > 0$ . Every proper two-generator form has determinant $1 - \cos^{2} (π / m) > 0$ for $m = 3, 4$ , so its group is finite by [F8]; the whole group is infinite by that same criterion. Thus infiniteness together with finite proper parabolics cannot replace positive semidefiniteness.

2.1F6F9F14F15step 1.3choosealgebra

Every point $x$ of the root-system Euclidean space lies in the closure of an alcove. By [F14] choose a compact closed ball $K$ about $x$ ; by [F15] only finitely many walls meet $K$ . The finitely many walls in $K$ not containing $x$ have positive distance from $x$ , so a smaller open ball $U$ about $x$ misses all of them. The direction hyperplanes of the finitely many walls through $x$ , together with the zero subspace, are proper linear subspaces because $\dim E \geq 1$ . By [F9] choose a nonzero direction $v$ outside their union. For all sufficiently small $ε > 0$ , the point $x + ε v$ lies in $U$ and lies on none of the walls through $x$ , so the short ray lies in one connected component of the wall complement. Its closure contains $x$ . By [F6], every alcove is a translate of $A_{Φ}$ , so the alcove closures cover the space. Their interiors are disjoint because they are distinct connected components. Hence the transported slice closures tile $E$ .

3.1F4F6F12step 1.3step 2.1algebra

The slice closed alcove meets each orbit in exactly one point. Coverage gives existence. If $x, y \in \overset{ˉ}{A}$ and $y = w x$ , then $y \in w \overset{ˉ}{A} \cap \overset{ˉ}{A}$ ; [F4]'s intersection formula gives $y (e_{s}) = 0$ for every $s \in S (w)$ . Every corresponding facet reflection fixes $y$ , and [F12] gives $w \in W_{S (w)}$ , so $w y = y$ . Consequently $x = w^{- 1} y = y$ . This proves the strict closed fundamental-domain assertion, which transfers to $\overline{A_{Φ}}$ . Simple transitivity on open alcoves and the separating-wall length formula transfer from [F6]; similarity sends the entire family of reflected walls bijectively to the slice walls, so it preserves which walls separate two alcoves.

3.2F10F11F13F16F17F18F20step 1.3step 2.1algebra

By [F16], each affine Weyl element has a unique form $t_{q} u$ with $q \in Q^{\lor}$ and $u \in W (Φ)$ . If $(t_{q} u) K \cap K \neq \emptyset$ for a compact $K$ , then $q \in K - u K$ . There are finitely many $u$ by [F17]. A compact subset of a metric space is bounded (cover it by unit balls and take a finite subcover), and $u$ preserves the metric, so each $K - u K$ is bounded. By [F18], the simple coroots form a finite basis and integrally generate $Q^{\lor}$ ; after a finite enumeration, each basis-coordinate function is linear and continuous by [F20]. Its values on $K$ and on $u K$ are bounded: continuity gives a neighborhood of each point on which the coordinate differs by less than $1$ from its value at that point, and compactness gives a finite subcover and hence a finite bound. Since $u$ is an isometry, $u K$ is compact. Linearity then bounds the coordinate of each difference $x - u y$ by the sum of the two coordinate bounds, so the coordinates of $K - u K$ are bounded. Since $Q^{\lor}$ consists of integer coordinate tuples in this basis, only finitely many $q$ occur. Thus only finitely many affine Weyl elements meet $K$ back to itself, proving proper discontinuity. By [F13], $\overline{A_{Φ}}$ is a geometric simplex, hence the convex hull of finitely many vertices by [F10]; it is compact by [F11]. Coverage by its translates proves cocompactness. Both properties transfer under the similarity in Step 1.3.

4.1F10F16F21step 1.2step 1.3step 3.2algebra∎

If $∣ S ∣ = 1$ , then $B = [1]$ is positive definite. No faithful realization by reflections in facets of a bounded Euclidean simplex is possible: a positive-dimensional bounded simplex has at least two facets, whereas a zero-dimensional simplex has no codimension-one reflecting hyperplane. Thus both sides of the stated equivalence are false in this case. Otherwise suppose the stated faithful Euclidean realization has a bounded fundamental Coxeter simplex of dimension $n \geq 1$ , with its $n + 1 = ∣ S ∣$ facets indexed by $S$ and labelled angles $π / m_{s t}$ (and disjoint endpoints in dimension one). Its inward unit-normal Gram matrix is the cosine matrix: in a two-dimensional normal section through a codimension-two face, the angle between inward normals is $π - π / m_{s t}$ ; in dimension one the two normals are opposite. The normals span the $n$ -dimensional direction space, since otherwise a nonzero perpendicular direction would leave all facet inequalities unchanged and make the simplex unbounded. There are $n + 1$ facets, so the Gram matrix is positive semidefinite of rank $n$ , hence corank one. This proves the reverse Euclidean implication; Steps 1.2–3.2 prove the forward one. The facet-generated translation subgroup is exactly $Q^{\lor}$ by [F16]. Therefore adjoining translations in $P^{\lor}$ gives a strictly larger group if $P^{\lor} \neq Q^{\lor}$ , and gives the same group if the lattices are equal by [F21]. No twisted-diagram claim is made. All arguments use finite bases and finite avoidance, without Choice.

5 · Examples, counterexamples and false statements

None yet.

Affine Coxeter Diagrams and Semidefinite Classification

1 · Prerequisites

2 · Summary

Items

3 · Logical flowchart

4 · Definitions, theorems and proofs

Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet

Statement

Facts & Assumptions

Proof

Irreducible affine Coxeter type: the corank-one form, the radical quotient, and the affine slice

Definition

Remarks

Positive radical, corank one, positive definiteness of proper submatrices, and domination exclusions

Statement

Facts & Assumptions

Proof

The standard affine diagrams A-tilde, B-tilde, C-tilde, D-tilde, E-tilde, F-tilde and G-tilde

Definition

Remarks

Remarks

The affine slice: faithful isometric action, the alcove simplex, and its facet reflections

Statement

Facts & Assumptions

Proof

Crystallographic alcove diagrams: the affine list realized by Weyl types A–G

Statement

Facts & Assumptions

Proof

Enumeration of the connected positive semidefinite corank-one diagrams

Statement

Facts & Assumptions

Proof

Classification of affine Coxeter diagrams and their Euclidean simplex realization

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources