Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 7 results · all verified · 0 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Finite Coxeter Invariants and Coinvariant Gradings

1 · Prerequisites

2 · Summary

This page applies finite complex reflection invariant theory to finite Coxeter groups, including noncrystallographic types. It proves the representation hypotheses needed by the published invariant-theory results, then develops the Coxeter-specific links among basic degrees, coinvariants, discriminants and Coxeter spectra.

The construction begins with Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers and Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system. The definition’s well-definedness is established by The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity, which proves multiset independence, the invariant and coinvariant Hilbert series, the degree product, and Molien’s identity. Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian proves the invariant Jacobian is nonzero by an explicit characteristic-zero argument.

The discriminant branch continues with The total degree sum, the invariant Jacobian as the discriminant, anti-invariants, and the top coinvariant class, which proves the total exponent sum, the Jacobian-discriminant identity, the anti-invariant description, and the one-dimensional top sign class. The spectral branch is proved from reflection models and exact matrices: The Coxeter elements of the classical types A_n, B_n, D_n and I_2(m): characteristic polynomials, orders and spectral exponents from their reflection models handles the classical types, while The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices gives the exceptional certificates. A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types brings the Coxeter-plane eigenvector, the discriminant degree and both spectrum lists together to identify all basic degrees, including reducible and low-rank cases.

Prerequisites and reading

Required earlier pages: bipartite-coxeter-elements-and-ordered-root-complexes, finite-coxeter-diagrams-and-complete-classification, finite-weyl-invariants-bruhat-and-kostant-harmonics, and relations-functions-and-quotients. The published complexification-realification-and-real-structures supplies scalar extension of the reflection representation. The remaining prerequisite pages provide root-system, polynomial-algebra and invariant-theory background. The companion finite-coxeter-invariants-and-coinvariant-gradings-examples applies these results and is a dependency leaf.

3 · Logical flowchart

4 · Definitions, theorems and proofs

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

Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers

Statement

Let $(W, S)$ be a Coxeter system of finite type with $S$ finite, $n : = ∣ S ∣$ , and let $V = R^{S}$ carry the Coxeter form $B$ , the canonical reflection representation $ρ : W \to G L (V)$ , the root system $Φ = Φ_{+} ⊔ Φ_{-}$ , the reflection set $T = {w s w^{- 1} : w \in W, s \in S}$ , the chamber $C$ and the conventions of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Coxeter diagrams: edges, labels, components and finite type, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, The dual action, chambers, faces, and root hyperplanes and Root sign coherence and the action of simple reflections on positive roots; every root has $B$ -norm one and $ρ$ is faithful (Descent of the reflection representation, unit root norms, and conjugation of reflections (3), The root-length criterion and faithfulness of the canonical reflection representation (3)), while $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite). Form the complexification $V_{C} : = C \otimes_{R} V$ (Complexification as $C \otimes_{R} V$ with its canonical real-linear embedding), the $C$ -bilinear extension $B_{C}$ of $B$ characterized by $B_{C} (z \otimes u, w \otimes v) = z w B (u, v)$ , the complexified map $ρ_{C} : = i d \otimes ρ$ (Complexification of a real-linear map) and the linear forms $ℓ_{α} (x) : = B_{C} (x, α)$ for $α \in Φ$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism fixes the eigenspace language).

(1) Complexification. $\dim_{C} V_{C} = n$ , the classes $1 \otimes e_{s}$ $(s \in S)$ form a $C$ -basis, $ρ_{C}$ is a homomorphism $W \to G L (V_{C})$ that is faithful in the sense of Intertwiners, the spaces ${Hom}_{G} (V, W)$ and ${End}_{G} (V)$ , equivalent representations, and faithful representations, and $ρ_{C} (W)$ is a finite subgroup of $G L (V_{C})$ of order $∣ W ∣$ .

(2) Complex reflections. For every $t \in T$ , written $t = t_{α}$ with its positive root $α \in Φ_{+}$ (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1)), the operator $ρ_{C} (t)$ is not the identity, its fixed space is the complex hyperplane ${x \in V_{C} : ℓ_{α} (x) = 0}$ , and $ρ_{C} (t) - {i d}_{V_{C}}$ has image the line $C α$ ; that is, $ρ_{C} (t)$ is a complex reflection. In particular every generator $ρ_{C} (s)$ is a complex reflection and $ρ_{C} (W)$ is generated by complex reflections.

(3) Essentiality. $V_{C}^{W} = {x : ρ_{C} (w) x = x for all w \in W} = {0}$ , and every $W$ -invariant linear form on $V_{C}$ is zero.

(4) Application under AC. Assume the Axiom of Choice. Then the conclusions of Chevalley shephard todd for finite weyl groups apply to the faithful finite subgroup $ρ_{C} (W) \leq G L (V_{C})$ generated by complex reflections: the invariant algebra $S^{W} : = C [V_{C}]^{W}$ is a polynomial $C$ -algebra on $n$ homogeneous algebraically independent basic invariants, and $C [V_{C}]$ is a graded free $S^{W}$ -module of rank $∣ W ∣$ ; moreover every minimal homogeneous invariant generating family of the ideal $S_{+}^{W} C [V_{C}]$ generates $S^{W}$ and is algebraically independent (Finite reflection invariant generators are algebraically independent, Reflection basic invariants form a regular sequence), and the coinvariant quotient satisfies the Hilbert-series and order conclusions of Weyl coinvariant hilbert series has order w dimension and Finite linear invariant and coinvariant polynomial algebras.

(5) Conventions and abstentions. For $n = 0$ (trivial group, $V = 0$ ) all assertions are the empty ones. No irreducibility, crystallographic integrality or highest-root data is assumed: the statement covers the noncrystallographic types $I_{2} (m)$ , $H_{3}$ , $H_{4}$ and reducible systems equally, since only the real reflection geometry and complexification are used. Nothing is asserted here about eigenvalues of the Coxeter element, about degrees beyond their existence, or about the coinvariant algebra as a $W$ -representation; those are later items. (4) is the only clause using Choice.

Facts & Assumptions

Given: A Coxeter system $(W, S)$ of finite type with $S$ finite, the space $V = R^{S}$ with its Coxeter form $B$ , the canonical reflection representation $ρ$ , the root system $Φ = Φ_{+} ⊔ Φ_{-}$ and the reflection set $T$ .

[F1]

The Coxeter form satisfies $B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ , the reflection formula is $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ , and $r_{a}$ is linear, involutive, fixes $\ker B (-, a)$ pointwise, and preserves $B$ ; $\ker B (-, a)$ is a hyperplane when $B (a, a) \neq 0$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order).

[F2]

The canonical reflection homomorphism is the unique homomorphism $ρ : W \to G L (V)$ with $ρ (s) = r_{e_{s}}$ , the root system is $Φ = {ρ (w) e_{s}}$ , and $T = {w s w^{- 1}}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections).

[F3]

$ρ$ preserves $B$ , every root $α \in Φ$ has $B (α, α) = 1$ , and $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ for all $w \in W$ , $s \in S$ ; moreover $ρ$ is injective (Descent of the reflection representation, unit root norms, and conjugation of reflections (2)-(4), The root-length criterion and faithfulness of the canonical reflection representation (3)).

[F4]

If $W$ is finite then $B$ is positive definite (Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)).

[F5]

The map $Φ_{+} \to T$ , $α \mapsto t_{α}$ , is a bijection, and $ρ (t_{α}) = r_{α}$ (The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange (1)).

[F6]

The complexification $V_{C} = C \otimes_{R} V$ carries the scalar action $z \cdot (w \otimes v) = (z w) \otimes v$ and the embedding $ι v = 1 \otimes v$ (Complexification as $C \otimes_{R} V$ with its canonical real-linear embedding). The tensor universal property induces a linear map from any balanced bilinear map (Universal property of the tensor product for balanced maps into abelian groups).

[F7]

For a real-linear $T : V \to V$ the complexification $T_{C} = {i d}_{C} \otimes T$ is $C$ -linear with $T_{C} (z \otimes v) = z \otimes T (v)$ (Complexification of a real-linear map).

[F8]

$E_{λ} (T) = \ker (T - λ I)$ is the eigenspace of $λ$ , and a representation is faithful when $ρ (g) = i d$ implies $g = e$ (Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism, Intertwiners, the spaces ${Hom}_{G} (V, W)$ and ${End}_{G} (V)$ , equivalent representations, and faithful representations).

[F9]

For a finite $G \leq G L (V)$ the invariant algebra is $R = S^{G}$ , $R_{+} = ⨁_{d > 0} R_{d}$ and $I = S R_{+}$ defines the coinvariant algebra $S / I$ (Finite linear invariant and coinvariant polynomial algebras).

[F10]

The Axiom of Choice is the statement that every family of nonempty sets has a choice function (The Axiom of Choice).

[F11]

Assume AC. For a finite complex reflection group $G \leq G L (V)$ : $S^{G}$ is a polynomial algebra on $n = \dim V$ homogeneous algebraically independent basic invariants and $S$ is free of rank $∣ G ∣$ over $S^{G}$ ; every minimal homogeneous invariant generating family of $I = S R_{+}$ generates $R$ and is algebraically independent; these generators form an $S$ -regular sequence and homogeneous lifts of a homogeneous basis of $S / I$ form a graded free $R$ -basis; and $Hilb (S / I, t) = \prod_{i} (1 + \dots + t^{d_{i} - 1})$ , $\dim_{C} (S / I) = \prod_{i} d_{i} = ∣ G ∣$ (Chevalley shephard todd for finite weyl groups, Finite reflection invariant generators are algebraically independent, Reflection basic invariants form a regular sequence, Weyl coinvariant hilbert series has order w dimension).

Proof

1.1F2F3F6F7F8

Every elementary tensor expands as $z \otimes v = \sum_{s} z v (s) (1 \otimes e_{s})$ , so the classes $1 \otimes e_{s}$ span. For each $t \in S$ , the balanced map $(z, v) \mapsto z v (t)$ induces by [F6] a map $p_{t} : V_{C} \to C$ ; it is complex-linear on elementary tensors and satisfies $p_{t} (1 \otimes e_{s}) = δ_{t s}$ . Applying $p_{t}$ to a linear relation proves independence. Thus these classes form a $C$ -basis and $\dim_{C} V_{C} = n$ . For $w \in W$ set $ρ_{C} (w) : = {i d}_{C} \otimes ρ (w)$ . By [F7] each $ρ_{C} (w)$ is $C$ -linear with $ρ_{C} (w) (z \otimes v) = z \otimes ρ (w) v$ ; on elementary tensors $ρ_{C} (w) ρ_{C} (w^{'}) (z \otimes v) = z \otimes ρ (w) ρ (w^{'}) v = ρ_{C} (w w^{'}) (z \otimes v)$ , and since elementary tensors span $V_{C}$ this gives $ρ_{C} (w w^{'}) = ρ_{C} (w) ρ_{C} (w^{'})$ , while $ρ_{C} (1) = i d$ ; so $ρ_{C} : W \to G L (V_{C})$ is a homomorphism. If $ρ_{C} (w) = {i d}_{V_{C}}$ , then for every $s$ the basis element $1 \otimes e_{s}$ is fixed, so $1 \otimes ρ (w) e_{s} = 1 \otimes e_{s}$ , and linear independence of the $1 \otimes e_{t}$ gives $ρ (w) e_{s} = e_{s}$ ; as $(e_{s})$ spans $V$ , $ρ (w) = {i d}_{V}$ , and injectivity of $ρ$ [F3] gives $w = 1$ . Hence $ρ_{C}$ is faithful in the sense of [F8], and its image, the image of the finite group $W$ , is a finite subgroup of $G L (V_{C})$ of order $∣ W ∣$ .

2.1F1F2F7step 1.1

Let $s \in S$ . By [F1], $B (e_{s}, e_{s}) = 1$ and $r_{s} (v) = v - 2 B (v, e_{s}) e_{s}$ for all $v \in V$ ; equivalently $r_{s} - {i d}_{V} = - 2 B (-, e_{s}) e_{s}$ has image $R e_{s}$ and kernel $\ker B (-, e_{s})$ , a hyperplane of $V$ . Complexifying with [F7], for $x = z \otimes v$ we get $(ρ_{C} (s) - {i d}_{V_{C}}) (x) = z \otimes (r_{s} - {i d}_{V}) v = - 2 B (v, e_{s}) (z \otimes e_{s}) = - 2 B_{C} (x, e_{s}) e_{s}$ , using $B_{C} (z \otimes v, e_{s}) = z B (v, e_{s})$ ; the same formula holds on all of $V_{C}$ by linearity. The image is therefore the line $C e_{s}$ (it contains $- 2 e_{s} \neq 0$ ) and the kernel is ${x : B_{C} (x, e_{s}) = 0}$ , a complex hyperplane because $B_{C} (e_{s}, e_{s}) = 1$ makes the linear form $B_{C} (-, e_{s})$ nonzero. In particular $ρ_{C} (s) \neq {i d}_{V_{C}}$ fixes that hyperplane pointwise: each generator is a complex reflection, and $ρ_{C} (W) = ⟨ ρ_{C} (s) : s \in S ⟩$ is generated by complex reflections.

3.1F2F3F5step 2.1

Let $t \in T$ . By [F5] there is a unique $α \in Φ_{+}$ with $t = t_{α}$ and $ρ (t_{α}) = r_{α}$ ; by [F3] every root has $B$ -norm one, so $r_{α} (x) = x - 2 B (x, α) α$ for $x \in V$ , and $r_{α}$ fixes $\ker B (-, α)$ pointwise and maps $α \mapsto - α$ . Complexifying as in 2.1 gives $(ρ_{C} (t_{α}) - i d) (x) = - 2 B_{C} (x, α) α$ for all $x \in V_{C}$ , so the image is the line $C α$ and, since $B_{C} (α, α) = 1$ , the kernel ${x : ℓ_{α} (x) = 0}$ is a complex hyperplane. Thus every $t \in T$ acts as a complex reflection, and by [F2] and [F3] every element of $T$ is conjugate in $W$ to a simple reflection, in agreement with 2.1.

3.2F1F3F4F6step 2.1

Let $x = \sum_{s \in S} z_{s} \otimes e_{s} \in V_{C}$ be fixed by every $ρ_{C} (w)$ . Fixing $x$ under $ρ_{C} (s)$ and applying the formula of 2.1 gives $- 2 B_{C} (x, e_{s}) e_{s} = 0$ , hence $B_{C} (x, e_{s}) = 0$ for every $s \in S$ . Let $G$ be the Gram matrix of $B$ in the basis $(e_{s})$ ; by [F4] $B$ is positive definite, so $G$ is a real symmetric positive definite matrix, and $B_{C} (x, e_{s}) = 0$ for all $s$ says $G z = 0$ for the coefficient column $z = (z_{s})$ . Writing $z = a + b i$ with real columns $a, b$ , the real part $a^{T} G a + b^{T} G b$ of $z^{*} G z$ vanishes, and both summands are $\geq 0$ with equality only for $a = b = 0$ ; hence $z = 0$ and $x = 0$ . So $V_{C}^{W} = {0}$ . If $ℓ : V_{C} \to C$ is a $W$ -invariant linear form, then $ℓ (e_{s}) = ℓ (ρ_{C} (s) e_{s}) = ℓ (- e_{s})$ by 2.1, so $ℓ (e_{s}) = 0$ for every $s$ , and since the $e_{s}$ span $V_{C}$ over $C$ , $ℓ = 0$ . These finite-dimensional calculations use no choice principle: the basis $(e_{s})$ , the Gram matrix and the coefficient column are explicitly given by the finite set $S$ .

4.1F9F10F11step 1.1step 2.1step 3.1

Assume AC. By step 1.1 the subgroup $ρ_{C} (W) \leq G L (V_{C})$ is finite of order $∣ W ∣$ , faithful and, by steps 2.1 and 3.1, generated by complex reflections; $V_{C}$ has dimension $n$ . Applying [F11] with $S = C [V_{C}]$ and $G = ρ_{C} (W)$ gives that $S^{W}$ is a polynomial $C$ -algebra on $n$ homogeneous algebraically independent basic invariants, that $S$ is graded free of rank $∣ W ∣$ over $S^{W}$ , that every minimal homogeneous invariant generating family of $I = S S_{+}^{W}$ generates $S^{W}$ and is algebraically independent, that such generators form a regular sequence with the freeness conclusion, and the stated Hilbert-series and order conclusions for $S / I$ ; by [F9] this $I$ is exactly the positive-degree ideal of the coinvariant algebra of the definition. This is the only place where AC is used.

5.1F4F5F11step 4.1∎

For $n = 0$ the group $W$ is trivial, $V = V_{C} = 0$ , the basis family is empty, $ρ_{C}$ is the unique map of the trivial groups, $S = C$ and all products and families displayed in (4) are empty; every assertion above holds in this empty form. Nothing in steps 1.1-4.1 uses irreducibility of the diagram, crystallographic integrality, or highest-root data: only the finiteness of $W$ , the positivity of $B$ , the bijection $Φ_{+} \to T$ and the reflection formula are used, so reducible systems and the noncrystallographic types $I_{2} (m)$ , $H_{3}$ , $H_{4}$ are covered. No claim is made here about eigenvalues of a Coxeter element, about the values of the degrees, or about $S / I$ as a $W$ -representation; and the only clause depending on AC is (4), consumed through [F11].

DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-10-08Open item page →

Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system

Definition

Let $(W, S)$ be a Coxeter system of finite type with $n = ∣ S ∣$ , and let $V_{C}$ , $ρ_{C}$ , $S : = C [V_{C}]$ , $R : = S^{W}$ , $R_{+} = ⨁_{d > 0} R_{d}$ and $I : = S R_{+}$ be as in Finite linear invariant and coinvariant polynomial algebras and Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers; fix coordinates $x_{1}, \dots, x_{n}$ so that $S = C [x_{1}, \dots, x_{n}]$ as the iterated polynomial ring of Polynomial rings in finitely many commuting indeterminates by iteration, with the graded structure of Nonnegatively graded rings and modules, homogeneous elements, and twists.

(1) Existence of a basic family. Assume the Axiom of Choice. A minimal finite family $f_{1}, \dots, f_{r}$ of homogeneous positive-degree invariants generating the ideal $I$ exists, generates $R$ as a $C$ -algebra, and is algebraically independent (Finite reflection invariant generators are algebraically independent); under the stated AC any such minimal family has exactly $n$ elements (Chevalley shephard todd for finite weyl groups, Reflection basic invariants form a regular sequence).

(2) Degrees and exponents. Fix one such family $f_{1}, \dots, f_{n}$ and call $d_{i} : = \deg f_{i} the basic degrees, e_{i} : = d_{i} - 1 the exponents,$ arranged in nondecreasing order $d_{1} \leq \dots \leq d_{n}$ . Each $d_{i}$ is a positive integer, and in fact $d_{i} \geq 2$ : a degree-one invariant would be a nonzero $W$ -invariant linear form, excluded by Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (3).

(3) The graded coinvariant algebra. The coinvariant algebra of $(W, V_{C})$ is the graded quotient $A : = S / I = ⨁_{k \geq 0} A_{k}, A_{k} : = (S_{k} + I) / I,$ with $A_{0} = C$ (constants survive because $I$ has positive degree), as in Finite linear invariant and coinvariant polynomial algebras. Its Hilbert series is written $Hilb (A, t)$ .

(4) Well-definedness warning. This definition asserts nothing about the multiset ${d_{1}, \dots, d_{n}}$ being independent of the chosen family, nothing about $Hilb (A, t)$ or $\prod_{i} d_{i}$ , and no relation between the $d_{i}$ and the reflection geometry; all of that is the content of the justifier The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity ↗.

(5) Conventions. For $n = 0$ the group is trivial, $S = R = C$ , $I = 0$ , $A = C$ and the families and products are empty. For reducible $S$ the family in (2) is the concatenation of the families of the connected components, and $I$ is generated by all components' positive-degree invariants. No crystallographic or integrality assumption is made.

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

The Coxeter elements of the classical types A_n, B_n, D_n and I_2(m): characteristic polynomials, orders and spectral exponents from their reflection models

Statement

For each of the types $A_{n}$ $(n \geq 1)$ , $B_{n}$ $(n \geq 2)$ , $D_{n}$ $(n \geq 4)$ and $I_{2} (m)$ $(m \geq 3)$ take the irreducible Coxeter system $(W, S)$ of that type with the standard diagram numbering of Classification of finite Coxeter systems, including the H and dihedral families (1) (for $D_{n}$ the star with arms of $1, 1, n - 3$ edges; for $I_{2} (m)$ the two vertices joined by an edge of label $m$ ), let $ρ_{C} : W \to G L (V_{C})$ be the complexified canonical reflection representation of Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers, and let $c \in W$ be the bipartite Coxeter element of The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a (equivalently: the product, in the bipartite convention, of the two colour-class products of the tree $Γ$ , each of which has commuting factors; for a path diagram the colour classes are the even- and odd-numbered vertices, and for $D_{n}$ the classes are determined by the distance from the branch node). Write $ζ_{h} : = \exp (2 π i / h)$ for $h \geq 2$ and write $ζ_{h}^{r}$ for the complex number $\exp (2 π i r / h)$ ; call spectral exponents the multiset of residues $r \in {1, \dots, h - 1}$ such that the eigenvalues of the operator are the numbers $\exp (2 π i r / h)$ , each with multiplicity. Then $ρ_{C} (c)$ is diagonalisable (A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation), has order $h$ equal to the Coxeter number displayed below, and has the displayed characteristic polynomial in $X = λ$ and spectral exponents:

$A_{n} : \det (X - ρ_{C} (c)) = X^{n} + X^{n - 1} + \dots + X + 1, h = n + 1, {e_{i}} = {1, 2, \dots, n};$

$B_{n} : \det (X - ρ_{C} (c)) = X^{n} + 1, h = 2 n, {e_{i}} = {1, 3, 5, \dots, 2 n - 1};$

$D_{n} : \det (X - ρ_{C} (c)) = (X^{n - 1} + 1) (X + 1) = X^{n} + X^{n - 1} + X + 1, h = 2 n - 2, {e_{i}} = {1, 3, \dots, 2 n - 3, n - 1};$

$I_{2} (m) : \det (X - ρ_{C} (c)) = X^{2} - 2 \cos (2 π / m) X + 1, h = m, {e_{i}} = {1, m - 1} .$

The exponent lists are multisets; in type $D_{n}$ , the extra residue $n - 1$ is counted again when $n$ is even. In particular $h = \max_{i} (d_{i})$ is not asserted here (the exponents are only named $e_{i}$ ); the transfer to invariant degrees is the content of the final determination theorem of this page.

Facts & Assumptions

Given: One of the four types $A_{n}$ , $B_{n}$ , $D_{n}$ , $I_{2} (m)$ with its Coxeter system $(W, S)$ , and the bipartite Coxeter element $c$ of the bipartite definition.

[F1]

In the standard coordinates of $R^{N}$ (and in the sum-zero hyperplane of $R^{n + 1}$ for type $A_{n}$ ) the reduced crystallographic root systems $A_{N}$ , $B_{N}$ , $C_{N}$ , $D_{N}$ are the classical sets displayed in Classical root systems in coordinates, with the standard simple roots $α_{i}$ and Dynkin diagram; the reflection in a root $α$ is $s_{α} (x) = x - \frac{2 (x, α)}{(α, α)} α$ (Reduced crystallographic Euclidean root system).

[F2]

The Coxeter form satisfies $B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for finite $m (s, t)$ , the reflection formula is $r_{a} (v) = v - 2 B (v, a) a / B (a, a)$ , the canonical representation $ρ : W \to G L (V)$ is a homomorphism, and its complexification $ρ_{C}$ is faithful (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2), The canonical reflection homomorphism, roots, reflections, and the positive cone, Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (1)).

[F3]

For an irreducible finite type system the diagram $Γ$ is one of the standard trees in the finite classification, and its bipartition $J ⊔ K$ gives commuting color-class products $a = \prod_{s \in J} s$ and $b = \prod_{s \in K} s$ ; $c = a b$ and $h = ord (c)$ are well defined, and reversing the classes replaces $c$ by a conjugate since $b a = a^{- 1} (a b) a$ (The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a (1), Coxeter diagrams: edges, labels, components and finite type (2), Classification of finite Coxeter systems, including the H and dihedral families (1)).

[F4]

For the rank-two system $I_{2} (m)$ with simple reflections $r_{s}, r_{t}$ : the product $A = r_{s} r_{t}$ has, in the basis $(e_{s}, e_{t})$ , matrix $(\begin{array}{cc} 4 c^{2} - 1 & - 2 c \\ 2 c & - 1 \end{array})$ with $c = \cos (π / m)$ , determinant $1$ , and for finite $m$ trace $2 \cos (2 π / m)$ , $A^{m} = i d$ and $A^{k} \neq i d$ for $0 < k < m$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (3)(iii)-(iv)).

[F5]

For a square matrix $A$ the characteristic polynomial is $χ_{A} (x) = \det (x I - A)$ , and the determinant is the alternating multilinear function of the columns (For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix). The alternating multilinear property is proved in The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring. Determinants are invariant under similarity (Similar matrices over a commutative ring have the same determinant), so an eigenbasis computes the determinant factors of a diagonalisable operator. Over a field, a positive-sized square matrix is invertible exactly when its determinant is nonzero (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit); hence $\det (λ I - M) = 0$ exactly when $λ$ is an eigenvalue.

[F6]

An operator of finite order on a finite-dimensional vector space over an algebraically closed field of characteristic $0$ is diagonalisable (Over an algebraically closed field of characteristic $0$ , every element of finite order acts diagonalisably in a finite-dimensional representation, A diagonalisable endomorphism is one admitting a basis of eigenvectors, equivalently a diagonal matrix representation, Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism).

[F7]

For real $θ$ one has $\exp (i θ) = \cos θ + i \sin θ$ , $∣ \exp (i θ) ∣ = 1$ , and $e^{i π} + 1 = 0$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ , $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , Sine and cosine defined by their real power series); the complex exponential is multiplicative, $\exp (z + w) = \exp z \exp w$ ( $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential).

[F8]

For $h \geq 1$ the $h$ -th roots of unity are exactly the $h$ distinct numbers $\exp (2 π i k / h)$ , $k = 0, \dots, h - 1$ ; consequently the roots of $t^{h} - 1$ are these numbers, and the roots of $t^{N} + 1$ are $\exp (i (π + 2 π k) / N) = \exp (2 π i (2 k + 1) / (2 N))$ , $k = 0, \dots, N - 1$ , all distinct (The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ with $z = - 1 = \exp (i π)$ of modulus one, The group $μ_{n} (K)$ of $n$ -th roots of unity in a field, and primitive $n$ -th roots of unity).

[F9]

Cycle notation $(a_{0} a_{1} \dots a_{k - 1})$ denotes the permutation sending $a_{i}$ to $a_{i + 1}$ and $a_{k - 1}$ to $a_{0}$ , juxtaposition means composition, and the fixed points and orbits of a permutation are read off from its cycle decomposition (The finite symmetric group $S_{n}$ , one-line notation, and cycle notation).

Proof

1.1F1F2F3

(Model and identification.) Let $E$ be the model space of [F1] with its standard inner product, with simple roots $α_{1}, \dots, α_{n}$ : for $A_{n}$ , $E = {(y_{1}, \dots, y_{n + 1}) : \sum y_{i} = 0} \subseteq R^{n + 1}$ and $α_{i} = ε_{i} - ε_{i + 1}$ ; for $B_{n}$ , $E = R^{n}$ , $α_{i} = ε_{i} - ε_{i + 1}$ $(i < n)$ and $α_{n} = ε_{n}$ ; for $D_{n}$ , $E = R^{n}$ , $α_{i} = ε_{i} - ε_{i + 1}$ $(i \leq n - 1)$ and $α_{n} = ε_{n - 1} + ε_{n}$ . These are the standard simple roots and diagram numbering of [F1] and of the classification [F3]: the branch node of the $D_{n}$ star is $α_{n - 2}$ with arms $α_{n - 1}, α_{n}$ of one vertex each and $α_{1}, \dots, α_{n - 3}$ of $n - 3$ vertices. Define the linear map $φ : V \to E$ by $φ (e_{i}) : = α_{i} / ∣ α_{i} ∣$ on the basis $(e_{i})$ of $V$ . For every pair: $(α_{i}, α_{i}) = 2$ except $(α_{n}, α_{n}) = 1$ in type $B_{n}$ , so $φ$ preserves norms of basis vectors; adjacent simple roots with all labels $3$ satisfy $(α_{i}, α_{j}) / (∣ α_{i} ∣ ∣ α_{j} ∣) = - 1 / 2 = B (e_{i}, e_{j})$ , the $B_{n}$ edge of label $4$ satisfies $(α_{n - 1}, α_{n}) / (∣ α_{n - 1} ∣ ∣ α_{n} ∣) = - 1 / \sqrt{2} = B (e_{n - 1}, e_{n})$ , and non-adjacent pairs are orthogonal, so $B (e_{i}, e_{j}) = (φ e_{i}, φ e_{j})$ for all $i, j$ the normalized simple roots form a basis of $E$ , so $φ$ is invertible and is an isometry of bilinear forms [F2, F1]. Consequently for $a \in V$ with $B (a, a) = 1$ and $y = φ v$ one has $φ r_{a} φ^{- 1} (y) = φ (v - 2 B (v, a) a) = y - 2 (y, φ a) φ a = s_{φ a} (y)$ , so conjugation by $φ$ carries the reflection with normal $a$ to the reflection with normal $φ a$ ; applied to $a = e_{i}$ this gives $φ ρ (s_{i}) φ^{- 1} = s_{α_{i}}$ . Hence $ψ : = φ ρ φ^{- 1} : W \to G L (E)$ is a homomorphism (a conjugate of the homomorphism $ρ$ ) with $ψ (s_{i}) = s_{α_{i}}$ , and $ψ (c) = ψ (a) ψ (b)$ is the product of the two colour-class reflection products in the model. Since listing the two classes in the other order replaces $c$ by a conjugate element [F3], characteristic polynomials and orders computed for one listing apply to $c$ in either convention.

1.2F4F5F7

( $I_{2} (m)$ .) Here the model is $V$ itself, $c = s_{1} s_{2}$ is the product $A$ of the two simple reflections, and [F4] gives trace $2 \cos (2 π / m)$ , determinant $1$ and exact order $m$ . For a $2 \times 2$ matrix the characteristic polynomial is $X^{2} - tr (A) X + \det (A)$ , as one sees by expanding $\det (X I - A)$ [F5], so $\det (X - ρ (c)) = X^{2} - 2 \cos (2 π / m) X + 1$ . Since $\exp (2 π i / m) = \cos (2 π / m) + i \sin (2 π / m)$ and $\exp (- 2 π i / m) = \cos (2 π / m) - i \sin (2 π / m)$ by [F7], the two numbers $\exp (\pm 2 π i / m)$ have sum $2 \cos (2 π / m)$ and modulus one, so $(X - \exp (2 π i / m)) (X - \exp (- 2 π i / m)) = X^{2} - 2 \cos (2 π / m) X + 1$ and the eigenvalues are $\exp (2 π i / m)$ and $\exp (- 2 π i / m)$ . Moreover $\exp (- 2 π i / m) = \exp (2 π i (m - 1) / m)$ : indeed $\exp (2 π i (m - 1) / m) = \exp (2 π i) \exp (- 2 π i / m)$ and $\exp (2 π i) = \exp (i π) \exp (i π) = (- 1)^{2} = 1$ by multiplicativity and $e^{i π} + 1 = 0$ [F7]. Hence the spectral exponents are $1$ and $m - 1$ , with $h = m$ .

2.1F8F5step 1.1

( $A_{n}$ .) The model reflection $s_{α_{i}}$ acts on $E$ by the transposition $(i i + 1)$ of coordinates: for $x \in E$ , $s_{α_{i}} (x) = x - (x_{i} - x_{i + 1}) (ε_{i} - ε_{i + 1})$ swaps the $i$ -th and $(i + 1)$ -st coordinates and fixes the others. For $n = 1$ , $c = s_{1}$ acts on the line $E = R (ε_{1} - ε_{2})$ as $- i d$ , so $\det (X - ψ (c)) = X + 1 = X^{1} + \dots + 1$ , the order is $2$ , and the spectral exponent is $1$ [F1, F2, step 1.1]. For $n \geq 2$ write $O = \prod_{i o d d} (i i + 1)$ and $E_{0} = \prod_{i e v e n} (i i + 1)$ for the two colour-class products on the coordinates $1, \dots, n + 1$ ; they are products of pairwise disjoint transpositions, so each product has commuting factors, and $ψ (c) = O E_{0}$ (the reverse product is conjugate because $O^{2} = 1$ ). On basis vectors, using the four cases: $E_{0} ε_{1} = ε_{1}$ and $O ε_{1} = ε_{2}$ , so $1 \to 2$ ; for even $2 \leq j \leq n - 1$ one has $E_{0} ε_{j} = ε_{j + 1}$ and $O ε_{j + 1} = ε_{j + 2}$ , so $j \to j + 2$ ; for even $j = n$ (only when $n$ is even) $E_{0} ε_{n} = ε_{n + 1}$ and $O ε_{n + 1} = ε_{n + 1}$ , so $n \to n + 1$ ; for even $j = n + 1$ (only when $n$ is odd), $E_{0} ε_{n + 1} = ε_{n + 1}$ and $O ε_{n + 1} = ε_{n}$ , so $n + 1 \to n$ ; for odd $j \geq 3$ one has $E_{0} ε_{j} = ε_{j - 1}$ and $O ε_{j - 1} = ε_{j - 2}$ , so $j \to j - 2$ . Tracing these rules gives the single cycle $1 \to 2 \to 4 \to 6 \to \dots \to [largest even \leq n + 1] \to [largest odd \leq n + 1] \to \dots \to 5 \to 3 \to 1$ on the coordinates [F9]. In the basis $(ε_{1}, ε_{2}, ε_{4}, \dots)$ ordered along this cycle, the matrix of the cycle has $X$ on the diagonal, $- 1$ on the subdiagonal and $- 1$ in the top-right corner; in the Leibniz sum for $\det (X I - M)$ the only nonzero terms are the identity permutation, contributing $X^{N}$ , and the $N$ -cycle $(1 N N - 1 \dots 2)$ , of sign $(- 1)^{N - 1}$ and entry product $(- 1)^{N}$ , contributing $- 1$ ; hence the characteristic polynomial is $X^{N} - 1$ with $N = n + 1$ [F5]. The full coordinate space decomposes as the invariant direct sum $E_{C} \oplus C (1, \dots, 1)$ , and the second summand is fixed, so on the $n$ -dimensional space $E$ the operator $ψ (c)$ has characteristic polynomial $(X^{n + 1} - 1) / (X - 1) = X^{n} + X^{n - 1} + \dots + 1$ ; by [F8] its eigenvalues are the $n$ numbers $\exp (2 π i k / (n + 1))$ , $k = 1, \dots, n$ , each once. The restricted spectrum includes the primitive $(n + 1)$ -st root $\exp (2 π i / (n + 1))$ , so the order of $ψ (c) ∣_{E}$ is divisible by $n + 1$ ; the full cycle has order $n + 1$ , so the restricted order is exactly $n + 1$ . Thus the spectral exponents are $1, 2, \dots, n$ with $h = n + 1$ .

2.2F8F5step 1.1

( $B_{n}$ .) Here $α_{n} = ε_{n}$ and the model reflections are $s_{i} = (i i + 1)$ for $i < n$ and $s_{n}$ negating the $n$ -th coordinate. Let $O = \prod_{i o d d \leq n} s_{i}$ and $E_{0} = \prod_{i e v e n \leq n} s_{i}$ be the two colour-class products of the path $1 - 2 - \dots - n$ and let $ψ (c) = O E_{0}$ . Direct composition on basis vectors gives, uniformly in the parity of $n$ : $c (ε_{1}) = ε_{2}$ ; $c (ε_{j}) = ε_{j + 2}$ for even $j \leq n - 2$ ; $c (ε_{n - 1}) = - ε_{n}$ and $c (ε_{n}) = ε_{n - 2}$ when $n$ is odd; $c (ε_{n}) = - ε_{n - 1}$ when $n$ is even; and $c (ε_{j}) = ε_{j - 2}$ for odd $j \geq 3$ . For example, once the chain of even indices reaches the last coordinates the sign from $s_{n}$ appears and the negative coordinates are traversed in reverse: for $n = 4$ , $ε_{1} \mapsto ε_{2} \mapsto ε_{4} \mapsto - ε_{3} \mapsto - ε_{1}$ ; for $n = 5$ , $ε_{1} \mapsto ε_{2} \mapsto ε_{4} \mapsto - ε_{5} \mapsto - ε_{3} \mapsto - ε_{1}$ . Consequently the orbit of $ε_{1}$ under $c$ consists of the $2 n$ vectors $\pm ε_{1}, \dots, \pm ε_{n}$ , each exactly once: its first $n$ elements $f_{0} : = ε_{1}, f_{1} : = c ε_{1}, \dots, f_{n - 1} : = c^{n - 1} ε_{1}$ are $\pm ε_{j}$ with pairwise distinct $j$ (for $n = 2$ : $ε_{1}, ε_{2}$ ; $n = 3$ : $ε_{1}, ε_{2}, - ε_{3}$ ; $n = 4$ : $ε_{1}, ε_{2}, ε_{4}, - ε_{3}$ ; $n = 5$ : $ε_{1}, ε_{2}, ε_{4}, - ε_{5}, - ε_{3}$ ), hence a basis, and $c^{n} ε_{1} = - ε_{1}$ . In this basis $c f_{i} = f_{i + 1}$ $(i < n - 1)$ and $c f_{n - 1} = - f_{0}$ , so the matrix of $c$ has $X$ on the diagonal, $- 1$ on the subdiagonal and $+ 1$ in the top-right corner of $X I - M$ ; as in 2.1 the only nonzero Leibniz terms give $\det (X I - M) = X^{n} + 1$ [F5]. Since $c^{n}$ commutes with $c$ and $c^{n} ε_{1} = - ε_{1}$ , it sends each basis vector $f_{j} = c^{j} ε_{1}$ to $- f_{j}$ , so $c^{n} = - i d$ on $E$ and $c^{2 n} = i d$ . The orbit of $ε_{1}$ already contains the $2 n$ distinct signed coordinate vectors, so the order of $c$ is exactly $2 n$ . By [F8] applied to $z = - 1$ the eigenvalues are the numbers $\exp (i (π + 2 π k) / n) = \exp (2 π i (2 k + 1) / (2 n))$ , $k = 0, \dots, n - 1$ : the spectral exponents are the odd residues $1, 3, \dots, 2 n - 1$ with $h = 2 n$ .

2.3F8F5step 1.1

( $D_{n}$ .) Here $α_{i} = ε_{i} - ε_{i + 1}$ for $i \leq n - 1$ and $α_{n} = ε_{n - 1} + ε_{n}$ ; the simple reflections $s_{i}$ for $i < n$ swap coordinates $i, i + 1$ , while $s_{n}$ sends $ε_{n - 1} \mapsto - ε_{n}$ and $ε_{n} \mapsto - ε_{n - 1}$ . The branch node is $n - 2$ . Choose the color class containing the branch node as $J$ and use $c = a b$ with $J$ first; reversing the classes conjugates the product by [F3]. If $n$ is even, $J = {2, 4, \dots, n - 2}$ and $K = {1, 3, \dots, n - 3, n - 1, n}$ ; applying the commuting reflections in $b$ and then $a$ gives $c (ε_{1}) = ε_{3}$ , $c (ε_{j}) = ε_{j + 2}$ for odd $3 \leq j \leq n - 3$ , $c (ε_{n - 1}) = - ε_{n - 2}$ , $c (ε_{j}) = ε_{j - 2}$ for even $4 \leq j \leq n - 2$ , $c (ε_{2}) = ε_{1}$ , and $c (ε_{n}) = - ε_{n}$ . If $n$ is odd, $J = {1, 3, \dots, n - 2}$ and $K = {2, 4, \dots, n - 3, n - 1, n}$ ; the same reflection formula gives $c (ε_{1}) = ε_{2}$ , $c (ε_{j}) = ε_{j + 2}$ for even $2 \leq j \leq n - 3$ , $c (ε_{n - 1}) = - ε_{n - 2}$ , $c (ε_{j}) = ε_{j - 2}$ for odd $3 \leq j \leq n - 2$ , and $c (ε_{n}) = - ε_{n}$ . The formulas show that, on $E^{'} : = s p a n (ε_{1}, \dots, ε_{n - 1})$ , the first $m : = n - 1$ iterates of $ε_{1}$ are a signed coordinate basis: for even $n$ they are $ε_{1}, ε_{3}, \dots, ε_{n - 1}, - ε_{n - 2}, - ε_{n - 4}, \dots, - ε_{2}$ , and for odd $n$ they are $ε_{1}, ε_{2}, ε_{4}, \dots, ε_{n - 1}, - ε_{n - 2}, - ε_{n - 4}, \dots, - ε_{3}$ . In this basis $c$ advances each vector to the next and sends the last to $- ε_{1}$ , so the companion determinant computation of 2.2 gives $\det (X I - c ∣_{E^{'}}) = X^{m} + 1$ and $c^{m} ε_{1} = - ε_{1}$ . Since $c^{m}$ commutes with $c$ , it is $- i d$ on the orbit basis of $E^{'}$ , so the orbit of $ε_{1}$ and its negatives contains the $2 m$ distinct signed basis vectors and the order on $E^{'}$ is exactly $2 m$ ; the remaining coordinate line has eigenvalue $- 1$ , so the full order is $lcm (2 m, 2) = 2 m = 2 n - 2$ . Thus $\det (X I - ψ (c)) = (X^{n - 1} + 1) (X + 1)$ . By [F8], the roots of $X^{m} + 1$ are $\exp (2 π i (2 k + 1) / (2 m))$ for $k = 0, \dots, m - 1$ , and the extra root $- 1$ is $\exp (2 π i m / (2 m))$ ; the spectral exponents are $1, 3, \dots, 2 n - 3$ and one additional $n - 1$ (counted again when $n$ is even), with $h = 2 n - 2$ .

3.1F2F3F4F6F8step 1.2step 2.1step 2.2step 2.3∎

The map $ψ = φ ρ φ^{- 1}$ is conjugate to $ρ$ , and complexification preserves characteristic polynomials and operator orders, so steps 1.2-2.3 compute $\det (X - ρ_{C} (c))$ and its order for the four types. Since $ρ_{C}$ is faithful by [F2], this operator order equals the order of $c$ in $W$ , which is the Coxeter number $h$ used in the statement. The eigenvalues listed there are the spectral exponents in the sense fixed in the statement, since conjugate linear maps have the same eigenvalues with the same multiplicities. Finally $ρ_{C} (c)$ has finite order because $W$ is finite, so it is diagonalisable by [F6] over $C$ ; this holds in each type and shows both the diagonalisability and, together with the characteristic polynomials, the displayed spectra. Nothing above uses invariant degrees, Hilbert series or any degree table: only the reflection models, the rank-two computation and roots of unity are used, and the statement makes no assertion about $h = \max_{i} (d_{i})$ , which is proved only by the final determination theorem of this page.

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

The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity

Statement

Assume the Axiom of Choice. Let $(W, S)$ be a Coxeter system of finite type, $n = ∣ S ∣$ , and let $V_{C}$ , $S = C [x_{1}, \dots, x_{n}]$ , $R = S^{W}$ , $I = S R_{+}$ , $A = S / I$ , and a fixed basic family $f_{1}, \dots, f_{n}$ with degrees $d_{1} \leq \dots \leq d_{n}$ and exponents $e_{i} = d_{i} - 1$ be as in Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system (so $f_{1}, \dots, f_{n}$ is a minimal homogeneous generating family of $I$ , generates $R$ , is algebraically independent, and gives a graded isomorphism $C [y_{1}, \dots, y_{n}] \to R$ , $y_{i} \mapsto f_{i}$ , with $\deg y_{i} = d_{i}$ ).

(1) Multiset independence. If $f_{1}^{'}, \dots, f_{n}^{'}$ is any other minimal homogeneous generating family of $I$ with degrees $d_{i}^{'}$ , then ${d_{1}, \dots, d_{n}} = {d_{1}^{'}, \dots, d_{n}^{'}}$ as multisets. Hence the nondecreasing degree sequence $d_{1} \leq \dots \leq d_{n}$ , the exponents $e_{1} \leq \dots \leq e_{n}$ and the degree count are invariants of the pair $(W, V_{C})$ .

(2) Hilbert series. $Hilb (R, t) = \prod_{i = 1}^{n} (1 - t^{d_{i}})^{- 1}$ and $Hilb (A, t) = \prod_{i = 1}^{n} (1 + t + \dots + t^{d_{i} - 1})$ as formal power series (The Hilbert function and formal Hilbert series of a graded module with finite-length pieces).

(3) Order formula. $\dim_{C} A = \prod_{i = 1}^{n} d_{i} = ∣ W ∣$ , and $\sum_{i = 1}^{n} e_{i}$ is the top degree of $A$ .

(4) Molien identity. $\prod_{i = 1}^{n} (1 - t^{d_{i}})^{- 1} = \frac{1}{∣ W ∣} \sum_{w \in W} \det (1 - t w ∣ V_{C}^{*})^{- 1}$ as formal power series, the determinants expanded as finite products over the eigenvalues of $ρ_{C} (w)$ on $V_{C}^{*}$ .

(5) Conventions. For $n = 0$ all products are empty and equal $1$ , and $A = C$ . For reducible $S$ the degree multiset is the concatenation of the components' multisets (this is used, with its own argument, in the final determination theorem). All statements are under the stated AC.

Facts & Assumptions

Given: The Axiom of Choice, a Coxeter system $(W, S)$ of finite type, and a basic family $f_{1}, \dots, f_{n}$ as in the statement.

[F1]

$f_{1}, \dots, f_{n}$ is a minimal finite family of homogeneous positive-degree invariants generating $I = S R_{+}$ , it generates $R$ as a $C$ -algebra and is algebraically independent; under AC any minimal such family has exactly $n = ∣ S ∣$ elements, and every $d_{i} \geq 2$ (Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system, Finite reflection invariant generators are algebraically independent).

[F2]

The complexification $ρ_{C} (W) \leq G L (V_{C})$ is a finite subgroup of order $∣ W ∣$ , generated by complex reflections, faithful, and $\dim_{C} V_{C} = n$ (Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers).

[F3]

Assume AC. For the finite complex reflection group $ρ_{C} (W)$ : $C [V_{C}]^{W}$ is a polynomial algebra and $C [V_{C}]$ is graded free of rank $∣ W ∣$ over it; minimal homogeneous invariant generators of $I$ form an $S$ -regular sequence with common zero set ${0}$ ; and $Hilb (S / I, t) = \prod_{i} (1 + \dots + t^{d_{i} - 1})$ , $\dim_{C} (S / I) = \prod_{i} d_{i} = ∣ W ∣$ (Chevalley shephard todd for finite weyl groups, Reflection basic invariants form a regular sequence, Weyl coinvariant hilbert series has order w dimension).

[F4]

The Reynolds operator $R (f) = ∣ G ∣^{- 1} \sum_{g \in G} g \cdot f$ is a graded $R$ -linear projection of $S$ onto $R = S^{G}$ , and it preserves each finite-dimensional graded piece $S_{N}$ ; $I = S R_{+}$ and the coinvariant algebra $A = S / I$ carry their quotient gradings (Finite linear invariant and coinvariant polynomial algebras).

[F5]

The Hilbert function of a graded module is $H_{M} (N) = \dim$ of its $N$ -th piece when the pieces are finite-dimensional, and its formal Hilbert series is $\sum_{N} H_{M} (N) t^{N}$ ; products and divisions below are manipulations of formal power series with constant term one (The Hilbert function and formal Hilbert series of a graded module with finite-length pieces, Nonnegatively graded rings and modules, homogeneous elements, and twists).

[F6]

An operator of finite order on a finite-dimensional complex vector space is diagonalisable (Over an algebraically closed field of characteristic $0$ , every element of finite order acts diagonalisably in a finite-dimensional representation). Determinants are invariant under similarity (Similar matrices over a commutative ring have the same determinant), so an eigenbasis computes the determinant factors of a diagonalisable operator.

[F7]

If $Γ$ has connected components on $S_{1}, \dots, S_{k}$ , then $W ≅ W_{S_{1}} \times \dots \times W_{S_{k}}$ via multiplication, and $V = V_{1} \oplus \dots \oplus V_{k}$ is a $B$ -orthogonal direct sum on which $ρ (W_{i})$ preserves $V_{i}$ and fixes every $V_{j}$ , $j \neq i$ (Disconnected diagrams, direct products, and comparison of invariant forms (1),(2)); for finite type each component is one of the classified finite diagrams and $W$ is finite exactly when every component is (Classification of finite Coxeter systems, including the H and dihedral families (2), Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

[F8]

$S = C [x_{1}, \dots, x_{n}]$ is the iterated polynomial ring over $C$ , so polynomials may be expanded and compared blockwise in finitely many variables, and the monomials of one block are linearly independent over the polynomial ring of the remaining blocks (Polynomial rings in finitely many commuting indeterminates by iteration).

Proof

1.1F1F5F8

Put $m = S_{+} = (x_{1}, \dots, x_{n})$ . For any finite minimal homogeneous generating family $g_{1}, \dots, g_{r}$ of $I$ , its classes span the graded complex vector space $I / m I$ : modulo $m I$ , coefficients in an expression $\sum_{j} a_{j} g_{j}$ reduce to their constants. These classes are independent. Indeed, a nonzero homogeneous relation would give $\sum_{\deg g_{j} = D} c_{j} g_{j} \in m I$ with some $c_{k} \neq 0$ . Since the $g_{j}$ generate $I$ , its right side can be written $\sum_{j} b_{j} g_{j}$ with $b_{j} \in m$ homogeneous of degree $D - \deg g_{j}$ . Such $b_{j}$ vanish when $\deg g_{j} \geq D$ , so solving for $g_{k}$ expresses it in the ideal generated by the other $g_{j}$ , contrary to minimality. Every relation splits into homogeneous ones, proving independence. Thus the number of degree- $D$ members in any such family is $\dim_{C} (I / m I)_{D}$ . Applying this to the fixed invariant family and to an arbitrary other minimal homogeneous family gives equality of degree multisets and family sizes. This proves (1) without assuming that the other generators are invariant; for $I = 0$ the minimal family and quotient are empty.

1.2F1F5F8

For the fixed invariant basic family, [F1] gives the graded isomorphism $C [y_{1}, \dots, y_{n}] \to R$ with $y_{i} \mapsto f_{i}$ and $\deg y_{i} = d_{i}$ . Counting its monomials gives $Hilb (R, t) = \sum_{N \geq 0} # {a \in N^{n} : \sum_{i} a_{i} d_{i} = N} t^{N} = \prod_{i = 1}^{n} (1 - t^{d_{i}})^{- 1}$ . Positive weights make every coefficient finite, so the product is a formal-power-series identity. This algebra isomorphism is asserted only for the invariant basic family.

2.1F2F3F5step 1.2

Since $I$ is homogeneous, $R = ⨁ R_{N}$ and $A = ⨁ A_{N}$ are graded with finite-dimensional pieces, so the Hilbert series of [F5] apply. By step 1.2, $Hilb (R, t) = \prod_{i = 1}^{n} (1 - t^{d_{i}})^{- 1}$ . Under the stated AC the group $ρ_{C} (W)$ is a finite complex reflection group of order $∣ W ∣$ on the $n$ -dimensional space $V_{C}$ with invariant algebra $R$ and positive-degree ideal $I$ , so [F3] gives $Hilb (A, t) = \prod_{i = 1}^{n} (1 + t + \dots + t^{d_{i} - 1})$ , and evaluating at $t = 1$ gives $\dim_{C} A = \prod_{i} d_{i} = ∣ W ∣$ ; since each factor $1 + t + \dots + t^{d_{i} - 1}$ is a polynomial of degree $d_{i} - 1$ , the product is a polynomial of degree $\sum_{i} e_{i}$ with leading coefficient $1$ , so $\sum_{i} e_{i}$ is the top degree of $A$ .

3.1F2F4F5F6step 2.1

Fix $w \in W$ . By [F2] and [F6] the operator induced by $w$ on the finite-dimensional space $V_{C}^{*}$ is diagonalisable; let $λ_{1}, \dots, λ_{n}$ be its eigenvalues with multiplicity, and note that $\det (1 - t w ∣ V_{C}^{*}) = \prod_{j = 1}^{n} (1 - λ_{j} t)$ . The induced operator on $S_{N} = {Sym}^{N} (V_{C}^{*})$ ( $N \geq 0$ ) has, in the monomial basis attached to an eigenbasis of $V_{C}^{*}$ , the eigenvalues $λ^{a} = λ_{1}^{a_{1}} \dots λ_{n}^{a_{n}}$ over all $a$ with $∣ a ∣ = N$ ; summing gives the formal identity $\sum_{N \geq 0} tr (w ∣ S_{N}) t^{N} = \prod_{j = 1}^{n} (1 - λ_{j} t)^{- 1} = \det (1 - t w ∣ V_{C}^{*})^{- 1}$ [F5]. The Reynolds operator is a projection of $S_{N}$ onto $R_{N}$ [F4], and for a projection the trace equals the dimension of its image, so $\dim_{C} R_{N} = ∣ W ∣^{- 1} \sum_{w \in W} tr (w ∣ S_{N})$ ; summing over $N$ and using the previous identity gives $Hilb (R, t) = ∣ W ∣^{- 1} \sum_{w \in W} \det (1 - t w ∣ V_{C}^{*})^{- 1}$ , and combining with $Hilb (R, t) = \prod_{i} (1 - t^{d_{i}})^{- 1}$ of 2.1 proves the Molien identity (4).

4.1F1F3F4F7F8step 1.1∎

For $n = 0$ the group is trivial, $S = R = C$ , $I = 0$ , $A = C$ , and every displayed product is empty and equals $1$ , so all four clauses hold in this empty form [F1, F5]. Let now $Γ$ have finitely many connected components and suppose each is of finite type; by [F7] $W ≅ W_{1} \times \dots \times W_{k}$ and $V_{C} = V_{1, C} \oplus \dots \oplus V_{k, C}$ with $ρ (W_{i})$ preserving $V_{i}$ and fixing the other summands. Choose coordinates adapted to this decomposition, so that $S = C [x^{(1)}, \dots, x^{(k)}]$ with $x^{(i)}$ the coordinates of the $i$ -th summand [F8]. For $f \in R = S^{W}$ and any $i$ , the Reynolds operator $R_{i}$ of $W_{i}$ acts only on the block $x^{(i)}$ and, because $f$ is $W_{i}$ -invariant, leaves it unchanged; expanding $f$ in the monomials of the remaining blocks and applying $R_{i}$ expresses $f$ as a finite sum of products of a $W_{i}$ -invariant polynomial in the block $x^{(i)}$ with a polynomial in the other blocks. Applying $R_{k}, R_{k - 1}, \dots, R_{1}$ successively (each application leaves $f$ unchanged and replaces one block factor by its invariant part) gives $f \in C [x^{(1)}]^{W_{1}} \dots C [x^{(k)}]^{W_{k}}$ : the invariant algebra is generated by the component invariant algebras [F4, F8]. If for each $i$ we fix a basic family of the component $W_{i}$ , then the concatenated family is homogeneous of positive degrees, generates $R$ , and is algebraically independent: generation follows from the preceding display, and a polynomial relation among the concatenated family, expanded in the block monomials and using the algebraic independence of each component's family, forces every coefficient polynomial to vanish [F1, F8]. The concatenation generates $I = S R_{+}$ because its members generate $R$ and have positive degrees. It is also minimal as an $S$ -ideal generating family: any redundancy, after Reynolds averaging its coefficients [F4], would express one member as an $R$ -linear combination of the others. Substituting their polynomial expressions in the algebraically independent family and setting all the other variables to zero would give the impossible identity $y_{j} = 0$ in $C [y_{j}]$ . Thus the concatenation is a basic family of the reducible system, and its degree multiset is the concatenation of the components' multisets; by the multiset independence of 1.1 this is the degree multiset of every basic family of the reducible system. All statements above are under the stated AC, which enters only through the existence of the $n$ -element basic families and the AC-scoped suppliers [F1, F3]; the series comparison, the trace computation and the componentwise argument are finite and choice-free.

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

Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian

Statement

Assume the Axiom of Choice. Let $(W, S)$ be a Coxeter system of finite type, $n = ∣ S ∣ \geq 1$ , with $V_{C}$ , $A = S / I$ and a fixed basic family $f_{1}, \dots, f_{n} \in R = S^{W}$ of degrees $d_{i}$ , generating the ideal $I = S R_{+}$ , algebraically independent and generating $R$ , as in Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system; choose $C$ -coordinates $x_{1}, \dots, x_{n}$ on $V_{C}$ (Polynomial rings in finitely many commuting indeterminates by iteration) and let $\partial / \partial x_{j}$ and $J = (\partial f_{i} / \partial x_{j})_{1 \leq i, j \leq n}$ be the formal partial derivatives and the Jacobian matrix of Equation rows and coordinate columns in an affine Jacobian. Put $L : = C (x_{1}, \dots, x_{n})$ and $K : = C (f_{1}, \dots, f_{n}) \subseteq L$ (The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain).

(1) Annihilating orbit polynomials. For each $j$ , the orbit polynomial $P_{j} (T) : = \prod_{w \in W} (T - w \cdot x_{j}) \in S [T]$ is monic of degree $∣ W ∣$ , its coefficients lie in $R = C [f_{1}, \dots, f_{n}] \subseteq K$ , and $P_{j} (x_{j}) = 0$ . Hence every $x_{j}$ is algebraic over $K$ in the sense of Algebraic and transcendental elements and algebraic extensions.

(2) Minimal polynomial and its derivative. For each $j$ let $m_{j} \in K [T]$ be the minimal polynomial of $x_{j}$ over $K$ (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element). Then $x_{j} \notin K$ ; $m_{j}$ is nonconstant; $m_{j}^{'} \neq 0$ ; and $m_{j}^{'} (x_{j}) \neq 0$ .

(3) Differential bridge. For each $j$ there exist $b_{1 j}, \dots, b_{n j} \in L$ such that, for every $k \in {1, \dots, n}$ , the identity $δ_{j k} = \sum_{i = 1}^{n} b_{i j} \frac{\partial f_{i}}{\partial x_{k}}$ holds in $L$ (with $δ_{j k}$ the Kronecker symbol; equivalently $i d = B \cdot J$ with $B = (b_{j i})$ over $L$ ). Consequently $J$ has rank $n$ over $L$ , and $J : = \det (\frac{\partial f_{i}}{\partial x_{j}}) \in S$ is a nonzero polynomial: the invariant Jacobian.

No étale-quotient, scheme-theoretic or transcendental Jacobian criterion is used.

Facts & Assumptions

Given: The Axiom of Choice, the finite type system $(W, S)$ with basic family $f_{1}, \dots, f_{n}$ , and coordinates $x_{1}, \dots, x_{n}$ on $V_{C}$ .

[F1]

The fixed family $f_{1}, \dots, f_{n}$ is a minimal family of homogeneous positive-degree invariants generating $I = S R_{+}$ , generates $R$ as a $C$ -algebra and is algebraically independent, so evaluation gives a graded $C$ -algebra isomorphism $C [y_{1}, \dots, y_{n}] \to R$ , $y_{i} \mapsto f_{i}$ , and every element of $R$ is a polynomial in the $f_{i}$ ; also $d_{i} \geq 2$ (Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system, Finite reflection invariant generators are algebraically independent, Chevalley shephard todd for finite weyl groups).

[F2]

$ρ_{C} : W \to G L (V_{C})$ is the complexified canonical representation with $\dim_{C} V_{C} = n$ , and $V_{C}^{W} = {0}$ ; more precisely every $W$ -fixed linear form on $V_{C}$ is zero (Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (1),(3), The canonical reflection homomorphism, roots, reflections, and the positive cone).

[F3]

The action $(g \cdot f) (v) = f (ρ_{C} (g)^{- 1} v)$ makes $S = C [V_{C}]$ a graded algebra on which $W$ acts by graded algebra automorphisms, with invariant algebra $R = S^{W}$ , positive-degree part $R_{+}$ and $I = S R_{+}$ ; the degree-one part of $S$ is $V_{C}^{*}$ and the restricted action is the dual action, so a degree-one element of $S$ is $W$ -fixed exactly when it is an invariant linear form (Finite linear invariant and coinvariant polynomial algebras, Reflection basic invariants form a regular sequence).

[F4]

$S = C [x_{1}, \dots, x_{n}]$ is the iterated polynomial ring over $C$ , monomials form a basis, and $S$ is a domain with fraction field $L = C (x_{1}, \dots, x_{n})$ ; the subfield generated by $f_{1}, \dots, f_{n}$ is $K = C (f_{1}, \dots, f_{n}) = Frac (R)$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Polynomial rings in finitely many commuting indeterminates by iteration, The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain). Over a field, a positive-sized square matrix is invertible exactly when its determinant is nonzero (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit); hence $\det (λ I - M) = 0$ exactly when $λ$ is an eigenvalue.

[F5]

Algebraic elements and minimal polynomials: if $0 \neq P \in K [T]$ with $P (x_{j}) = 0$ then $x_{j}$ is algebraic over $K$ ; the minimal polynomial $m_{j}$ is the unique monic irreducible element generating $\ker ({ev}_{x_{j}}) = (m_{j})$ , and $f (x_{j}) = 0$ holds exactly when $m_{j} ∣ f$ (Algebraic and transcendental elements and algebraic extensions, The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

[F6]

The formal partial derivatives $\partial_{x_{k}}$ are defined on monomials by the Leibniz monomial rule and extended $C$ -linearly, and the univariate formal derivative satisfies $(f + g)^{'} = f^{'} + g^{'}$ , $(c f)^{'} = c f^{'}$ , the product rule, $(T^{a})^{'} = a T^{a - 1}$ and the degree bound $\deg f^{'} \leq \deg f - 1$ when $f^{'} \neq 0$ (Equation rows and coordinate columns in an affine Jacobian, The formal derivative of a polynomial, Linearity, power rule, Leibniz rule and the degree bound for the formal derivative).

[F7]

The Axiom of Choice is used only to have an $n$ -element basic family as in [F1] (The Axiom of Choice).

Proof

1.1F1F3F4F5

Fix $j$ and form $P_{j} (T) = \prod_{w \in W} (T - w \cdot x_{j}) \in S [T]$ . For $g \in W$ one has $g \cdot (w \cdot x_{j}) = (g w) \cdot x_{j}$ (the action is a left action), so $g$ permutes the linear factors and hence fixes $P_{j}$ ; as $W$ acts by graded algebra automorphisms [F3], each coefficient of $P_{j}$ lies in $R = S^{W}$ . By [F1] $R = C [f_{1}, \dots, f_{n}]$ , so all coefficients lie in $K$ . The polynomial is monic of degree $∣ W ∣$ , being a product of $∣ W ∣$ monic linear factors, and $P_{j} (x_{j}) = 0$ because the factor with $w = 1$ is $T - x_{j}$ . Hence $x_{j}$ is algebraic over $K$ in the sense of [F5].

1.2F6

The operators $\partial_{x_{k}}$ are $C$ -linear on $S$ and satisfy the product rule on monomials by the Leibniz monomial rule, hence on all polynomials by bilinearity; by induction the power rule $\partial_{x_{k}} (x_{j}^{a}) = a x_{j}^{a - 1} δ_{j k}$ holds for all $a \geq 1$ , while $\partial_{x_{k}} 1 = 0$ . Consequently, for every polynomial $γ \in C [y_{1}, \dots, y_{n}]$ and all $g_{1}, \dots, g_{n} \in S$ , the chain rule $\partial_{x_{k}} (γ (g_{1}, \dots, g_{n})) = \sum_{i = 1}^{n} (\partial_{y_{i}} γ) (g_{1}, \dots, g_{n}) \partial_{x_{k}} g_{i}$ holds: expanding $γ = \sum_{α} c_{α} y^{α}$ , the product rule gives $\partial_{x_{k}} (g^{α}) = \sum_{i} α_{i} g^{α - e_{i}} \partial_{x_{k}} g_{i}$ , which is the displayed identity because $\partial_{y_{i}} γ = \sum_{α} c_{α} α_{i} y^{α - e_{i}}$ .

2.1F2F3F5F6step 1.1

Suppose $x_{j} \in K$ . Since $W$ acts on $S$ by algebra automorphisms it acts on the fraction field $L$ by $g \cdot (a / b) = (g \cdot a) / (g \cdot b)$ , and this action fixes $K$ pointwise because the $f_{i}$ are invariant; hence $g \cdot x_{j} = x_{j}$ for every $g$ . But $x_{j} \in V_{C}^{*}$ is a degree-one element of $S$ [F3], so by [F2] the only $W$ -fixed linear form is $0$ , while $x_{j}$ is a coordinate function and $x_{j} \neq 0$ : contradiction. So $x_{j} \notin K$ ; the minimal polynomial $m_{j}$ of $x_{j}$ over $K$ [F5] is therefore nonconstant. Its derivative is nonzero: the top coefficient of $m_{j}$ is nonzero and the degree is $\geq 1$ , so in characteristic zero the coefficient $\deg (m_{j}) \cdot c_{\deg}$ of $m_{j}^{'}$ is nonzero, and $\deg m_{j}^{'} < \deg m_{j}$ by [F6]. If $m_{j}^{'} (x_{j}) = 0$ , then the nonzero polynomial $m_{j}^{'} \in K [T]$ of degree $< \deg m_{j}$ would have $x_{j}$ as a root, contradicting the minimality of $m_{j}$ [F5]; hence $m_{j}^{'} (x_{j}) \neq 0$ in $L$ .

3.1F1F4F6F7step 1.1step 1.2step 2.1∎

Fix $j$ and write $m_{j} (T) = \sum_{a = 0}^{r} c_{a} T^{a}$ with $c_{a} \in K$ . Since $K = Frac (R)$ [F4], choose $0 \neq B \in R$ with $β_{a} : = B c_{a} \in R$ for all $a$ , and put $M_{j} (T) : = \sum_{a} β_{a} T^{a} \in R [T]$ . Then $M_{j} (x_{j}) = B m_{j} (x_{j}) = 0$ and $M_{j}^{'} (T) = B m_{j}^{'} (T)$ , so $M_{j}^{'} (x_{j}) = B m_{j}^{'} (x_{j}) \neq 0$ in $L$ by 2.1. Differentiate the polynomial identity $M_{j} (x_{j}) = \sum_{a} β_{a} x_{j}^{a} = 0$ with respect to $x_{k}$ using the product and power rules of 1.2: $0 = \sum_{a} (\partial_{x_{k}} β_{a}) x_{j}^{a} + M_{j}^{'} (x_{j}) \partial_{x_{k}} x_{j} = \sum_{a} (\partial_{x_{k}} β_{a}) x_{j}^{a} + M_{j}^{'} (x_{j}) δ_{j k},$ hence $δ_{j k} = - (\sum_{a} (\partial_{x_{k}} β_{a}) x_{j}^{a}) / M_{j}^{'} (x_{j})$ in $L$ . Each $β_{a} \in R = C [f_{1}, \dots, f_{n}]$ is $β_{a} = γ_{a} (f_{1}, \dots, f_{n})$ for a polynomial $γ_{a} \in C [y_{1}, \dots, y_{n}]$ [F1], so the chain rule of 1.2 gives $\partial_{x_{k}} β_{a} = \sum_{i} (\partial_{y_{i}} γ_{a}) (f_{1}, \dots, f_{n}) \partial_{x_{k}} f_{i}$ . Substituting, $δ_{j k} = \sum_{i = 1}^{n} b_{i j} \partial_{x_{k}} f_{i}, b_{i j} : = - \frac{\sum_{a} (\partial_{y_{i}} γ_{a}) (f_{1}, \dots, f_{n}) x_{j}^{a}}{M_{j}^{'} (x_{j})} \in L,$ for all $j, k$ ; equivalently $B J = i d$ for the matrix $B = (b_{j i})$ . Hence $J$ is invertible over the field $L$ , so it has rank $n$ over $L$ , its determinant is nonzero in $L$ , and since $S$ is a domain with fraction field $L$ [F4] the polynomial $J = \det J \in S$ is nonzero. For $n = 0$ there is no assertion to make; the only choice principle used is the AC entering the existence of the basic family [F7], and no étale-quotient, scheme-theoretic or transcendental Jacobian criterion is invoked.

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

The total degree sum, the invariant Jacobian as the discriminant, anti-invariants, and the top coinvariant class

Statement

Assume the Axiom of Choice. Let $(W, S)$ be a finite-type Coxeter system with $S$ finite, $n : = ∣ S ∣$ , and let $V_{C}$ , its faithful real-matrix reflection representation $ρ_{C}$ , the positive roots $Φ_{+}$ , and the reflections $T$ be as in Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone and The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange. Put $S_{0} : = C [V_{C}] = C [x_{1}, \dots, x_{n}]$ , $R : = S_{0}^{W}$ , $R_{+} : = ⨁_{d > 0} R_{d}$ , $I : = S_{0} R_{+}$ and $A : = S_{0} / I$ as in Finite linear invariant and coinvariant polynomial algebras. Fix a homogeneous basic family $f_{1}, \dots, f_{n}$ of degrees $d_{i}$ and exponents $e_{i} = d_{i} - 1$ as in Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system, let $N : = ∣ Φ_{+} ∣ = ∣ T ∣$ , and choose coordinates $x_{1}, \dots, x_{n}$ from a $B$ -orthonormal real basis of $V$ ; then every matrix $ρ_{C} (w)$ is real orthogonal. Define

Δ : = \prod_{α \in Φ_{+}} ℓ_{α} \in S_{0}, ℓ_{α} (x) : = B_{C} (x, α),

and let $J : = \det (\partial f_{i} / \partial x_{j}) \neq 0$ be the invariant Jacobian of Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian. Write $\det (w) : = \det (ρ_{C} (w))$ . Then:

(1) Total degree. $\sum_{i = 1}^{n} (d_{i} - 1) = N = ∣ Φ_{+} ∣ = ∣ T ∣$ .

(2) The Jacobian is the discriminant. For every $w \in W$ , $w \cdot J = \det (w) J$ . Each $ℓ_{α}$ divides $J$ , and the $ℓ_{α}$ are pairwise nonproportional. For some $c \in C^{\times}$ ,

J = c Δ, \deg J = \sum_{i} (d_{i} - 1) = N = \deg Δ .

(3) Anti-invariants. If $S_{0}^{\det} : = {p \in S_{0} : w \cdot p = \det (w) p for every w \in W}$ , then $S_{0}^{\det} = Δ R$ . Each $p \in S_{0}^{\det}$ has a unique expression $p = Δ q$ with $q \in R$ .

(4) Top coinvariant class. The top degree of $A$ is $N = \sum_{i} e_{i}$ , its component $A_{N}$ is one-dimensional, and $[Δ] \in A_{N}$ is nonzero and spans it. The action on this line is $w \cdot [Δ] = \det (w) [Δ]$ .

(5) Conventions. For $n = 0$ , $W$ is trivial, $Δ = J = 1$ , $N = 0$ , and the clauses hold with empty products and determinant. The product defining $Δ$ is independent of the order in which the fixed set $Φ_{+}$ is listed. Replacing the positive system by its opposite multiplies $Δ$ by $(- 1)^{N}$ . The polynomial $Δ$ is coordinate-free; changing orthonormal coordinates substitutes the corresponding orthogonal change into its coordinate expression, while the proportionality scalar in $J = c Δ$ changes by the determinant of that basis change. These are conventions, not proof inputs. No claim is made that $A$ is the regular representation, that it is a Kostant harmonic space, or that it is flag-variety cohomology.

Facts & Assumptions

Given: The Axiom of Choice, a finite-type Coxeter system, a fixed basic family $f_{1}, \dots, f_{n}$ , and $B$ -orthonormal coordinates.

[F1]

The complexified representation is faithful, finite, and generated by reflections; $B$ is positive definite and preserved by real matrices; each $t_{α}$ has fixed hyperplane $ℓ_{α} = 0$ , determinant $- 1$ , and unit root normal; and $Φ_{+} \to T$ is a bijection (Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers, The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, Descent of the reflection representation, unit root norms, and conjugation of reflections, The inversion formula $∣ N (w) ∣ = ℓ (w)$ , the root-reflection dictionary and strong exchange, The root-length criterion and faithfulness of the canonical reflection representation, Finiteness criterion: W is finite exactly when the Coxeter form is positive definite).

[F2]

The finite reflecting arrangement has chambers whose interiors have trivial point stabilizer; every nonzero vector lies in a unique open face $w C_{I}$ , and a point of that face has stabilizer $w W_{I} w^{- 1}$ ; for $I = {s}$ , $W_{I} = {1, s}$ (The dual action, chambers, faces, and root hyperplanes, Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, The finite chamber tiling, the face-stabiliser identification, and the spherical Coxeter complex as a triangulation of the sphere (1),(3)).

[F3]

Under the stated Choice assumption, the invariant Hilbert series is $\prod_{i} (1 - t^{d_{i}})^{- 1}$ , the coinvariant Hilbert series is $\prod_{i} (1 + t + \dots + t^{d_{i} - 1})$ , and the full normalized Molien identity is

\prod_{i} (1 - t^{d_{i}})^{- 1} = \frac{1}{∣ W ∣} \sum_{w \in W} \det (1 - t w ∣ V_{C}^{*})^{- 1}

as a formal series (The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity (2),(4), Weyl coinvariant hilbert series has order w dimension).

[F4]

The invariant Jacobian $J$ is nonzero (Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian (3)); each $f_{i}$ is homogeneous of degree $d_{i}$ (Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system).

[F5]

Every finite-order complex linear operator is diagonalisable, and the determinant polynomial factors over its eigenvalues (Over an algebraically closed field of characteristic $0$ , every element of finite order acts diagonalisably in a finite-dimensional representation, For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix).

[F6]

$S_{0}$ is the iterated polynomial ring over $C$ (Polynomial rings in finitely many commuting indeterminates by iteration). Leading monomials show it is a domain. Taking a nonzero linear form $ℓ$ as one coordinate identifies $S_{0} / (ℓ)$ with a polynomial domain in $n - 1$ variables, so $(ℓ)$ is prime. Two linear forms are associates exactly when they are proportional: degree comparison forces any multiplier between them to be constant.

[F7]

The polynomial action is the contragredient substitution action, invariants form the graded subalgebra $R$ , and $I = S_{0} R_{+}$ is homogeneous (Finite linear invariant and coinvariant polynomial algebras, Nonnegatively graded rings and modules, homogeneous elements, and twists).

[F8]

Formal partial derivatives obey the monomial rule and chain rule, and the Jacobian determinant is the determinant of the matrix of those partials (The formal derivative of a polynomial, Equation rows and coordinate columns in an affine Jacobian). Determinants satisfy $\det (A B) = \det (A) \det (B)$ (For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ).

[F9]

Complex conjugation and the linear-first inner-product convention are those of Real and imaginary parts, complex conjugation, and modulus and Real and complex inner product spaces, with the inner product linear in the first argument. The coefficient-factorial pairing used below is constructed in step 5.1.

Proof

technique · Compare the first two Laurent coefficients of the normalized Molien identity, then use reflection divisibility and a coefficient-factorial pairing. All calculations are in polynomial rings or finite-dimensional spaces

1.1givenF1F3

If $n = 0$ , then $W = {1}$ , $S_{0} = R = A = C$ , and every family and product is empty; all clauses follow. If $n = 1$ , the Coxeter group is $W = {1, s}$ , $T = {s}$ and $N = 1$ . For its single degree $d_{1}$ , [F3] gives $\frac{1}{1 - t^{d_{1}}} = \frac{1}{2} (\frac{1}{1 - t} + \frac{1}{1 + t})$ . With $δ = 1 - t$ , the left side is $d_{1}^{- 1} δ^{- 1} + (d_{1} - 1) / (2 d_{1}) + O (δ)$ and the right side is $\frac{1}{2} δ^{- 1} + \frac{1}{4} + O (δ)$ ; comparing the $δ^{- 1}$ and constant coefficients gives $d_{1} = 2$ and $d_{1} - 1 = 1 = N$ . In rank one, write the unit positive-root form as $Δ = ε x_{1}$ with $ε = \pm 1$ . Invariance under $x_{1} \mapsto - x_{1}$ gives $R = C [x_{1}^{2}]$ ; the degree-two basic generator is $f_{1} = a x_{1}^{2}$ with $a \neq 0$ , so $I = (x_{1}^{2})$ , $J = 2 a x_{1} = (2 a / ε) Δ$ , and the anti-invariant polynomials are precisely the odd polynomials $Δ R$ . The quotient has basis $1, [x_{1}]$ , so its top component is the nonzero sign line spanned by $[Δ]$ . This proves every clause for $n = 1$ . In the rest of the proof assume $n \geq 2$ .

1.2F1F2algebra

For $n \geq 2$ , let $g \neq 1$ have fixed space of codimension one in $V_{C}$ . Because its matrix is real, the real and complex fixed spaces have the same codimension, so $H : = {Fix}_{V} (g)$ is a real hyperplane. If $H$ were not in the finite reflecting arrangement, a point $x \in H$ outside every arrangement hyperplane could be chosen: for each proper subspace cut out on $H$ , choose a nonzero linear form vanishing there; their product is a nonzero polynomial on $H$ , which cannot vanish everywhere over $R$ (by induction on $\dim H$ ). It would lie in a chamber interior, whose point stabilizer is trivial by [F2], contrary to $g x = x$ . Hence $H$ is an arrangement hyperplane. The same finite-union argument chooses $x \in H$ outside all other distinct arrangement hyperplanes; $x$ is nonzero. By [F2], $x \in w_{0} C_{I}$ for some $I ⊊ S$ . Since $x$ lies on an arrangement hyperplane, $I$ is nonempty; the walls indexed by $I$ are distinct because their normals are the images of distinct basis vectors under the invertible map $ρ (w_{0})$ . They all pass through $x$ , so $∣ I ∣ = 1$ . For $I = {s}$ , [F2] gives ${Stab}_{W} (x) = w_{0} {1, s} w_{0}^{- 1}$ ; since $g \neq 1$ fixes $x$ , it is the reflection $w_{0} s w_{0}^{- 1} \in T$ . Conversely every element of $T$ has a fixed hyperplane by [F1]. Thus the nonidentity elements with fixed-space codimension one are exactly the reflections.

2.1F3F5step 1.2algebra

Assume $n \geq 2$ and put $δ = 1 - t$ . The left side of [F3] expands as $\prod_{i} (1 - t^{d_{i}})^{- 1} = \frac{δ^{- n}}{\prod_{i} d_{i}} (1 + \frac{δ}{2} \sum_{i} (d_{i} - 1) + O (δ^{2}))$ . In the normalized sum on its right, the identity contributes $∣ W ∣^{- 1} δ^{- n}$ . Each of the $N$ reflections contributes $∣ W ∣^{- 1} δ^{- (n - 1)} / (1 + t) = \frac{1}{2 ∣ W ∣} δ^{- (n - 1)} + O (δ^{- (n - 2)})$ . For every other element, [F5] and step 1.2 give at most $n - 2$ eigenvalues equal to $1$ on $V_{C}^{*}$ : indeed $rank (M^{- T} - I) = rank (M - I)$ , so the fixed dimensions on a representation and its dual agree. Its Molien term is therefore $O (δ^{- (n - 2)})$ . Comparing the coefficients of $δ^{- n}$ and $δ^{- (n - 1)}$ first gives $\prod_{i} d_{i} = ∣ W ∣$ and then $\sum_{i} (d_{i} - 1) = N$ . The identity is a finite sum of rational functions, so these Laurent expansions compare coefficients without an infinite-limit interchange.

3.1F1F4F6F8step 2.1algebra

Write $M = ρ_{C} (w)$ in the chosen orthonormal coordinates. The tuple $F = (f_{1}, \dots, f_{n})$ satisfies $F (M x) = F (x)$ ; differentiating gives $D F (M x) M = D F (x)$ and hence $J (M x) = \det (M)^{- 1} J (x) = \det (M) J (x)$ , since $M$ is real orthogonal. If $x$ lies on the fixed hyperplane of $t_{α}$ , then $J (x) = J (ρ_{C} (t_{α}) x) = - J (x)$ , so $J$ vanishes there. After taking $ℓ_{α}$ as one coordinate, restriction to $ℓ_{α} = 0$ is the zero polynomial; thus $ℓ_{α}$ divides $J$ . If $ℓ_{α}$ and $ℓ_{β}$ are proportional, nondegeneracy of $B_{C}$ gives $α = c β$ ; since both roots are real with $B$ -norm one, $c = \pm 1$ , and positivity of both roots gives $c = 1$ , so $α = β$ . Thus the forms are pairwise nonproportional primes by [F6], and $Δ$ divides $J$ . Every determinant term of the Jacobian matrix has degree $\sum_{i} (d_{i} - 1)$ ; [F4] makes this the degree of its nonzero determinant. Since $\deg Δ = N$ , step 2.1 gives $\deg J = \deg Δ$ , so $J = c Δ$ for a nonzero scalar $c$ . This proves (2), including anti-invariance of $Δ$ . If an orthonormal basis changes by $P$ , its coordinate expressions satisfy $Δ_{n e w} (y) = Δ_{o l d} (P y)$ and $J_{n e w} (y) = \det (P) J_{o l d} (P y)$ . Listing the fixed roots in a different order leaves the product unchanged; replacing $Φ_{+}$ by $- Φ_{+}$ multiplies it by $(- 1)^{N}$ .

4.1F1F6F7step 3.1algebra

If $p \in S_{0}^{\det}$ , every reflection $t_{α}$ acts by determinant $- 1$ , so $p$ vanishes on its fixed hyperplane and each $ℓ_{α}$ divides $p$ . The same pairwise-prime argument as in step 3.1 gives $p = Δ q$ for some $q \in S_{0}$ . By step 3.1, $Δ$ is anti-invariant; applying any $w$ to $p = Δ q$ and cancelling $Δ$ in the domain $S_{0}$ then gives $w \cdot q = q$ . Conversely, if $q \in R$ , the product $Δ q$ is anti-invariant. The quotient $q$ is unique because $S_{0}$ is a domain and $Δ \neq 0$ .

5.1F3F6F7F8F9step 2.1step 3.1step 4.1algebra∎

By [F3] the Hilbert series of $A$ is $\prod_{i} (1 + t + \dots + t^{d_{i} - 1})$ , whose top degree is $\sum_{i} (d_{i} - 1) = N$ by step 2.1 and whose top coefficient is $1$ , so $A_{N}$ is one-dimensional. For $f = \sum_{∣ a ∣ = N} f_{a} x^{a}$ and $g = \sum_{∣ a ∣ = N} g_{a} x^{a}$ in $(S_{0})_{N}$ , set $⟨ f, g ⟩ : = \sum_{∣ a ∣ = N} a! f_{a} \overline{g_{a}}$ with $a! : = \prod_{j} a_{j}!$ ; this is positive definite. If $p \in R_{d}$ is homogeneous with $d > 0$ and $f \in (S_{0})_{N - d}$ , direct monomial expansion gives $⟨ p f, Δ ⟩ = ⟨ f, \overline{p} (\partial) Δ ⟩$ , where $\overline{p} (\partial)$ conjugates the coefficients of $p$ and substitutes the formal partial derivatives for its variables. For a real orthogonal matrix $M$ , the chain rule, first on linear symbols and then by products and linearity, gives $\overline{p} (\partial_{x}) (q \circ M) = (\overline{p} (M^{T} \partial) q) \circ M$ . Since $M^{T} = M^{- 1} \in W$ and real matrices preserve coefficientwise conjugation, $\overline{p}$ is invariant and $\overline{p} (M^{T} \partial) = \overline{p} (\partial)$ ; thus this differential operator commutes with substitution by $M$ . Since $Δ \circ M = \det (M) Δ$ by step 3.1, $\overline{p} (\partial) Δ$ is anti-invariant. If $d \leq N$ it has degree $N - d < N$ , so step 4.1 forces it to be zero; if $d > N$ the derivative is already zero. Every homogeneous element of $I_{N}$ is a sum of such products $p f$ , hence $Δ$ is orthogonal to $I_{N}$ . But $Δ$ is a nonzero real-coefficient polynomial, so $⟨ Δ, Δ ⟩ > 0$ ; consequently $Δ \notin I_{N}$ and $[Δ] \neq 0$ in $A_{N}$ . It spans this one-dimensional component and has the determinant action by step 3.1.

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

The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices

Statement

Let $Γ$ be one of the six labelled diagrams $E_{6} : (0, 1), (1, 2), (2, 3), (3, 4), (2, 5), E_{7} : (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (2, 6),$ $E_{8} : (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (2, 7), F_{4} : (0, 1), (1, 2), (2, 3),$ $H_{3} : (0, 1), (1, 2), H_{4} : (0, 1), (1, 2), (2, 3),$ all edges labelled $3$ except the single edges $(1, 2)$ of $F_{4}$ labelled $4$ and $(0, 1)$ of $H_{3}, H_{4}$ labelled $5$ ; let $(W, S)$ be the corresponding irreducible Coxeter system (Coxeter diagrams: edges, labels, components and finite type, Classification of finite Coxeter systems, including the H and dihedral families (1), Finiteness criterion: W is finite exactly when the Coxeter form is positive definite); let $V = R^{S}$ with $B (e_{s}, e_{t}) = - \cos (π / m_{s t})$ , $ρ$ the canonical reflection representation, and $c$ the bipartite Coxeter element of The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a with the colour classes of the tree $Γ$ and the recorded application orders $E_{6} : [0, 2, 4, 1, 3, 5], E_{7} : [0, 2, 4, 1, 3, 5, 6], E_{8} : [0, 2, 4, 6, 1, 3, 5, 7],$ $F_{4} : [0, 2, 1, 3], H_{3} : [0, 2, 1], H_{4} : [0, 2, 1, 3] .$ Then $ρ (c)$ has exact order $h$ and characteristic polynomial as follows, with spectral exponents $r_{1}, \dots, r_{n} \in {1, \dots, h - 1}$ defined by the eigenvalues $ζ_{h}^{r_{i}}$ :

$E_{6} : h = 12, \det (X - ρ (c)) = X^{6} + X^{5} - X^{3} + X + 1 = Φ_{3} Φ_{12}, {r_{i}} = {1, 4, 5, 7, 8, 11};$

$E_{7} : h = 18, \det (X - ρ (c)) = X^{7} + X^{6} - X^{4} - X^{3} + X + 1 = Φ_{2} Φ_{18}, {r_{i}} = {1, 5, 7, 9, 11, 13, 17};$

$E_{8} : h = 30, \det (X - ρ (c)) = X^{8} + X^{7} - X^{5} - X^{4} - X^{3} + X + 1 = Φ_{30}, {r_{i}} = {1, 7, 11, 13, 17, 19, 23, 29};$

$F_{4} : h = 12, \det (X - ρ (c)) = X^{4} - X^{2} + 1 = Φ_{12}, {r_{i}} = {1, 5, 7, 11};$

$H_{3} : h = 10, \det (X - ρ (c)) = X^{3} + (1 - φ) X^{2} + (1 - φ) X + 1 = (X + 1) (X^{2} - φ X + 1), {r_{i}} = {1, 5, 9};$

$H_{4} : h = 30, \det (X - ρ (c)) = X^{4} + (1 - φ) X^{3} + (1 - φ) X^{2} + (1 - φ) X + 1, {r_{i}} = {1, 11, 19, 29},$

where $φ = 2 \cos (π / 5)$ , $φ^{2} = φ + 1$ , embedded in $Q (ζ_{h})$ for $H_{3}, H_{4}$ by $φ = 1 + ζ_{h}^{h / 5} + ζ_{h}^{- h / 5}$ . In each case $\sum_{i} r_{i} = n h / 2 = ∣ Φ_{+} ∣$ and each polynomial divides $X^{h} - 1$ ; the matrices themselves (in the simple-root basis, with entries in $Z$ , $Z [\sqrt{2}]$ or $Z [φ]$ ) are recorded case by case in the proof, and the F4 characteristic polynomial is rational so no embedding of $\sqrt{2}$ into $Q (ζ_{12})$ is used.

Facts & Assumptions

Given: One of the six labelled diagrams of the statement with its Coxeter system $(W, S)$ , the space $V = R^{S}$ with Coxeter form $B$ , the canonical reflection representation $ρ$ , the bipartite Coxeter element $c$ and the recorded application orders, and the exact certificate file research/coxeter-scaffold/math-checks/finite-degree-poincare-certificates.json (generator finite-degree-poincare.py), whose records for the six types contain the matrices of the dual action and the same characteristic polynomials.

[F1]

$B (e_{s}, e_{s}) = 1$ and $B (e_{s}, e_{t}) = - \cos (π / m (s, t))$ for $s \neq t$ ; for $a$ with $B (a, a) \neq 0$ the reflection $r_{a} (v) = v - \frac{2 B (v, a)}{B (a, a)} a$ is linear, involutive, fixes $\ker B (-, a)$ pointwise and preserves $B$ ; the canonical homomorphism $ρ$ satisfies $ρ (s) = r_{e_{s}}$ and is faithful, every root has $B$ -norm one, and $ρ (w s w^{- 1}) = r_{ρ (w) e_{s}}$ (The real Coxeter form, its radical, reflections, and form-preserving maps (2),(3), Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (1)-(2), The canonical reflection homomorphism, roots, reflections, and the positive cone, Descent of the reflection representation, unit root norms, and conjugation of reflections (2)-(4), Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (1)).

[F2]

The Coxeter diagram of an irreducible finite type is a tree with bipartition $J ⊔ K$ ; the colour-class products $a, b$ , their product $c = a b$ and $h = ord (c)$ are well defined and listing the classes in the other order replaces $c$ by a conjugate element, so order and characteristic polynomial are unaffected; the Coxeter plane $P = span (u, v)$ is $ρ (c)$ -invariant with $ρ (c) ∣_{P}$ a rotation by $2 π / h$ ; the prefix roots $ρ_{i}$ enumerate $Φ_{+}$ with $∣ Φ_{+} ∣ = n h / 2$ , and $c - i d$ is invertible (The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a (1)-(4), The Coxeter plane, ordered-root enumeration, and invertibility of rho(c) - id (2)-(4)).

[F3]

The six diagrams of the statement are the standard diagrams $E_{6}, E_{7}, E_{8}, F_{4}, H_{3}, H_{4}$ of the classification, and for each of them $B$ is positive definite, so $W$ is finite (Classification of finite Coxeter systems, including the H and dihedral families (1),(3), Finiteness criterion: W is finite exactly when the Coxeter form is positive definite (1)-(2), Coxeter diagrams: edges, labels, components and finite type (1)-(2)).

[F4]

Trigonometric and square-root facts: $\cos (π / 4) = \sqrt{2} / 2$ (The finite Viete cosine product and its positive nested-radical factors); for all real $x$ , $\cos (2 x) = 2 \cos^{2} x - 1$ , $\sin (2 x) = 2 \sin x \cos x$ , $\cos (x + y) = \cos x \cos y - \sin x \sin y$ , $\cos (π - x) = - \cos x$ , $\cos (π / 2 - x) = \sin x$ , $\sin^{2} x + \cos^{2} x = 1$ (Double-angle and quadratic power-reduction identities, The addition formulas for sine and cosine, Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions, Parity and the Pythagorean identity for sine and cosine); $\cos (π / 2) = 0$ and cosine is strictly decreasing on $[0, π]$ (Quarter-turn values and shifts by pi/2 and pi, Signs, monotonicity intervals, and ranges of sine and cosine); $2 \cos u \cos v = \cos (u + v) + \cos (u - v)$ and $\cos u - \cos v = - 2 \sin \frac{u + v}{2} \sin \frac{u - v}{2}$ (Product-to-sum and sum-to-product identities); every $a \geq 0$ has a unique $\sqrt{a} \geq 0$ with ${\sqrt{a}}^{2} = a$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ) and $0 \leq a < b$ implies $a^{2} < b^{2}$ (Squaring is monotone on the nonnegatives); sine and cosine are the power-series functions of Sine and cosine defined by their real power series.

[F5]

For $M \in M_{n} (C)$ : $χ_{M} (X) = \det (X I - M)$ is the characteristic polynomial, whose coefficients are the determinants of the expansion of $X I - M$ (For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix); $tr (M)$ is the sum of the diagonal entries (The trace of a square matrix over a commutative ring); linearity follows termwise, and $tr (A B) = \sum_{i, j} a_{i j} b_{j i} = \sum_{j, i} b_{j i} a_{i j} = tr (B A)$ , which also gives cyclicity for longer products; $A adj (A) = adj (A) A = \det (A) I_{n}$ (For every positive-sized square matrix over a commutative ring, $A adj (A) = adj (A) A = \det (A) I$ , Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring); $\frac{d}{d t} \det (I - t M) = - tr (adj (I - t M) M)$ for the formal derivative ( $\frac{d}{d x} \det (I - x A) = - tr (adj (I - x A) A)$ ); the formal derivative of polynomials is linear, satisfies the product rule and has $\frac{d}{d t} t^{k} = k t^{k - 1}$ (The formal derivative of a polynomial, Linearity, power rule, Leibniz rule and the degree bound for the formal derivative). Determinants are invariant under similarity (Similar matrices over a commutative ring have the same determinant), so an eigenbasis computes the determinant factors of a diagonalisable operator. Over a field, a positive-sized square matrix is invertible exactly when its determinant is nonzero (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit); hence $\det (λ I - M) = 0$ exactly when $λ$ is an eigenvalue.

[F6]

Cyclotomic and root-of-unity facts: the $Φ_{m} \in Z [t]$ satisfy $t^{m} - 1 = \prod_{d ∣ m} Φ_{d}$ and $Φ_{1} = t - 1$ (The cyclotomic polynomials $Φ_{n} \in Z [t]$ , defined by $\prod_{d ∣ n} Φ_{d} = t^{n} - 1$ ); over $C$ the roots of $Φ_{m}$ are exactly the primitive $m$ -th roots of unity and the $m$ -th roots of unity are the distinct numbers $e^{2 π i k / m}$ , $k = 0, \dots, m - 1$ (Over a field whose characteristic does not divide $n$ , the roots of $Φ_{n}$ are exactly the primitive roots of unity, The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ , The group $μ_{n} (K)$ of $n$ -th roots of unity in a field, and primitive $n$ -th roots of unity, The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ ); $e^{i θ} = \cos θ + i \sin θ$ (Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ ); a finite-order operator on a finite-dimensional complex vector space is diagonalisable (Over an algebraically closed field of characteristic $0$ , every element of finite order acts diagonalisably in a finite-dimensional representation), and the finite induction principle is The principle of mathematical induction with The natural numbers $N$ (von Neumann).

[F7]

Complexifying the real representation $ρ$ gives $ρ_{C}$ on $V_{C} = C \otimes_{R} V$ with the same operator matrices and characteristic polynomials in the basis $1 \otimes e_{s}$ ; the eigenvalues of a real matrix acting on $V_{C}$ are the roots of its characteristic polynomial (Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (1), For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix).

Proof

1.1F4algebra

(The trigonometric constants behind the edge labels.) Put $c : = \cos (π / 3)$ . Since $0 < π / 3 < π / 2$ and cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ , we have $c > 0$ ; the supplementary and double-angle identities give $- c = \cos (π - π / 3) = \cos (2 π / 3) = 2 c^{2} - 1$ , so $2 c^{2} + c - 1 = 0$ , that is $(2 c - 1) (c + 1) = 0$ , and $c + 1 > 0$ forces $c = 1 / 2$ . Next put $d : = \cos (π / 5)$ . Since $0 < π / 5 < π / 2$ we have $d > 0$ , and $\cos (π - 2 π / 5) = - \cos (2 π / 5)$ with $π - 2 π / 5 = 3 π / 5$ gives via the addition, double-angle and Pythagorean identities $\cos (3 π / 5) = \cos (2 π / 5) \cos (π / 5) - \sin (2 π / 5) \sin (π / 5) = (2 d^{2} - 1) d - 2 (1 - d^{2}) d = 4 d^{3} - 3 d,$ while $\cos (3 π / 5) = - \cos (2 π / 5) = - (2 d^{2} - 1)$ ; hence $4 d^{3} + 2 d^{2} - 3 d - 1 = 0$ , that is $(d + 1) (4 d^{2} - 2 d - 1) = 0$ , and $d + 1 > 0$ leaves $4 d^{2} - 2 d - 1 = 0$ . Completing the square, $(d - \frac{1}{4})^{2} = d^{2} - \frac{d}{2} + \frac{1}{16} = \frac{1}{4} + \frac{1}{16} = \frac{5}{16}$ , and $(\frac{\sqrt{5}}{4})^{2} = \frac{5}{16}$ with $\frac{\sqrt{5}}{4} \geq 0$ (here $\sqrt{5}$ is the unique nonnegative square root of $5$ and $\sqrt{5} > 0$ because $5 > 0$ is a nonzero square), so by uniqueness of nonnegative square roots $d - \frac{1}{4} = \pm \frac{\sqrt{5}}{4}$ and $d = \frac{1 \pm \sqrt{5}}{4}$ ; since $\sqrt{5} > 1$ by monotonicity of squaring on $[0, \infty)$ and $1 < 5$ , the root $\frac{1 - \sqrt{5}}{4}$ is negative while $d > 0$ , so $d = \frac{1 + \sqrt{5}}{4}$ . Define $φ : = \frac{1 + \sqrt{5}}{2} = 2 d = 2 \cos (π / 5)$ and $p : = 2 \cos (π / 4)$ ; then $\cos (π / 4) = \sqrt{2} / 2$ gives $p = \sqrt{2}$ and $p^{2} = 2$ , while $φ^{2} = \frac{6 + 2 \sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} = φ + 1$ ; finally the double-angle identity gives $\cos (2 π / 5) = 2 d^{2} - 1 = \frac{φ + 1}{2} - 1 = \frac{φ - 1}{2}$ , so $1 + 2 \cos (2 π / 5) = φ$ .

2.1F1F2F3F7step 1.1

(The six matrices.) For $t \in S$ let $R_{t}$ be the matrix of $ρ (s_{t}) = r_{e_{t}}$ in the basis $(e_{s})$ : $r_{e_{t}} (e_{j}) = e_{j} - 2 B (e_{j}, e_{t}) e_{t}$ , so the $j$ -th column of $R_{t}$ is $e_{j} - 2 B (e_{j}, e_{t}) e_{t}$ , and in particular the only nonzero off-diagonal entries of $R_{t}$ lie in row $t$ . Order the reflections as in the recorded application order $i_{1}, \dots, i_{n}$ and let $M : = R_{i_{n}} \dots R_{i_{1}}$ be the matrix obtained by applying $r_{i_{1}}$ first and then successively $r_{i_{2}}, \dots, r_{i_{n}}$ to a vector; equivalently $M$ is the matrix of $ρ (c)$ for $c : = s_{i_{n}} \dots s_{i_{1}}$ , the product of the two colour-class products in the order $K$ then $J$ (the classes commute internally), which is one of the two admissible orders of the bipartite definition and is conjugate to the element $a b$ of [F2] since $b a = a^{- 1} (a b) a$ . With $\cos (π / 3) = 1 / 2$ , $\cos (π / 4) = \sqrt{2} / 2$ , $\cos (π / 5) = φ / 2$ from 1.1, so that $B (e_{j}, e_{t})$ takes the values $- \frac{1}{2}$ , $- \frac{p}{2} = - \frac{\sqrt{2}}{2}$ (which we write with the symbol $p = \sqrt{2}$ ), $- \frac{φ}{2}$ , the resulting matrices are $E_{6} : (\begin{array}{cccccc} - 1 & 1 & 0 & 0 & 0 & 0 \\ - 1 & 1 & - 1 & 1 & 0 & 1 \\ 0 & 1 & - 1 & 1 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 1 \\ 0 & 0 & 0 & 1 & - 1 & 0 \\ 0 & 1 & - 1 & 1 & 0 & 0 \end{array}), E_{7} : (\begin{array}{ccccccc} - 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 1 & - 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & - 1 & 0 & 0 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 \end{array}),$ $E_{8} : (\begin{array}{cccccccc} - 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1 & 1 & - 1 & 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & - 1 & 1 & - 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & - 1 & 0 \\ 0 & 1 & - 1 & 1 & 0 & 0 & 0 & 0 \end{array}), F_{4} : (\begin{array}{cccc} - 1 & 1 & 0 & 0 \\ - 1 & 2 & - p & p \\ 0 & p & - 1 & 1 \\ 0 & p & - 1 & 0 \end{array}),$ $H_{3} : (\begin{array}{ccc} - 1 & p & 0 \\ - p & 1 + p & - 1 \\ 0 & 1 & - 1 \end{array}), H_{4} : (\begin{array}{cccc} - 1 & p & 0 & 0 \\ - p & 1 + p & - 1 & 1 \\ 0 & 1 & - 1 & 1 \\ 0 & 1 & - 1 & 0 \end{array}),$ with $p = \sqrt{2}$ in the F4 matrix and $p = φ$ in the H3 and H4 matrices; the entries lie in $Z$ , $Z [\sqrt{2}]$ or $Z [φ]$ , and $M$ is obtained from these entries over $C$ through the embeddings $\sqrt{2} \mapsto \sqrt{2}$ , $φ \mapsto φ$ . Since each $R_{t}$ preserves $B$ and is an involution [F1], each $R_{t}$ satisfies $R_{t}^{T} G R_{t} = G$ for the Gram matrix $G = (B (e_{s}, e_{t}))$ , and so does the product $M$ ; in particular $M$ has finite order because $W$ is finite [F3] and $ρ$ is a homomorphism, and $M$ is the matrix of a Coxeter element of the type.

3.1F5step 2.1

(Newton's identities from the cofactor identity.) Let $M \in M_{n} (C)$ be any of the six matrices of step 2.1 and write the expansion $χ (t) : = \det (I - t M) = \sum_{k = 0}^{n} (- 1)^{k} e_{k} t^{k}$ with $e_{0} : = 1$ , so that $\det (X I - M) = \sum_{k} (- 1)^{k} e_{k} X^{n - k}$ . Put $A (t) : = adj (I - t M)$ and $p_{i} : = tr (M^{i})$ for $i \geq 1$ . Then for every $1 \leq k \leq n$ , $k e_{k} = \sum_{i = 1}^{k} (- 1)^{i - 1} e_{k - i} p_{i} .$ Indeed the entries of $A (t)$ are cofactors of $I - t M$ , hence determinants of $(n - 1) \times (n - 1)$ matrices whose entries are affine in $t$ , so $A (t) = \sum_{k = 0}^{n - 1} C_{k} t^{k}$ with $C_{k} \in M_{n} (C)$ ; the adjugate identity $(I - t M) A (t) = χ (t) I_{n}$ gives, on comparing coefficients of $t^{k}$ for $0 \leq k \leq n$ , the recursion $C_{0} = I_{n}$ and $C_{k} = M C_{k - 1} + (- 1)^{k} e_{k} I_{n}$ for $1 \leq k \leq n - 1$ (the coefficient of $t^{n}$ only records $0 = C_{n} = M C_{n - 1} + (- 1)^{n} e_{n} I_{n}$ and is not used); induction on $k$ gives $C_{k} = \sum_{j = 0}^{k} (- 1)^{j} e_{j} M^{k - j}$ for $0 \leq k \leq n - 1$ . Differentiating the adjugate identity gives $χ^{'} (t) = - tr (A (t) M)$ by [F5]; the left side is $\sum_{k = 1}^{n} k (- 1)^{k} e_{k} t^{k - 1}$ and the right side $- \sum_{k = 0}^{n - 1} tr (C_{k} M) t^{k}$ , so comparing coefficients of $t^{k}$ for $0 \leq k \leq n - 1$ gives $(k + 1) (- 1)^{k + 1} e_{k + 1} = - tr (C_{k} M)$ . Taking traces in the displayed expansion of $C_{k}$ and using linearity, cyclicity and $p_{i} = tr (M^{i})$ gives $tr (C_{k} M) = \sum_{j = 0}^{k} (- 1)^{j} e_{j} p_{k + 1 - j}$ ; substituting and multiplying by $(- 1)^{k + 1}$ yields $(k + 1) e_{k + 1} = \sum_{i = 1}^{k + 1} (- 1)^{i - 1} e_{k + 1 - i} p_{i}$ , which is the displayed identity with $k$ replaced by $k + 1$ .

4.1step 2.1step 3.1algebra

(Traces and characteristic polynomials of the six matrices.) Computing the powers $M^{i}$ by exact matrix multiplication of the matrices of step 2.1 gives the following power sums, with $p = \sqrt{2}$ in the F4 case and $p = φ$ in the H3, H4 cases: $E_{6} : (p_{1}, \dots, p_{6}) = (- 1, 1, 2, - 3, - 1, - 2); E_{7} : (p_{1}, \dots, p_{7}) = (- 1, 1, 2, 1, - 1, - 2, - 1);$ $E_{8} : (p_{1}, \dots, p_{8}) = (- 1, 1, 2, 1, 4, - 2, - 1, 1); F_{4} : (p_{1}, \dots, p_{4}) = (0, 2, 0, - 2);$ $H_{3} : (p_{1}, p_{2}, p_{3}) = (φ - 1, φ, - φ); H_{4} : (p_{1}, \dots, p_{4}) = (φ - 1, φ, 2 φ, 1 - φ) .$ (For instance the E6 matrix has diagonal entries $- 1, 1, - 1, 1, - 1, 0$ , whose sum is $- 1 = p_{1}$ ; the higher power sums are the same finite computation.) Applying the Newton identities of step 3.1 successively for $k = 1, 2, \dots, n$ determines the coefficients $e_{k}$ in the order $k = 1, 2, \dots$ alone, and gives: $E_{6} : (e_{1}, \dots, e_{6}) = (- 1, 0, 1, 0, - 1, 1), \det (X I - M) = X^{6} + X^{5} - X^{3} + X + 1;$ $E_{7} : (e_{1}, \dots, e_{7}) = (- 1, 0, 1, - 1, 0, 1, - 1), \det (X I - M) = X^{7} + X^{6} - X^{4} - X^{3} + X + 1;$ $E_{8} : (e_{1}, \dots, e_{8}) = (- 1, 0, 1, - 1, 1, 0, - 1, 1), \det (X I - M) = X^{8} + X^{7} - X^{5} - X^{4} - X^{3} + X + 1;$ $F_{4} : (e_{1}, \dots, e_{4}) = (0, - 1, 0, 1), \det (X I - M) = X^{4} - X^{2} + 1;$ $H_{3} : (e_{1}, e_{2}, e_{3}) = (φ - 1, 1 - φ, - 1), \det (X I - M) = X^{3} + (1 - φ) X^{2} + (1 - φ) X + 1;$ $H_{4} : (e_{1}, \dots, e_{4}) = (φ - 1, 1 - φ, φ - 1, 1), \det (X I - M) = X^{4} + (1 - φ) X^{3} + (1 - φ) X^{2} + (1 - φ) X + 1,$ where each coefficient is obtained from the displayed Newton identity by the finite exact arithmetic in $Z$ , $Z [\sqrt{2}]$ or $Z [φ]$ , using $φ^{2} = φ + 1$ for the H3 and H4 cases.

5.1F2F6F7step 2.1step 4.1algebra

(Cyclotomic factorisations, E6, E7, E8, F4.) The computable cyclotomic polynomials are $Φ_{2} = X + 1$ , $Φ_{3} = X^{2} + X + 1$ , $Φ_{12} = X^{4} - X^{2} + 1$ , $Φ_{18} = X^{6} - X^{3} + 1$ and $Φ_{30} = X^{8} + X^{7} - X^{5} - X^{4} - X^{3} + X + 1$ ; expanding the products coefficientwise gives $Φ_{3} Φ_{12} = X^{6} + X^{5} - X^{3} + X + 1$ , $Φ_{2} Φ_{18} = X^{7} + X^{6} - X^{4} - X^{3} + X + 1$ and $Φ_{12} = X^{4} - X^{2} + 1$ , so by step 4.1 the E6, E7, E8 and F4 characteristic polynomials equal respectively $Φ_{3} Φ_{12}$ , $Φ_{2} Φ_{18}$ , $Φ_{30}$ and $Φ_{12}$ . Since $M$ is the matrix of the finite-order operator $ρ_{C} (c)$ [F7], $M$ is diagonalisable and its eigenvalue multiset is the root multiset of its characteristic polynomial; by [F6] the roots of $Φ_{m}$ are the primitive $m$ -th roots of unity, namely $ζ_{12}^{r}$ with $r \in {1, 5, 7, 11}$ for $Φ_{12}$ , $ζ_{12}^{4}, ζ_{12}^{8}$ for $Φ_{3}$ , $ζ_{18}^{9} = - 1$ for $Φ_{2}$ , $ζ_{18}^{r}$ with $r \in {1, 5, 7, 11, 13, 17}$ for $Φ_{18}$ , and $ζ_{30}^{r}$ with $r \in {1, 7, 11, 13, 17, 19, 23, 29}$ for $Φ_{30}$ . Hence the eigenvalue multisets are ${ζ_{12}^{r} : r \in {1, 4, 5, 7, 8, 11}}$ for E6, ${ζ_{18}^{r} : r \in {1, 5, 7, 9, 11, 13, 17}}$ for E7, ${ζ_{30}^{r} : r \in {1, 7, 11, 13, 17, 19, 23, 29}}$ for E8 and ${ζ_{12}^{r} : r \in {1, 5, 7, 11}}$ for F4; in particular each characteristic polynomial divides the corresponding $X^{h} - 1$ (because $Φ_{m} ∣ X^{m} - 1 ∣ X^{h} - 1$ when $m ∣ h$ ), and in each case $ζ_{h}$ is a root, so the order of $M$ is a multiple of $h$ ; since every eigenvalue is an $h$ -th root of unity and $M$ is diagonalisable, $M^{h} = I$ , so the order of $M$ is exactly the displayed integer $h$ . Faithfulness from [F7] implies $ρ_{C} (c)^{k} = I$ if and only if $c^{k} = 1$ , hence this is also the group order of $c$ used in [F2].

5.2F2F4F6F7step 4.1algebra

(H3 and H4.) For H3 the right side of step 4.1 expands as $(X + 1) (X^{2} - φ X + 1) = X^{3} + (1 - φ) X^{2} + (1 - φ) X + 1$ ; by [F6] the roots of $X^{2} - φ X + 1$ are the two complex numbers with sum $φ$ and product $1$ , and $e^{i π / 5}$ and $e^{- i π / 5}$ have sum $2 \cos (π / 5) = φ$ and product $1$ , so they are the roots; with $ζ_{10} = e^{2 π i / 10} = e^{i π / 5}$ and $- 1 = ζ_{10}^{5}$ the eigenvalue multiset of $M$ is ${ζ_{10}^{1}, ζ_{10}^{5}, ζ_{10}^{9}}$ . For H4 the four numbers $ζ_{30}^{1}, ζ_{30}^{11}, ζ_{30}^{19}, ζ_{30}^{29}$ lie in inverse pairs, so the monic quartic with these roots is $(X^{2} - a X + 1) (X^{2} - b X + 1) = X^{4} - (a + b) X^{3} + (a b + 2) X^{2} - (a + b) X + 1$ with $a = ζ_{30} + ζ_{30}^{- 1} = 2 \cos (π / 15)$ and $b = ζ_{30}^{11} + ζ_{30}^{- 11} = 2 \cos (11 π / 15) = - 2 \cos (4 π / 15)$ ; then $a b = - 4 \cos (π / 15) \cos (4 π / 15) = - 2 (\cos (π / 3) + \cos (- π / 5)) = - (1 + φ) = - φ^{2}$ and $a + b = 2 (\cos (π / 15) - \cos (4 π / 15)) = 2 \cdot 2 \sin (π / 6) \sin (π / 10) = 2 \sin (π / 10) = 2 \cos (2 π / 5) = φ - 1$ , using the product-to-sum and sum-to-product identities, $\cos (- π / 5) = \cos (π / 5) = φ / 2$ , $\cos (π / 3) = 1 / 2$ , $\sin (π / 6) = \cos (π / 3) = 1 / 2$ and $\sin (π / 10) = \cos (2 π / 5)$ ; hence the monic quartic with roots $ζ_{30}^{\pm 1}, ζ_{30}^{\pm 11}$ equals $X^{4} - (φ - 1) X^{3} + (- φ^{2} + 2) X^{2} - (φ - 1) X + 1 = X^{4} + (1 - φ) X^{3} + (1 - φ) X^{2} + (1 - φ) X + 1$ , because $- φ^{2} + 2 = 1 - φ$ and $- (φ - 1) = 1 - φ$ ; this is exactly the H4 polynomial computed in step 4.1, so the eigenvalue multiset of $M$ is ${ζ_{30}^{1}, ζ_{30}^{11}, ζ_{30}^{19}, ζ_{30}^{29}}$ , and the relation $φ = 1 + ζ_{30}^{6} + ζ_{30}^{- 6} = 1 + 2 \cos (2 π / 5)$ is the embedding recorded in the statement (for H3 the corresponding relation is $φ = 1 + ζ_{10}^{2} + ζ_{10}^{- 2}$ ). In both cases all eigenvalues are $h$ -th roots of unity, $M$ is diagonalisable and $ζ_{h}$ occurs among them, so the order of $M$ is exactly the displayed integer $h$ (the same argument as in 5.1); faithfulness from [F7] again identifies it with $ord (c)$ . Finally the displayed residue lists have sums $36, 63, 120, 24, 15, 60$ for $E_{6}, E_{7}, E_{8}, F_{4}, H_{3}, H_{4}$ , which are exactly $n h / 2$ and hence equal $∣ Φ_{+} ∣$ by [F2].

6.1F5step 1.1step 2.1step 3.1step 4.1step 5.1step 5.2∎

(Summary and conventions.) Steps 1.1-5.2 prove all assertions of the statement: for each of the six types the displayed matrix is the matrix of a Coxeter element of that type in the simple-root basis over $Z$ , $Z [\sqrt{2}]$ or $Z [φ]$ , its characteristic polynomial is computed from the exact power sums by Newton's identities and equals the displayed polynomial, the cyclotomic and quadratic factorisations identify the eigenvalue multisets with the displayed residue lists, each polynomial divides $X^{h} - 1$ and has $ζ_{h}$ as a root, the orders are exactly $h$ , and the residue sums equal $n h / 2 = ∣ Φ_{+} ∣$ . The computation uses no floating-point approximation, no enumeration of a Coxeter group and no invariant-degree table; the F4 characteristic polynomial is rational, so no embedding of $\sqrt{2}$ into $Q (ζ_{12})$ is used; the element whose matrix is displayed is the product of the colour-class products in the recorded order, conjugate to the element $a b$ of the bipartite definition [F2], so the conclusions hold for $c$ as well; and the dual-action matrices in the local certificate research/coxeter-scaffold/math-checks/finite-degree-poincare-certificates.json are $M^{- T}$ , not $M^{T}$ . Indeed the dual of each involutive reflection has matrix $R_{t}^{T}$ , and the same application order gives $M^{- T}$ . From $M^{T} G M = G$ we obtain $M^{- T} = G M G^{- 1}$ ; hence its characteristic polynomial agrees with that of $M$ .

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

A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types

Statement

Assume the Axiom of Choice. Let $(W, S)$ be an irreducible Coxeter system of finite type, $n = ∣ S ∣$ , with $V$ , $B$ , $ρ_{C}$ , $Φ_{+}$ , $T$ , chamber and longest-element conventions (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Root sign coherence and the action of simple reflections on positive roots, The dual action, chambers, faces, and root hyperplanes), let $c$ be the bipartite Coxeter element with order $h$ and Coxeter plane $P = s p a n (u, v)$ , prefix roots $ρ_{i}$ and the conclusion $∣ Φ_{+} ∣ = n h / 2$ (The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a, The Coxeter plane, ordered-root enumeration, and invertibility of rho(c) - id), and let $d_{1} \leq \dots \leq d_{n}$ and $e_{i} = d_{i} - 1$ be the basic degrees and exponents of Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system. Put $N : = ∣ Φ_{+} ∣$ . Then:

(1) Exponent-residue identification. The multiset of exponents equals the multiset of spectral exponents of $ρ_{C} (c)$ : after relabelling, $e_{i} = r_{i}$ , where $r_{1}, \dots, r_{n} \in {1, \dots, h - 1}$ are the residues with $ρ_{C} (c)$ -eigenvalues $ζ_{h}^{r_{1}}, \dots, ζ_{h}^{r_{n}}$ (each eigenvalue counted with multiplicity). Consequently $\sum_{i} e_{i} = N = ∣ Φ_{+} ∣ = n h / 2$ , the eigenvalues of $ρ_{C} (c)$ are exactly $ζ_{h}^{e_{1}}, \dots, ζ_{h}^{e_{n}}$ , and $h = \max_{i} d_{i}$ , $\min_{i} d_{i} = 2$ .

(2) Complete degree tables. Combining (1) with the classical and exceptional spectra: $A_{n} : 2, 3, \dots, n + 1; B_{n} : 2, 4, \dots, 2 n; D_{n} : 2, 4, \dots, 2 n - 2 and n;$ $I_{2} (m) : 2, m; E_{6} : 2, 5, 6, 8, 9, 12; E_{7} : 2, 6, 8, 10, 12, 14, 18; E_{8} : 2, 8, 12, 14, 18, 20, 24, 30;$ $F_{4} : 2, 6, 8, 12; H_{3} : 2, 6, 10; H_{4} : 2, 12, 20, 30.$

These degree lists are multisets; when $n$ is even, the degree $n$ in type $D_{n}$ is counted twice. With the conventions $A_{1}$ : degree $2$ ; the coincidences $A_{2} = I_{2} (3)$ , $B_{2} = I_{2} (4)$ , $G_{2} = I_{2} (6)$ , $A_{3} = D_{3}$ and $A_{1} \times A_{1} = I_{2} (2)$ of Classification of finite Coxeter systems, including the H and dihedral families (4) (for the extended convention $I_{2} (2) = A_{1} \times A_{1}$ the multiset is ${2, 2}$ , covered by (3)); and products of these degree multisets for reducible types as in (3). In each case $\prod_{i} d_{i} = ∣ W ∣$ .

(3) Reducible systems. If $(W, S) = (W_{S_{1}} \times \dots \times W_{S_{k}})$ with connected components $S_{1}, \dots, S_{k}$ and $V_{C} = ⨁_{j} V_{C, j}$ the $B_{C}$ -orthogonal decomposition (Disconnected diagrams, direct products, and comparison of invariant forms), then $C [V_{C}]^{W} = ⨂_{j} C [V_{C, j}]^{W_{S_{j}}}$ as graded algebras, the degree multiset of $W$ is the union of the component degree multisets, and the coinvariant algebra is the tensor product of the component coinvariant algebras. There is no single global Coxeter number in general.

(4) Conventions and abstentions. For $n = 1$ ( $A_{1}$ ) the assertions are direct; for $n = 0$ they are empty. Nothing is imported about a common length function, about Poincaré polynomial growth, or about the regular-representation structure of the coinvariant algebra; the tables are derived from the spectral data, not assumed.

Facts & Assumptions

Given: The Axiom of Choice; for the irreducible clause, a finite irreducible Coxeter system $(W, S)$ of rank $n \geq 2$ , its complex reflection representation, bipartite Coxeter element $c$ of order $h$ , Coxeter plane, positive roots, reflecting hyperplanes and fixed homogeneous basic invariants $p_{1}, \dots, p_{n}$ with degrees $d_{1} \leq \dots \leq d_{n}$ ; rank zero and rank one are treated separately below.

[F1]

Under AC, the fixed basic family exists, its degrees satisfy $d_{i} \geq 2$ , and its exponents are $e_{i} = d_{i} - 1 > 0$ ; each $p_{i}$ is homogeneous, $W$ -invariant, and the invariant ring is generated by these algebraically independent polynomials (Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system (1)-(3), Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (4), Finite linear invariant and coinvariant polynomial algebras, The Axiom of Choice).

[F2]

The canonical representation is a real group homomorphism and its complexification is faithful; since $c^{h} = 1$ , its complexified operator $C = ρ_{C} (c)$ satisfies $C^{h} = I$ . The root-sign and chamber conventions specify the positive roots and reflecting hyperplanes, and the bipartite element has the ordered root data of its definition (The real Coxeter form, its radical, reflections, and form-preserving maps, The canonical reflection homomorphism, roots, reflections, and the positive cone, Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers (1), Coxeter diagrams: edges, labels, components and finite type, The dual action, chambers, faces, and root hyperplanes, Root sign coherence and the action of simple reflections on positive roots, The bipartite Coxeter element, its ordered prefix roots, and the conditional vector map mu(a) = -2(c-1)^{-1}a).

[F3]

For irreducible finite type, $c$ acts on its Coxeter plane $P$ as rotation by $2 π / h$ ; the traces $H_{α} \cap P$ are exactly the $h$ reflection lines, $c - i d$ is invertible (so its complexification $C - I$ is invertible), and $Φ_{+} = {ρ_{1}, \dots, ρ_{n h / 2}}$ , so $∣ Φ_{+} ∣ = n h / 2$ (The Coxeter plane, ordered-root enumeration, and invertibility of rho(c) - id (2)-(4)).

[F4]

The invariant Jacobian $J = \det (d p_{i} / d x_{j})$ is nonzero; it is a nonzero scalar multiple of the discriminant $Δ = \prod_{α \in Φ_{+}} ℓ_{α}$ , whose zero set is the union of the complexified reflecting hyperplanes; and $\sum_{i} e_{i} = ∣ Φ_{+} ∣ = ∣ T ∣$ (Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian (3), The total degree sum, the invariant Jacobian as the discriminant, anti-invariants, and the top coinvariant class (1)-(2)).

[F5]

For a finite type Coxeter system under AC, $\dim_{C} A = \prod_{i} d_{i} = ∣ W ∣$ , and for a reducible diagram the basic-degree multiset is the union of the component multisets (The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity (3),(5)).

[F6]

The real matrix $ρ_{C} (c)$ has characteristic polynomial with real coefficients; its nonreal eigenvalues therefore occur with their complex conjugates and with equal algebraic multiplicities. A finite-order complex operator is diagonalisable. For any matrix $M$ , $χ_{M^{T}} (X) = \det ((X I - M)^{T}) = \det (X I - M) = χ_{M} (X)$ , by $\det (A^{T}) = \det (A)$ , so the eigenvalues of $M$ and $M^{T}$ agree with multiplicity (Over an algebraically closed field of characteristic $0$ , every element of finite order acts diagonalisably in a finite-dimensional representation, For $A \in M_{n} (F)$ , the characteristic polynomial is $χ_{A} (x) = \det (x I_{n} - A)$ when $n \geq 1$ , with $χ_{A} (x) = 1$ for the unique $0 \times 0$ matrix, For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, Eigenvalues, eigenvectors, eigenspaces $E_{λ} (T) = \ker (T - λ I)$ , and the spectrum $σ_{F} (T)$ of an endomorphism, The group $μ_{n} (K)$ of $n$ -th roots of unity in a field, and primitive $n$ -th roots of unity, The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ ). Over a field, a positive-sized square matrix is invertible exactly when its determinant is nonzero (A positive-sized square matrix over a commutative ring is invertible if and only if its determinant is a unit); hence $\det (λ I - M) = 0$ exactly when $λ$ is an eigenvalue.

[F7]

The exact irreducible finite diagrams, standard coincidences, direct-product decomposition, and orthogonal decomposition of the representation are as stated in the finite classification and product theorem (Classification of finite Coxeter systems, including the H and dihedral families (1)-(4), Disconnected diagrams, direct products, and comparison of invariant forms (1)-(2)). Polynomial functions on a direct sum decompose into the tensor product of the coordinate polynomial algebras; expansion in the block monomial basis is finite (Polynomial rings in finitely many commuting indeterminates by iteration, Finite linear invariant and coinvariant polynomial algebras).

[F8]

The exact classical and exceptional Coxeter characteristic polynomials and spectral-exponent multisets are those proved in the two preceding spectrum items (The Coxeter elements of the classical types A_n, B_n, D_n and I_2(m): characteristic polynomials, orders and spectral exponents from their reflection models, The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices).

Proof

technique · At a regular Coxeter eigenvector the gradients of the basic invariants form a basis. Their covariance gives the degrees modulo $h$; a real-spectrum sum and the discriminant degree remove all possible multiples of $h$

1.1F2F3F4F6algebra

Assume first that $W$ is irreducible and $n \geq 2$ . Let $P = {span}_{R} (u, v)$ be the Coxeter plane. Choose real vectors $ξ, η$ forming a basis of $P$ so that $x = ξ + i η$ is an eigenvector of $C : = ρ_{C} (c)$ with eigenvalue $ζ_{h}^{- 1}$ ; this is possible because $C ∣_{P}$ is rotation by $2 π / h$ [F2, F3]. If $x$ lay in a complexified reflecting hyperplane $H_{α} \otimes_{R} C$ , then its real and imaginary parts $ξ, η$ would both lie in $H_{α}$ , so $P \subseteq H_{α}$ . But $H_{α} \cap P$ is a line by [F3], so no reflecting hyperplane contains $P$ . Therefore every factor of $Δ (x)$ is nonzero and $J (x) \neq 0$ by [F4]. The matrix $(d p_{i} (x))_{i = 1}^{n}$ is thus invertible, and $d p_{1} (x), \dots, d p_{n} (x)$ form a basis of $V_{C}^{*}$ [F4].

1.2F5F7algebra

Let the connected components of a reducible diagram be $S_{1}, \dots, S_{k}$ , with corresponding spaces $V_{C, j}$ and groups $W_{j}$ . By [F7], $W = \prod_{j} W_{j}$ acts on $V_{C} = ⨁_{j} V_{C, j}$ factorwise, and the coordinate polynomial algebra is $C [V_{C}] = ⨂_{j} C [V_{C, j}]$ . To compute invariants, expand any polynomial as a finite sum of block monomials with coefficients in the other blocks. Invariance under $W_{j}$ forces every coefficient in the $j$ -th block to be $W_{j}$ -invariant; doing this for each factor proves $C [V_{C}]^{W} = ⨂_{j} C [V_{C, j}]^{W_{j}}$ , and the reverse inclusion follows because each factor acts trivially on the other blocks [F7]. Each component invariant ring is polynomial on its homogeneous basic family, so the tensor product is polynomial on the union of these families. Its degrees are therefore the union of the component degree multisets, also as in [F5]. If $I_{j}$ is the ideal generated by positive-degree invariants in the $j$ -th block, then the positive-degree ideal of $⨂_{j} C [V_{C, j}]^{W_{j}}$ generates precisely $I = \sum_{j} I_{j} C [V_{C}]$ : every positive-degree pure tensor has at least one positive-degree factor, and each $I_{j}$ is generated by invariants of the full product. The tensor-product quotient is consequently $C [V_{C}] / I ≅ ⨂_{j} (C [V_{C, j}] / I_{j})$ , which is the asserted tensor decomposition of coinvariant algebras. By [F5] the product of all degrees is $∣ W ∣ = \prod_{j} ∣ W_{j} ∣$ . There is no common Coxeter number for unequal component orders, and this clause uses no global $h$ .

2.1F1F2F3F6step 1.1

For each $i$ and all $y \in V_{C}$ , $p_{i} (C y) = p_{i} (y)$ because $p_{i}$ is invariant [F1]. Differentiate this identity at $y = x$ in an arbitrary direction $v$ to get $d p_{i} (C x) (C v) = d p_{i} (x) (v)$ , or $d p_{i} (C x) \circ C = d p_{i} (x)$ . Since $C x = ζ_{h}^{- 1} x$ and $p_{i}$ is homogeneous of degree $d_{i}$ , $d p_{i} (ζ_{h}^{- 1} x) = ζ_{h}^{- (d_{i} - 1)} d p_{i} (x)$ ; hence $d p_{i} (x) \circ C = ζ_{h}^{d_{i} - 1} d p_{i} (x) = ζ_{h}^{e_{i}} d p_{i} (x)$ . These covectors form a basis by 1.1, so the eigenvalue multiset of the transpose action is ${ζ_{h}^{e_{i}}}_{i = 1}^{n}$ . A matrix and its transpose have the same characteristic polynomial, so this is also the eigenvalue multiset of $C$ , with multiplicity [F1, F2, F6, step 1.1, algebra]. By [F3], $C - I$ is invertible, so $1$ is not an eigenvalue. Thus each $e_{i}$ is congruent modulo $h$ to a unique residue in ${1, \dots, h - 1}$ , and these residues, counted with multiplicity, are exactly the spectral exponents $r_{1}, \dots, r_{n}$ of the statement [F2, F3, F6].

3.1F1F3F4step 2.1

The matrix $C$ is real by [F2], so its nonreal eigenvalues pair as $ζ_{h}^{r}, ζ_{h}^{- r} = ζ_{h}^{h - r}$ with equal multiplicity; each such pair contributes $h$ to the sum of residues. The only real $h$ -th roots of unity are $1$ and, when $h$ is even, $- 1$ ; $1$ is excluded by 2.1, while each occurrence of $- 1 = ζ_{h}^{h / 2}$ contributes $h / 2$ . Partitioning the full eigenvalue multiset into these pairs and occurrences of $- 1$ therefore gives $\sum_{i} r_{i} = n h / 2$ [F2, F3, F6, algebra]. By [F4], $\sum_{i} e_{i} = ∣ Φ_{+} ∣ = n h / 2$ as well. Since each positive integer $e_{i}$ is congruent to a residue $r$ in ${1, \dots, h - 1}$ , it has the form $e_{i} = r + h q_{i}$ with an integer $q_{i} \geq 0$ ; the residue multiset and exponent multiset have the same cardinality, and the equal sums force every $q_{i} = 0$ . Thus the exponent multiset is exactly the spectral-residue multiset, proving (1)'s identification and sum claim.

4.1F3step 3.1

The rotation on $P$ has eigenvalues $ζ_{h}$ and $ζ_{h}^{- 1} = ζ_{h}^{h - 1}$ , so the spectral residues $1$ and $h - 1$ both occur, counted with multiplicity (when $h = 2$ these are the same residue and the eigenvalue occurs with multiplicity two on $P$ ). By 3.1 the same residues occur among the $e_{i}$ . Therefore $\min_{i} e_{i} = 1$ and $\max_{i} e_{i} = h - 1$ , which gives $\min_{i} d_{i} = 2$ and $\max_{i} d_{i} = h$ .

4.2F5F7

Add $1$ to each spectral exponent from the two computed spectrum suppliers. The classical lists give $A_{n} : 2, 3, \dots, n + 1$ , $B_{n} : 2, 4, \dots, 2 n$ , $D_{n} : 2, 4, \dots, 2 n - 2$ together with $n$ , and $I_{2} (m) : 2, m$ . The exceptional lists give $E_{6} : 2, 5, 6, 8, 9, 12$ , $E_{7} : 2, 6, 8, 10, 12, 14, 18$ , $E_{8} : 2, 8, 12, 14, 18, 20, 24, 30$ , $F_{4} : 2, 6, 8, 12$ , $H_{3} : 2, 6, 10$ , and $H_{4} : 2, 12, 20, 30$ [F8, step 3.1]. The classification's coincidence conventions identify $A_{2} = I_{2} (3)$ , $B_{2} = I_{2} (4)$ , $G_{2} = I_{2} (6)$ , $A_{3} = D_{3}$ , and, with the extended notation, $A_{1} \times A_{1} = I_{2} (2)$ ; for the last case the degrees are the union ${2, 2}$ [F7]. The product-degree formula $\prod_{i} d_{i} = ∣ W ∣$ is [F5], so it holds for every listed irreducible type and for the reducible unions.

5.1F1F5F7algebra∎

If $n = 0$ , then $W$ is trivial, the invariant and coinvariant algebras are $C$ , and all degree, exponent and product assertions are empty, with empty products equal to $1$ [F1, F5]. If $n = 1$ , the irreducible system is $A_{1}$ : its generator acts by $x \mapsto - x$ , so the invariant ring is $C [x^{2}]$ , the basic degree is $2$ , $c = s$ has order $h = 2$ and eigenvalue $- 1 = ζ_{2}$ , the sole exponent/residue is $1$ , $∣ Φ_{+} ∣ = 1 = n h / 2$ , and the coinvariant algebra is $C [x] / (x^{2})$ . Thus all claims hold directly; for reducible rank one factors the componentwise argument of 1.2 applies [F1, F5, F7, algebra]. The Axiom of Choice is used only for the basic-family and invariant-theory supplier conclusions in [F1] and [F5]; the plane, derivative, residue and tensor calculations above use no further choice [F1, F5, def-axiom-of-choice].

5 · Examples, counterexamples and false statements

None yet.

Finite Coxeter Invariants and Coinvariant Gradings

1 · Prerequisites

2 · Summary

Prerequisites and reading

3 · Logical flowchart

4 · Definitions, theorems and proofs

Complexifying a finite Coxeter reflection representation: faithfulness, complex reflections, and the hypotheses of the invariant-theory suppliers

Statement

Facts & Assumptions

Proof

Basic degrees, exponents, and the graded coinvariant algebra of a finite Coxeter system

Definition

The Coxeter elements of the classical types A_n, B_n, D_n and I_2(m): characteristic polynomials, orders and spectral exponents from their reflection models

Statement

Facts & Assumptions

Proof

The basic degrees are independent of the chosen family; Hilbert series of the invariants and of the coinvariant algebra; the order formula and the Molien identity

Statement

Facts & Assumptions

Proof

Algebraicity of the coordinates over the invariant field and non-vanishing of the invariant Jacobian

Statement

Facts & Assumptions

Proof

The total degree sum, the invariant Jacobian as the discriminant, anti-invariants, and the top coinvariant class

Statement

Facts & Assumptions

Proof

The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices

Statement

Facts & Assumptions

Proof

A regular Coxeter eigenvector determines the basic degrees: the exponent-residue identification and the complete degree tables for all finite Coxeter types

Statement

Facts & Assumptions

Proof

5 · Examples, counterexamples and false statements

Sources