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An indefinite Coxeter form with a faithful canonical reflection representation

Example

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

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

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

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

Facts & Assumptions

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

[F1]

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

[F2]

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

[F3]

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

[F4]

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

[F5]

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

Verification

technique · direct

1.1F1F4algebra

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

1.2F1F2F3algebra

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

1.3F1F4algebra

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

1.4F1algebra

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

1.5F2F5algebra

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

2.1F3step 1.1step 1.2step 1.3step 1.4∎

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

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Sources

Michael W. Davis, The Geometry and Topology of Coxeter Groups (Princeton University Press, 2008; author's complete institutional PDF) (standard reference, not scraped)
Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups (Graduate Texts in Mathematics 231, Springer 2005; author-hosted full PDF) (standard reference, not scraped)