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The finite dihedral rotation and the infinite unipotent rank-two product

Example

Let $S = {s, t}$ , $m = m (s, t)$ , $V = R^{S}$ , $B$ the Coxeter form, $P = R e_{s} + R e_{t}$ and $c$ as in The real Coxeter form, its radical, reflections, and form-preserving maps ( $c = 1$ for $m = \infty$ ). By Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order the product $A : = r_{s} r_{t}$ acts on $P$ with matrix $A = (\begin{array}{cc} 4 c^{2} - 1 & - 2 c \\ 2 c & - 1 \end{array})$ in the basis $(e_{s}, e_{t})$ (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases).

(i) Finite dihedral rotation. For $m = 3$ one has $c = \cos (π / 3) = \frac{1}{2}$ and $A = (\begin{array}{cc} 0 & - 1 \\ 1 & - 1 \end{array}), A^{2} = (\begin{array}{cc} - 1 & 1 \\ - 1 & 0 \end{array}), A^{3} = I_{2},$ so $A$ has order $3 = m$ ; its trace is $- 1 = 2 \cos (2 π / 3)$ , consistent with a rotation through $2 π / 3$ of the positive definite plane $(P, B ∣_{P})$ , and $A \neq I_{2} \neq A^{2}$ . For $m = 2$ the same formula gives $c = 0$ and $A = - I_{2}$ , of order $2 = m$ .

(ii) Infinite unipotent product. For $m = \infty$ one has $c = 1$ and $A = (\begin{array}{cc} 3 & - 2 \\ 2 & - 1 \end{array}) = I_{2} + N, N = (\begin{array}{cc} 2 & - 2 \\ 2 & - 2 \end{array}) \neq 0, N^{2} = 0.$ Hence $A^{k} = I_{2} + k N$ for every $k \in Z$ , so $A^{k} (e_{s}) = e_{s} + 2 k (e_{s} + e_{t})$ and no nonzero power of $A$ is the identity: the product $r_{s} r_{t}$ has infinite order, in contrast to the finite cases where $A$ has order $m$ . In the abstract group $W$ of Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, the element $s t$ likewise has infinite order when $m = \infty$ , and has order $m$ in the displayed finite cases $m = 2, 3$ .

Facts & Assumptions

Given: $S = {s, t}$ with $s \neq t$ , a Coxeter matrix value $m = m (s, t) \in {2, 3, \dots} \cup {\infty}$ , the space $V = R^{S}$ , the Coxeter form $B$ , the plane $P = R e_{s} + R e_{t}$ and the number $c$ of The real Coxeter form, its radical, reflections, and form-preserving maps ( $c = \cos (π / m)$ for finite $m$ , and $c = 1$ for $m = \infty$ ).

[F1]

In the ordered basis $(e_{s}, e_{t})$ of $P$ the product $A = r_{s} r_{t}$ has matrix $[A] = (\begin{array}{cc} 4 c^{2} - 1 & - 2 c \\ 2 c & - 1 \end{array})$ of determinant $1$ , and $A$ acts on $P$ by that matrix (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, clause (3)(iii)); the same item's clause (3)(iv) records the order conclusions for finite $m$ and the unipotent shape for $m = \infty$ , which the computations below verify directly.

[F2]

$B (e_{s}, e_{s}) = B (e_{t}, e_{t}) = 1$ and $B (e_{s}, e_{t}) = - c$ ; $r_{s}$ and $r_{t}$ are $B$ -preserving involutions (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order, clause (2)).

[F3]

Trigonometric facts: the addition formulas and the resulting triple-angle identity $\cos 3 x = 4 \cos^{3} x - 3 \cos x$ ; $\sin^{2} x + \cos^{2} x = 1$ ; $\cos π = - 1$ and $\cos (π / 2) = 0$ ; $\sin x = 0$ if and only if $x \in π Z$ (The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Quarter-turn values and shifts by pi/2 and pi).

[F4]

Matrices of linear maps in an ordered basis, products of matrices and the identity matrix are as defined entrywise; $[T^{2}] = [T]^{2}$ for a linear endomorphism $T$ and an ordered basis ( $S$ finite, $V$ finite-dimensional) (Coordinate columns $[v]_{B}$ and matrices $[T]_{B}^{C}$ of linear maps relative to ordered bases, $[S \circ T]_{B}^{D} = [S]_{C}^{D} [T]_{B}^{C}$ , Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F5]

The group $W$ is presented by $s^{2} = t^{2} = 1$ and, when $m < \infty$ , $(s t)^{m} = 1$ . Any assignment of $s, t$ to involutions in a group satisfying the finite relator extends to a homomorphism from $W$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups, Universal property).

Verification

1.1givenF1F2F3F4algebra

The case $m = 3$ . Here $c = \cos (π / 3)$ . The triple-angle identity of [F3] at $x = π / 3$ gives $\cos π = 4 c^{3} - 3 c$ ; with $\cos π = - 1$ this is $4 c^{3} - 3 c + 1 = 0$ , that is $(c + 1) (2 c - 1)^{2} = 0$ . The factor $c + 1$ is nonzero: $\cos x = - 1$ forces $\sin^{2} x = 1 - \cos^{2} x = 0$ , hence $\sin x = 0$ and $x \in π Z$ , whereas $π / 3$ is not an integer multiple of $π$ . Hence $2 c - 1 = 0$ and $c = \frac{1}{2}$ . Substituting into [F1], $[A] = (\begin{array}{cc} 0 & - 1 \\ 1 & - 1 \end{array}), [A]^{2} = (\begin{array}{cc} - 1 & 1 \\ - 1 & 0 \end{array}), [A]^{3} = [A]^{2} [A] = I_{2},$ and $[A] \neq I_{2} \neq [A]^{2}$ , so $A$ has order exactly $3 = m$ . Its trace is $- 1$ , and $2 \cos (2 π / 3) = 2 (2 c^{2} - 1) = 2 (\frac{1}{2} - 1) = - 1$ , consistent with the rotation through $2 π / 3$ of the positive definite plane: $A$ preserves $B ∣_{P}$ and has determinant $1$ .

1.2givenF1F3F4algebra

The case $m = 2$ . Here $c = \cos (π / 2) = 0$ , so [F1] gives $[A] = (\begin{array}{cc} - 1 & 0 \\ 0 & - 1 \end{array}) = - I_{2}$ , and $A^{2} = {i d}_{P}$ while $A \neq {i d}_{P}$ ; thus $A$ has order $2 = m$ , again a rotation through $2 π / 2 = π$ of the positive definite plane.

1.3givenF1F2F4algebra

The case $m = \infty$ . Here $c = 1$ , so [F1] gives $[A] = (\begin{array}{cc} 3 & - 2 \\ 2 & - 1 \end{array}) = I_{2} + N$ with $N = (\begin{array}{cc} 2 & - 2 \\ 2 & - 2 \end{array}) \neq 0$ , and direct multiplication gives $N^{2} = 0$ . The binomial theorem in a ring with $N^{2} = 0$ gives $(I_{2} + N)^{k} = I_{2} + k N$ for every $k \geq 0$ , and $A^{- 1} = I_{2} - N$ because $(I_{2} + N) (I_{2} - N) = I_{2} - N^{2} = I_{2}$ , so $(I_{2} - N)^{j} = I_{2} + (- j) N$ for $j \geq 0$ and $A^{k} = I_{2} + k N for every k \in Z .$ Therefore $A^{k} (e_{s}) = e_{s} + k N e_{s} = e_{s} + 2 k (e_{s} + e_{t})$ , and since $k N$ has first entry $2 k$ , which is nonzero for $k \neq 0$ , no nonzero power of $A$ is the identity: the product $r_{s} r_{t}$ has infinite order in $G L (V)$ .

2.1givenF1F2F5step 1.1step 1.2step 1.3∎

Conclusion in the abstract group. By [F2], $r_{s}, r_{t}$ are invertible involutions. If $m < \infty$ , [F1] gives $(r_{s} r_{t})^{m} = {i d}_{V}$ ; the reversed finite relator holds too, since $r_{t} r_{s} = (r_{s} r_{t})^{- 1}$ . If $m = \infty$ , there is no finite pair relator to check. Thus [F5] supplies a homomorphism $ρ : W \to G L (V)$ with $ρ (s) = r_{s}$ , $ρ (t) = r_{t}$ , and $ρ (s t) = A$ . For $m = 2, 3$ , the relator gives $(s t)^{m} = 1$ , and steps 1.1–1.2 show that no smaller positive power can be $1$ : it would map to the corresponding nonidentity power of $A$ . Hence $s t$ has order exactly $m$ in these finite cases. For $m = \infty$ , if $(s t)^{k} = 1$ for any nonzero integer $k$ , applying $ρ$ would give $A^{k} = {i d}_{V}$ , contradicting step 1.3. Hence $s t$ has infinite order. This conclusion uses the homomorphism and the computed nonidentity powers, rather than inferring element order from the absence of a relator.

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Sources

Michael W. Davis, The Geometry and Topology of Coxeter Groups (Princeton University Press; author's full institutional PDF) (standard reference, not scraped)
Anders Björner and Francesco Brenti, Combinatorics of Coxeter Groups (Graduate Texts in Mathematics 231, Springer 2005) (standard reference, not scraped)
George Lusztig, Hecke Algebras with Unequal Parameters (revised 2014 text, arXiv:math/0208154v2) (standard reference, not scraped)