ON CENTRALIZERS OF REFLECTIONS IN COXETER GROUPS

BRIGITTE BRINK

Abstract

We show that the centralizer of a reflection in a Coxeter group is the semidirect product of a Coxeter group by a free group.

1. Summary of background material

Let (W, R) be a Coxeter system, as defined in [1] or [5], and for r, s ∈ R let m_{rs} be the order of rs. Further, let V be an ℝ-vector space with basis Π = {a_{r} | r ∈ R}, and let ⟨ , ⟩: V × V → ℝ be a symmetrical bilinear form which satisfies ⟨a_{r}, a_{s}⟩ = − cos(π/m_{rs}) for all r, s ∈ R such that m_{rs} is finite, and ⟨a_{r}, a_{s}⟩ ≤ −1 if m_{rs} is infinite. Then r · v = v − 2⟨v, a_{r}⟩a_{r} (for all r ∈ R and v ∈ V) defines a faithful action of W on V. The set Φ = {w · a_{r} | r ∈ R, w ∈ W } is the root system of W in V, and the elements of Π are called simple roots. A root α is said to be positive (negative), if it can be written as ∑_{a∈Π} λ_{a}a with all λ_{a} ≥ 0 (all λ_{a} ≤ 0). The sets of positive and negative roots are denoted by Φ^{+} and Φ^{−} respectively, and it is well known that Φ = Φ^{+} ∪ Φ^{−}.

Elements of W which are conjugate to elements of R are called re�ections. For each α ∈ Φ, there is a reflection r_{α} ∈ W which satisfies r_{α} · v = v − 2⟨α, v⟩v for all v ∈ V. Furthermore, r_{α} = wrw^{−1} whenever r ∈ R and w ∈ W satisfy α = w · α_{r}.

For w ∈ W, the length l(w) is defined to be the minimal integer n such that w equals r_{1} ⋯ r_{n} for some r_{i} ∈ R. If w ≠ 1, it is clear that there exists an r ∈ R with l(wr) = l(w) − 1. We also define N(w) to be the set of all α ∈ Φ^{+} such that w · α ∈ Φ^{−}.

The following proposition summarizes some well known properties.

Proposition 1. (i) Let α and β be roots, and for each n ∈ ℤ, let λ_{n}, μ_{n} ∈ ℝ be such that (r_{α}r_{β})^{n} · α = λ_{n}α + μ_{n}β. Then (r_{α}r_{β})^{n} · β = −μ_{n}α − λ_{n−1}β for all n ∈ ℤ, and

(a) λ_{n} = sin(\frac{(2n+1)π}{m})/ sin(\frac{π}{m}) and μ_{n} = sin(\frac{2nπ}{m})/ sin(\frac{π}{m}) if ⟨α, β⟩ = − cos(\frac{π}{m});

(b) λ_{n} − 1 ≥ μ_{n} ≥ λ_{n−1} + 1 if ⟨α, β⟩ ≤ −1.
(ii) Let α ∈ Φ^{+} and w ∈ W. Then l(wr_{α}) < l(w) if and only if w · α ∈ Φ^{−}.
(iii) If w ∈ W, then |N(w)| = l(w) and N(w^{−1}) = −w · N(w).

(iv) If u, w ∈ W, then N(uw) ⊆ N(w) ∪ w^{−1} · N(u), and the following are equivalent: (a) l(uw) = l(u) + l(w),

(b) N(uw) = N(w) ∪ w^{−1} · N(u),

(c) N(u) ∩ N(w^{−1}) = ∅.

For Γ ⊆ Φ, define W_{Γ} to be the group generated by r_{γ} with γ ∈ Γ, and Φ_{Γ} to be the set of w · γ with γ ∈ Γ and w ∈ W_{Γ}. The next theorem was proved independently by Deodhar in [3] and Dyer in [4].

Theorem 2. Let Γ ⊆ Φ. Then W_{Γ} is a Coxeter group, and Φ_{Γ} is a root system for W_{Γ}; in particular, if w ∈ W_{Γ} with Φ_{Γ} ∩ N(w) = ∅, then w = 1.

2. The centralizer of a re�ection

In this section we will prove a structure theorem for the centralizer of a reflection. It is clear that the centralizer of the reflection wrw^{−1} equals wC(r)w^{−1}, where C(r) is the centralizer of r (and w ∈ W, r ∈ R). Hence it suffices to describe the structure of C(r) for r ∈ R.

For r ∈ R, define Γ(r) to be the set of roots γ with ⟨γ, a_{r}⟩ = 0, and for r, s ∈ R, let Y_{s,r} be the set of u ∈ W with a_{s} = u · a_{r} and N(u) ∩ Γ(r) = ∅. It follows from Proposition 1(iii) that w^{−1} ∈ Y_{r,s} for w ∈ Y_{s,r}, and from Proposition 1(iv) that uw ∈ Y_{t,r} for u ∈ Y_{t,s} and w ∈ Y_{s,r}. Since clearly 1 ∈ Y_{r,r}, we deduce that Y_{r,r} is a group.

We also define the odd Coxeter graph of W to be the graph with vertex set R and edge set {{r, s} | r, s ∈ R such that r ≠ s and m_{rs} is odd}.

Theorem. Let r ∈ R. Then the centralizer of r in W is the semidirect product of W_{Γ(r)∪{±a_{r}}} by Y_{r,r}; that is,

C(r) = W_{Γ(r)∪{±a_{r}}} ⋊ Y_{r,r}.

Moreover, W_{Γ(r)∪{±a_{r}}} is a Coxeter group, and Y_{r,r} is free of rank e(r) − n(r) + 1, where e(r) denotes the number of edges and n(r) the number of vertices of the connected component of the odd Coxeter graph of W containing r.

Proof. Since uru^{−1} = r for u ∈ C(r), we find that
−a_{r} = r · a_{r} = uru^{−1} · a_{r} = r_{u·a_{r}} · a_{r} = a_{r} − 2⟨u · a_{r}, a_{r}⟩u · a_{r}.

Thus a_{r} = ⟨u · a_{r} , a_{r}⟩u · a_{r}, and as ⟨α, α⟩ = 1 for α ∈ Φ, we deduce that u · a_{r} = ±a_{r}. So if γ ∈ Γ(r), then ⟨u · γ, a_{r}⟩ = ⟨γ, u^{−1} · a_{r}⟩ = ⟨γ, ±a_{r}⟩ = 0. Therefore

u · Γ(r) = Γ(r) and u · {±a_{r}} = {±a_{r}},

and it follows readily that W_{Γ(r)∪{±a_{r}}} is normal in C(r). The above also yields that

Φ_{Γ(r)∪{±a_{r}}} = Γ(r) ∪ {±a_{r}};  (∗)

hence W_{Γ(r)∪{±a_{r}}} is a Coxeter group with root system Γ(r) ∪ {±a_{r}} by Theorem 2. It remains to show that Y_{r,r} is a complement of W_{Γ(r)∪{±a_{r}}} in C(r), and that Y_{r,r} is free of rank e(r) − n(r) + 1.

Let X_{r} be a set of coset representatives of W_{Γ(r)∪{±a_{r}}} in C(r) of minimal length. For u ∈ X_{r}, the minimality of l(u) yields that l(ur) ≥ l(u); so u · a_{r} ∈ Φ^{+} by Proposition 1(ii), and hence u · a_{r} = a_{r} (as u · a_{r} = ±a_{r} by the discussion above). Minimality of l(u) yields further that l(ur_{γ}) ≥ l(u) for γ ∈ Γ(r); so if γ is positive, then u · γ ∈ Φ^{+} by Proposition 1(ii). Hence N(u) ∩ Γ(r) = ∅, and thus X_{r} ⊆ Y_{r,r}.

To show that X_{r} = Y_{r,r}, let w ∈ Y_{r,r}, and let u ∈ X_{r} be the representative of the coset wW_{Γ(r)∪{±a_{r}}}. Then u ∈ Y_{r,r}, and thus w^{−1}u ∈ Y_{r,r}; that is, (w^{−1}u) · a_{r} = a_{r} and Γ(r) ∩ N(w^{−1}u) = ∅. Thus by (∗),

Φ_{Γ(r)∪{±a_{r}}} ∩ N(u^{−1}w) = (Γ(r) ∩ N(u^{−1}w)) ∪ ({±a_{r}} ∩ N(u^{−1}w)) = ∅.

As w^{−1}u ∈ W_{Γ(r)∪{±a_{r}}} by construction, Theorem 2 now yields that w^{−1}u = 1. Hence X_{r} = Y_{r,r}, and it remains to show that Y_{r,r} is free of rank e(r) − n(r) + 1.

It is clear that the elements of the fundamental group of a connected graph Ξ can be identified with paths in Ξ which start and end at a fixed vertex, and which never back-track upon themselves; that is, at no stage does the path traverse an edge and then immediately traverse it again in the opposite direction. It is well known (see, for example, [6], Chapter 6, Theorem (5.2), p.198) that this fundamental group is free of rank ϵ − ν + 1, where ϵ is the number of edges, and ν is the number of vertices of Ξ. (This follows from the fact that Ξ is homotopy equivalent to a graph on one vertex with ϵ−ν +1 edges - topologically a bouquet of circles - as can be seen by shrinking a spanning tree of Ξ to a single vertex.)

Suppose now that the graph Ξ is the connected component of the odd Coxeter graph containing the vertex r. Define F to be the set of all (l + 1)-tuples (r_{0}, …, r_{l}) with r_{0}, …, r_{l} ∈ R such that r_{i−1} and r_{i} are adjoined in the odd Coxeter graph for all i ∈ {1, …, l}, and r_{i−1} ≠ r_{i+1} for all i ∈ {1, …, l − 1}. Further, let F_{r} be the set of (r_{0}, …, r_{l}) ∈ F with r_{0} = r_{l} = r. The fundamental group of Ξ can be identified with F_{r}, multiplication on F_{r} being defined by the rule

(r_{0}, …, r_{l}) ∗ (s_{0}, …, s_{m}) = (r_{0}, …, r_{l−i−1}, s_{i}, …, s_{m}),

where i is the maximal integer such that s_{j} = r_{l−j} for all j ∈ {0, …, i}. Note that the identity of this group is (r). We will now construct an isomorphism π_{r}: F_{r} → Y_{r,r}.

If s, t ∈ R are adjoined by a bond of weight 2n + 1, define π(s, t) = (ts)^{n}. Then sπ(s, t) is the uniquely determined word in W_{{a_{s},a_{t}}} of maximal length; it follows from Proposition 1(i)(a) that N(π(s, t)) = Φ_{{a_{s},a_{t}}}^{+} \ {a_{t}} and that π(s, t) ∈ Y_{s,t}. Define π: F → W by π(r_{0}) = 1, and for l ≥ 1,

π(r_{0}, …, r_{l}) = π(r_{0}, r_{1})π(r_{1}, r_{2})π(r_{2}, r_{3}) ⋯ π(r_{l−2}, r_{l−1})π(r_{l−1}, r_{l}).

Then π(r_{0}, …, r_{l}) ∈ Y_{r_{0},r_{l}}, since π(r_{i−1}, r_{i}) ∈ Y_{r_{i−1},r_{i}} for all i, and we conclude that π induces a homomorphism π_{r}: F_{r} → Y_{r,r}. The next two propositions yield that π_{r} is surjective and injective respectively, and therefore a group isomorphism, and this finishes the proof.

Proposition. Let r, s ∈ R and w ∈ Y_{s,r}. Then there exists an (r_{0}, …, r_{l}) ∈ F with r_{0} = s and r_{l} = r such that w = π(r_{0}, …, r_{l}).

Proof. If w = 1, the assertion is trivially true. So assume that w ≠ 1, and proceed by induction on l(w). Let t ∈ R with l(wt) < l(w); then t ≠ r by Proposition 1(ii) as w · a_{r} = a_{s} is positive. Next, let d ∈ W be of minimal length such that w = du for some u ∈ W_{{a_{r},a_{t}}}. Since w = (wt)t, we know that l(d) ≤ l(wt) < l(w), and thus u ≠ 1. Minimality of d forces l(dr), l(dt) ≥ l(d), and thus d · a_{r}, d · a_{t} ∈ Φ^{+} by Proposition 1(ii). Further, let λ, μ ∈ ℝ such that u · a_{r} = λa_{r} + μa_{t}; then λ, μ ≥ 0 or λ, μ ≤ 0 by Proposition 1(i). Now

a_{s} = w · a_{r} = d · (u · a_{r}) = λ(d · a_{r}) + μ(d · a_{t}),

and since a_{s} is simple, we deduce that λ = 0 and a_{s} = d · a_{t}, or μ = 0 and a_{s} = d · a_{r}. So u · a_{r} equals a_{t} or a_{r}, and we denote u · a_{r} by a_{z}.

Next, let x ∈ R with l(ux) < l(u); that is, u · a_{x} ∈ Φ^{−}. Since u ∈ W_{{a_{r},a_{t}}} and u · a_{r} = a_{z} is positive, we find that x = t; that is, u · a_{t} ∈ Φ^{−}. Linear independence of a_{r} and a_{t} yields that u · a_{r} and u · a_{t} are linearly independent; in particular, u · a_{t} does not equal −u · a_{r} = −a_{z}, and thus (zu) · a_{t} ∈ Φ^{−}. Since (zu) · a_{r} = −a_{z} ∈ Φ^{−}, it follows that N(zu) includes all roots of the form ν_{r}a_{r} + ν_{t}a_{t} with ν_{r}, ν_{t} ≥ 0. But N(zu) is finite by Proposition 1(iii), and so Proposition 1(i)(b) implies that rt must have finite order. Furthermore, zu is the uniquely determined word in W_{{a_{r},a_{t}}} of maximal length.

Assume for a contradiction that the order of rt is even and equals 2m. Then u equals t(rt)^{m−1}, and (tr)^{\frac{m−1}{2}} · a_{t} ∈ N(u) if m is odd, while t(rt)^{\frac{m}{2} −1} · a_{r} ∈ N(u) if m is even. It can be easily verified using Proposition 1(i) that these roots are perpendicular to a_{r}, and since N(u) ⊆ N(w) by Proposition 1(iv), this contradicts w ∈ Y_{s,r}. Therefore rt must have odd order; moreover, u = π(t, r) and d · a_{t} = d · (u · a_{r}) = w · a_{r} = a_{s}.

If β ∈ N(d), then ⟨β, a_{t}⟩ = ⟨u^{−1} · β, u^{−1} · a_{t}⟩ = ⟨u^{−1} · β, a_{r}⟩. Since u^{−1} · β ∈ N(w) by Proposition 1(iv), while w ∈ Y_{s,r}, we find that ⟨β, a_{t}⟩ ≠ 0, and thus d ∈ Y_{s,t}. By induction, there exists an (r_{0}, …, r_{l}) ∈ F with r_{0} = s and r_{l} = t such that d equals π(r_{0}, …, r_{l}). Since l(dr) ≥ l(d) by minimality of d, it follows that r_{l−1} ≠ r (if l ≥ 1); hence (r_{0}, …, r_{l}, r) ∈ F and w = π(r_{0}, …, r_{l}, r), as required.

Proposition. Let (r_{0}, …, r_{l}) ∈ F. Then l(π(r_{0}, …, r_{l})) = ∑_{i=1}^{l} l(π(r_{i−1}, r_{i})).

Proof. Assume for a contradiction that the assertion is not true, and let l be minimal such that there exists an (r_{0}, …, r_{l}) ∈ F with l(π(r_{0}, …, r_{l})) ≠ ∑_{i=1}^{l} l(π(r_{i−1}, r_{i})). Clearly l ≥ 2, and we define u_{1} = π(r_{0}, r_{1}), w = π(r_{1}, …, r_{l−1}) and u_{2} = π(r_{l−1}, r_{l}). Then l(u_{1}w) = ∑_{i=1}^{l−1} l(π(r_{i−1}, r_{i})) by minimality of l, and our assumption becomes

l(u_{1}wu_{2}) = l(π(r_{0}, …, r_{l})) ≠ ∑_{i=1}^{l} l(π(r_{i−1}, r_{i})) = l(u_{1}w) + l(u_{2}).

Therefore N(u_{1}w) ∩ N(u_{2}^{−1}) ≠ ∅ by Proposition 1(iv), and - again by Proposition 1(iv) - we deduce that

w^{−1} · N(u_{1}) ∩ N(u_{2}^{−1}) ≠ ∅ or N(w) ∩ N(u_{2}^{−1}) ≠ ∅.

But minimality of l also yields that l(w) = ∑_{i=2}^{l−1} l(π(r_{i−1}, r_{i})) and l(wu_{2}) = l(w)+l(u_{2}), and so N(w)∩N(u_{2}^{−1}) = ∅ by Proposition 1(iv). In order to avoid double indices, denote the simple root corresponding to r_{i} by a_{i} for all i. Then N(u_{1}) = Φ_{{a_{0},a_{1}}}^{+} \ {a_{1}} and N(u_{2}^{−1}) = Φ_{{a_{l−1},a_{l}}}^{+} \ {a_{l−1}}, and thus there exist λ > 0, μ ≥ 0 and x ≥ 0, y > 0 with

w^{−1} · (λa_{0} + μa_{1}) = xa_{l−1} + ya_{l}.

Now w^{−1} · a_{1} = a_{l−1} implies that w^{−1} · a_{0} = (\frac{x−μ}{λ})a_{l−1} + (\frac{y}{λ})a_{l}. Since λ, y > 0, we find that w^{−1} · a_{1} is positive, and thus x ≥ μ. Symmetrically, w · a_{l−1} = a_{1} yields that w · a_{l} = (\frac{λ}{y})a_{0} + (\frac{μ−x}{λ})a_{1}, and thus μ ≥ x. Therefore μ = x and w · a_{l} = a_{0}.

It follows that l > 2, since otherwise w = 1 and a_{0} = w · a_{l} = a_{2}, contradicting (r_{0}, …, r_{l}) ∈ F. Set u = π(r_{2}, …, r_{l−1}) and n = \frac{m_{r_{1}r_{2}} −1}{2}. Then π(r_{1}, r_{2}) = (r_{2}r_{1})^{n}, and thus w = (r_{2}r_{1})^{n}u with l(w) = l((r_{2}r_{1})^{n}) + l(u); hence by Proposition 1(iv),

N((r_{2}r_{1})^{n}) ∩ N(u^{−1}) = ∅.  (1)

Next, define α = (r_{1}r_{2})^{n} ·a_{0} and β = (r_{1}r_{2})^{n−2}r_{1} ·a_{0}. Since r_{0} ≠ r_{2} (as (r_{0}, …, r_{l}) ∈ F), the coefficients of a_{0} in α and β equal 1, and so α and β are positive. Furthermore,

(r_{l}u^{−1}) · α = r_{l} · (u^{−1} · ((r_{1}r_{2})^{n} · a_{0})) = r_{l} · (w^{−1} · a_{0}) = r_{l} · a_{l} = −a_{l}.

We will show that (r_{l}u^{−1}) · β is also negative, and that ⟨α, β⟩ ≤ −1. This completes the proof, for then each root of the form ν_{α}α + ν_{β}β with ν_{α}, ν_{β} ≥ 0 is an element of N(r_{l}u^{−1}); but Proposition 1(i)(b) yields that there are infinitely many roots of this form, and thus N(r_{l}u^{−1}) must have infinitely many elements, contradicting Proposition 1(iii).

Let γ = (r_{1}r_{2})^{n−1} · a_{1}. A straightforward calculation yields that β = r_{γ} · α, and thus

(r_{l}u^{−1}) · β = (r_{l}u^{−1}) · (α − 2⟨α, γ⟩γ) = −a_{l} − 2⟨α, γ⟩(r_{l}u^{−1}) · γ.

So in order to show that (r_{l}u^{−1}) · β is negative, it suffices to show that (r_{l}u^{−1}) · γ is positive, and ⟨α, γ⟩ ≥ 0. Proposition 1(i)(a) implies that γ ∈ N((r_{2}r_{1})^{n}), and so u^{−1} · γ is positive by (1); since clearly γ ≠ α = u · a_{l}, and thus u^{−1} · γ ≠ a_{l}, it follows that (r_{l}u^{−1}) · γ is positive. Next,

⟨α, γ⟩ = ⟨(r_{1}r_{2})^{n} · a_{0}, (r_{1}r_{2})^{n−1} · a_{1}⟩ = ⟨r_{1}r_{2} · a_{0}, a_{1}⟩ = −⟨a_{0}, a_{1}⟩ + 2⟨a_{0}, a_{2}⟩⟨a_{1}, a_{2}⟩,

and since (r_{0}, …, r_{l}) ∈ F, we know that ⟨a_{0}, a_{1}⟩, ⟨a_{1}, a_{2}⟩ ≤ − cos(\frac{π}{3} ) = −\frac{1}{2}, and ⟨a_{0}, a_{2}⟩ ≤ 0. Therefore

⟨α, γ⟩ ≥ \frac{1}{2} − ⟨a_{0}, a_{2}⟩,  (2)

and ⟨α, γ⟩ > 0, as required; it remains to show that ⟨α, β⟩ ≤ −1.

A straightforward induction yields that α = a_{0} + Aa_{1} + Ba_{2}, with

A = −⟨a_{0}, a_{2}⟩(\frac{2}{sin(\frac{π}{m})} ∑_{i=1}^{n} sin(\frac{2iπ}{m} )) − ⟨a_{0}, a_{1}⟩(\frac{2}{sin(\frac{π}{m})} ∑_{i=0}^{n−1} sin(\frac{(2i + 1)π}{m} )),

where m = m_{r_{1}r_{2}}; therefore A = \frac{−⟨a_{0},a_{1}⟩−⟨a_{0},a_{2}⟩}{1+⟨a_{1},a_{2}⟩}, and thus A > 0. Further,

\begin{aligned}a_{l} = u^{−1} · α & = u^{−1} · a_{0} + A(u^{−1} · a_{1}) + B(u^{−1} · a_{2}) \\ & = u^{−1} · a_{0} + A(u^{−1} · a_{1}) + Ba_{l−1}, \end{aligned}

and since a_{l} is simple, we deduce that u^{−1} · a_{0} and u^{−1} · a_{1} cannot both be positive. But a_{1} ∈ N((r_{2}r_{1})^{n}) by Proposition 1(i)(a), and thus u^{−1} · a_{1} is positive by (1); hence a_{0} ∈ N(u^{−1}). Since u^{−1} ∈ Y_{r_{l−2},r_{2}}, while r_{0} ≠ r_{2} (as (r_{0}, …, r_{l}) ∈ F), this yields that ⟨a_{0}, a_{2}⟩ ≤ − cos(\frac{π}{3} ) = −\frac{1}{2}. Therefore ⟨α, γ⟩ ≥ 1 by (2), and thus

⟨α, β⟩ = ⟨α, r_{γ} · α⟩ = ⟨α, α⟩ − 2⟨α, γ⟩^{2} = 1 − 2⟨α, γ⟩^{2} ≤ −1,

as required.

Acknowledgements. This paper is taken from my dissertation. I would like to thank my supervisor Dr. R. B. Howlett for his support throughout my candidature and the writing of this paper. The work was completed while I was supported by an Overseas Postgraduate Research Award and a University Postgraduate Research Award.

References

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3. V. V. Deodhar, ‘A Note on Subgroups Generated by Reflections in Coxeter Groups’, Archiv der Mathematik 53 (1989) 543–546.

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5. J. E. Humphreys, Re�ection Groups and Coxeter Groups (Cambridge University Press, 1990).

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School of Mathematics and Statistics University of Sydney NSW 2006 Australia