Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 39 (1998), No. 1, 7-18.

Generating Series for the Coxeter Groups and Applications

Pierrick Meignen

Université de Caen, Département de Mathématiques et Mécanique 14032 Caen France

Abstract. By forming generating series associated to convenient linear representations of Coxeter groups acting on spaces of constant curvature, we show that the mean values of some related quantities are given by linear recurrence sequences. The tools are the ones used to count words with a given length. MSC 1991: 20F55, 51B10, 11J70

1. Introduction

Let W be a discrete group of Euclidean affine motions generated by the reflections in hyperplanes of a bounded fundamental domain A. It is clear that for O ∈ A and w ∈ W_{a}:

∥\overrightarrow{Ow(O)}∥ ≤ l(w)δ

where δ = sup_{x,y∈A} ∥x − y∥ and l(w) is the length of the word w. We shall see that we can say a little more about the ∥\overrightarrow{Ow(O)}∥. In particular, we can get some mean values for a given length, Theorem 2 giving their asymptotic behavior.

Similarly, let W be a discrete group of hyperbolic motions generated by the inversions in hyperspheres of an unbounded fundamental domain C in a Poincaré half-space of R^{m}. We suppose that the only point at infinity of C is ∞. We describe a multidimensional generalization of Hermite’s algorithm [4] due to the author [7]. For z in the floor R^{m−1}, let w_{n}(C) be a sequence of adjacent domains through which the geodesic (z∞) passes and such that l(w_{n}) = n. We know that:

|z − p_{n}q_{n}^{−1}| ≤ \frac{1}{γ |q_{n}|^{2}} .

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1998 Heldermann Verlag

where p_{n}q_{n}^{−1} = w_{n}(∞), and γ is a constant depending only on W. Here, p_{n} and q_{n} are in a Clifford algebra with (as vector space) a Euclidean norm | | which restricts to ∥ ∥ on R^{m−1}. In the framework of the Diophantine approximation, it is interesting to have some results about the size of the denominators q_{n}. We show that it is possible to get the mean value of the |q_{n}|^{2} produced by Hermite’s algorithm at the step n. The Theorem 3 gives a general result.

In the two cases previously described, W is a Coxeter group. The results of Section 2 are all aimed at an explicit evaluation of the formal power series ∑_{w} ρ(w)X^{l(w)} when ρ is a linear representation of W. The method is a generalization of the one which permits to get the number of words of a given length. For suitable choices of ρ (in symmetric algebras) we deduce the moments at any order of ∥\overrightarrow{Ow(O)}∥^{2} and |q_{n}|^{2}.

2. Generating series for Coxeter groups

For a general survey about the classical results for Coxeter groups, the reader is referred to [2] or [5]. But we give here the statements that are sufficient for the understanding of the sequel.

In this paper, W_{S} denotes a Coxeter group generated by a finite subset S = {s_{1}, …, s_{p}} of distinct involutions with the defining relations

(s_{i}s_{j})^{m_{i,j}} = Id with 1 ≤ m_{i,j} ≤ ∞.

For w in W_{S}, the length l_{S}(w) is the smallest integer q such that w is the product of q elements of S with the particular case l_{S}(Id) = 0. It is clear that we have l_{S}(sw) = l_{S}(w) ± 1 for all s in S. It is well known that when W_{S} is finite, there exists a unique element w_{S} of maximal length. This element verifies w_{S}^{2} = Id.

For T ⊂ S, the subgroup of W_{S} generated by all s in T is also a Coxeter group denoted by W_{T}. Such a subgroup is called a parabolic subgroup. A Coxeter group which is not a direct product of non-trivial parabolic subgroups is said to be irreducible. For all w in W_{T}, we have l_{S}(w) = l_{T} (w). So we shall simply write l instead of l_{S}. We get a set of coset representatives for W_{S}/W_{T} by introducing the set W^{T} of w in W_{S} verifying l(ws) > l(w) for all s in T. A main result for our purpose is that for a given w in W_{S}, there are unique u ∈ W^{T} and v ∈ W_{T} such that w = uv. Moreover, we have l(w) = l(u) + l(v) and u is the unique element of smallest length in the coset wW_{T}. We have:

W_{S} = W^{T} · W_{T} .

We denote the group algebra of W_{S} over a commutative ring R by R[W_{S}].

Let K be a commutative field. We denote the quotient field of the polynomial ring K[X] in one variable by K(X), and the ring of the formal power series by K[[X]]. For any subset H ⊂ W_{S}, we denote by H(X) the formal power series

∑_{w∈H} wX^{l(w)} in K[[X]][W_{S}].

In order to get an explicit expression for W_{S}(X), we generalize a method which gives the number of elements of a given length [5]. The key formula is a consequence of the decomposition previously described :

W_{S}(X) = W^{T} (X) · W_{T} (X) for all T ⊂ S.  (1)

We note that

∑_{T ⊂S}(−1)^{|T |}W^{T} = {w ∈ W_{S}, l(ws) = l(w) − 1 for all s in S}.

This set will be denoted by B_{S}. If W_{S} is infinite then B_{S} = ∅ , else B_{S} = {w_{S}}. It follows that:

B_{S}(X) = ∑_{T ⊂S}(−1)^{|T |}W^{T} (X).

From (1), we can write

(B_{S}(X) − (−1)^{|S|})(W_{S}(X))^{−1} = ∑_{T ⊊S}(−1)^{|T |}(W_{T} (X))^{−1}.  (2)

We set C_{T} (X) = (1 − (−1)^{|S|}B_{S}(X))^{−1}. According to whether W_{T} is finite or not, we have:

C_{T} (X) = (1 − X^{2l(w_{T})})^{−1}(1 + (−1)^{|T |}w_{T} X^{l(w_{T})}) or C_{T} (X) = 1.

Hence (2) becomes:

(W_{S}(X))^{−1} = (−1)^{|S|+1}C_{S}(X) ∑_{T ⊊S}(−1)^{|T |}(W_{T} (X))^{−1}.

By induction on |S|, we obtain

(W_{S}(X))^{−1} = (−1)^{|S|+1}C_{S}(X) ∑(−1)^{i}C_{T_{i}}(X) ⋯ C_{T_{1}}(X)

where the sum is taken over the (T_{1}, …, T_{i}) such that ∅ ≠ T_{1} ⊊ ⋯ ⊊ T_{i} ⊊ S.

Let P_{0} ∈ K[X] be the generator of the ideal

{P ∈ K[X], P (X)(W_{S}(X))^{−1} ∈ K[X][W_{S}]}

such that P_{0}(0) = 1. There are unique polynomials P_{i} verifying the main relation

P_{0}(X) = (∑_{i≥1} P_{i}(X)w_{i})W_{S}(X)  (3)

where the w_{i} run over the finite set of the different products w_{T_{i}} ⋯ w_{T_{1}}.

Example 1. Let W_{{s_{1},s_{2},s_{3}}} be the Coxeter group defined by:

s_{1}^{2} = s_{2}^{2} = s_{3}^{2} = (s_{1}s_{2})^{3} = (s_{1}s_{3})^{2} = 1.

This group is isomorphic to P GL_{2}(Z) acting on R ∪ {∞} by

\{ \begin{matrix}s_{1}(x) = 1/x \\ s_{2}(x) = 1 − x \\ s_{3}(x) = −x \end{matrix}

The main relation (3) is

\begin{matrix}(1 − X^{2})(1 − X^{6}) = \{1 + 2X^{2} + X^{4} + X^{6} − (X + X^{3} + X^{5} + X^{7})s_{1} \\ −Xs_{2} − (X + X^{3} + X^{5})s_{3} + (X^{2} + X^{4} + X^{6})s_{1}s_{3} \\ −X^{4}(s_{1}s_{2} + s_{2}s_{1}) + (X^{3} + X^{5})s_{1}s_{2}s_{1}\} · P GL_{2}(Z)(X). \end{matrix}  (4)

3. Linear recurrence sequences

Let A be a K-algebra. We say that a sequence (u_{n})_{n≥0} of elements in A is a linear recurrence sequence of length k > 0 over A (resp. over K) if there is an integer N and some A_{0} ≠ 0, A_{1}, …, A_{k} = 1 in A (resp. in K) such that

∑_{i=0}^{k} A_{i}u_{n+i} = 0 for n ≥ N.  (5)

To any sequence (u_{n})_{n≥0} in A, we associate the formal power series

u(X) = ∑_{n≥0} u_{n}X^{n} in A[[X]],

which is invertible if and only if u_{0} is invertible. If (u_{n})_{n≥0} verifies (5) then

(∑_{i=0}^{k} A_{k−i}X^{i}) u(X) ∈ A[X].  (6)

Reciprocally, let u(X) be in A[[X]] and v(X) be in A[X] such that v(0) is invertible and v(X) u(X) ∈ A[X]. It is easy to see that (u_{n})_{n≥0} is a linear recurrence sequence in A. Hence, from the main relation (3), the sequence defined by

u_{n} = ∑_{\begin{matrix}w∈W_{S} \\ l(w)=n \end{matrix}} w

is a linear recurrence sequence over K[W_{S}].

We shall work with some linear representations of W_{S}. So we consider the case where A is the algebra End_{K}(E) of the K-endomorphisms of a vector space E of dimension p over K. One can see End_{K}(E) as the algebra Mat_{p}(K) of the square p × p matrices over K. For R = K[X], K[[X]] or K(X), we have also the isomorphisms:

R ⊗_{K} End_{K}(E) ≈ Mat_{p}(R).

Let (u_{n})_{n≥0} be a recurrence sequence verifying (5) over End_{K}(E). For all n ≥ N, it is possible to get (u_{n+1}, …, u_{n+k}) from (u_{n}, …, u_{n+k−1}) using a linear operator D_{A} that we can write in the form

D_{A} = ( \begin{matrix}0 & Id \\ & 0 & ... \\ & & ... & ... \\ & & & 0 & Id \\ −A_{0} & −A_{1} & ⋯ & ⋯ & −A_{k−1} \end{matrix} ) , A_{i} ∈ End_{K}(E) ≈ Mat_{p}(K).

Viewing D_{A} as an element of End_{K}(E^{k}) ≈ Mat_{p^{k}}(K), we define its characteristic polynomial in K[X] by

P_{A}(X) = det(X · Id − D_{A}) = det(∑_{i=0}^{k} A_{i}X^{i}).

For P (X) ≠ 0 in K[X], we set P^{∗}(X) = X^{deg(P)}P (X^{−1}). From (6), the ideal

I(u) = {P (X) ∈ K[X], P (X)u(X) ∈ K[X] ⊗_{K} End_{K}(E)} ,

generated by a l.c.m. of the denominators of the entries of u(X) ∈ Mat_{p}(K(X)), contains P_{A}^{∗}. We see that a linear recurrence sequence over End_{K}(E) is a linear recurrence sequence over K :

Theorem 1. Let E be a finite dimensional vector space over K and a representation ρ : W_{S} → Aut_{K}(E). The sequence defined by

n ↦ ∑_{\begin{matrix}w∈W_{S} \\ l(w)=n \end{matrix}} ρ(w)

is a linear recurrence sequence over K.

4. Some representations of Coxeter groups

Let χ : W_{S} → {1} be the trivial character. Applying the results of Section 2, we can get the number N_{l} of elements with length l. Indeed N_{l} is the coefficient of X^{l} in χ(W_{S})(X) = ∑_{w} χ(w)X^{l(w)}. There are general results for some Coxeter groups:

• If W_{S} is finite and irreducible, then χ(W_{S})(X) is the Poincaré polynomial of W_{S} [3] and we have:

χ(W_{S})(X) = (1 − X)^{−p} ∏_{i=1}^{p}(1 − X^{m_{i}+1})

where the m_{i} are the exponents of W_{S}.

• If W_{S} is an affine Weyl group (see §5) then, using Morse theory [1], we have:

χ(W_{S})(X) = (1 − X)^{1−p} ∏_{i=1}^{p−1} \frac{1 − X^{m_{i}+1}}{1 − X^{m_{i}}}

where the m_{i} are the exponents of the corresponding Weyl group. In this case, we obtain

N_{l} = \frac{l^{p−2} ∏_{i}(m_{i} + 1)}{(p − 1)! ∏_{i} m_{i}} + O(l^{p−3})  (7)

Now, we introduce the geometric representation. We consider a bilinear symmetric form B on E = R^{p} verifying

B(e_{i}, e_{j}) = \{ \begin{matrix}− cos(π/m_{i,j}) & if & m_{i,j} < ∞ \\ r_{i,j} ≤ −1 & if & m_{i,j} = ∞ \end{matrix}

where (e_{i})_{1≤i≤p} is the canonical basis of E. We obtain an injective representation of W_{S} as a discrete subgroup of the orthogonal group O_{B}(E) by mapping each s_{i} on the reflection

x ↦ x − 2B(e_{i}, x)e_{i}.

It is an important fact that the stabilizer {w ∈ W_{S}, w(x) = x} of x ∈ E is a Coxeter group generated by S(x) = {w ∈ S, s(x) = x}.

When B is positive, we can identify W_{S} with a discrete subgroup of an affine Euclidean group generated by orthogonal reflections. If B is hyperbolic, i.e. of signature (∗, 1), it is possible to identify W_{S} with a discrete group of Möbius transformations. These are the two cases that we will encounter in the sequel of this paper.

The geometric representation is sufficient to compute the mean value of the quantities which are of interest to us. But to get the moments of any orders, we shall use representations in symmetric algebras.

Let E^{∗} be the dual space of a vector space E of dimension p over K. Let ρ : W_{S} → Aut_{K}(E) be a representation of W_{S}. First we get from ρ the contragredient representation ρ^{∗} : W_{S} → Aut_{K}(E^{∗}) following the rule

⟨ρ^{∗}(w)x, y⟩ = ⟨x, ρ(w^{−1})y⟩

where ⟨ , ⟩ is the duality bracket on E^{∗} × E. In practice, we have

ρ^{∗}(w) = ρ(w^{−1})^{t}

where A^{t} denotes the transpose of a linear operator A. From the fact that l(w^{−1}) = l(w), we get

ρ^{∗}(W_{S})(X) = ρ(W_{S})^{t}(X).

We denote by S(E^{∗}) the symmetric algebra of E^{∗}. This Z -graded K-algebra is the direct sum of the components S^{k}(E^{∗}) of degree k formed by the commutative products x_{1} ×⋯×x_{k} with x_{i} in E^{∗}. Hence, the contragredient representation ρ^{∗} induces a finite dimensional representation

ρ_{k}^{∗} : W_{S} → Aut_{K}(S^{k}(E^{∗})) for all k ≥ 0

by

ρ_{k}^{∗}(w)(x_{1} × ⋯ × x_{k}) = ρ^{∗}(w)(x_{1}) × ⋯ × ρ^{∗}(w)(x_{k}).

Let (ε_{i}) be the canonical basis of E and (ε_{i}^{∗}) be its dual basis. The algebra S(E^{∗}) is isomorphic to the algebra K [X_{1}, ⋯, X_{p}] of the polynomials in p variables over K and S^{k}(E^{∗}) corresponds to the vector space of the homogeneous polynomials of degree k. By identifying ε_{i}^{∗} with X_{i}, the action of W_{S} on K[X_{1}, …, X_{p}] is simply given by

P (X_{1}, …, X_{p}) ↦ P (∑_{i} a_{i,1}X_{i}, ⋯, ∑_{i} a_{i,n}X_{i})

where (a_{i,j}) is the matrix of ρ^{∗}(w) with respect to the dual basis.

Remark 1. One can show that the geometric representation ρ : W_{S} → Aut_{K}(E) induces a faithful representation K[W_{S}] → End_{K}(S(E^{∗})). This means that, working in S(E^{∗}), it is impossible to get a relation which is simpler than the main relation (3).

5. Affine Weyl groups

An irreducible Coxeter group W_{S} which admits a positive degenerate form B is isomorphic to a discrete group W_{a} of Euclidean transformations by mapping the s_{i} on orthogonal reflections of the affine space R^{p−1} where p = |S|. The group W_{a} is termed an affine Weyl group. We describe briefly the situation in terms of root systems.

The simplex A bounded by the hyperplanes of the s_{i} is a fundamental domain for the action of W_{a} on R^{p−1}. There is a vertex O of A such that W_{a} is the semi-direct product of the stabilizer W_{O} of O with the subgroup T of the translations of W_{a}. The group W_{O} is a Weyl group (i.e. a finite crystallographic reflection group).

Let H_{i}, i = 1, …, p − 1, be the walls of A which contain O. We choose the vectors α_{i} of the H_{i} such that A is in the cone C_{O} = {O + x, (x, α_{i}) ≥ 0 for i < p} and that (α_{i}^{∨}) is a basis

of the lattice L of T where α_{i}^{∨} = \frac{2α_{i}}{(α_{i}, α_{i})}. The set R = W_{O} \{α_{i}, i < p\} is the root system of

W_{a} and R^{∨} = W_{O} {α_{i}^{∨}, i < p} is the inverse system of R. The roots are linear combinations of the α_{i} with integer coefficients. A root α = ∑_{i} a_{i}α_{i} is said to be positive if a_{i} ≥ 0 for all i. Denoting by R_{+} the set of the positive roots, we have R = R_{+} ∪ (−R_{+}). There is a unique positive root α̃ = ∑_{i} a_{i}α_{i} such that ∑_{i} a_{i} is maximum. This highest root α̃ is a vector of the wall H_{p} of A which does not contain O. We can suppose that the scalar product ( , ) is such that H_{p} is given by (x, α̃) = 1. This implies that there is a 1-1 correspondence between the hyperplanes H_{α,k} of equation (x, α) = k, α ∈ R and k ∈ Z, and the reflections of W_{a}. It is known that the length of w is the number of hyperplanes H_{α,k} which separate A and w(A). In particular, the maximum length in W_{O} is |R_{+}|.

To establish the theorem 1, we shall work with real linear recurrence sequences and we need the following result Lemma 1. Let a ∈ C^{m}, a ≠ (0), and u_{n} = ℜ(∑_{j} a_{j} exp(inθ_{j})) with distinct θ_{j} in ]0, π]. Then we have

lim inf_{n} u_{n} < 0 < lim sup_{n} u_{n}.

Proof. We consider the mean values m_{n} = \frac{1}{n} ∑_{0}^{n−1} u_{k} and v_{n} = \frac{1}{n} ∑_{0}^{n−1} u_{k}^{2}. From

lim_{n} \frac{1}{n} ∑_{0≤k<n} exp(ikθ) = \{ \begin{matrix}0 if θ ≠ 0 \\ 1 if θ = 0 \end{matrix}

we deduce lim_{n} m_{n} = 0 and lim_{n} v_{n} = \frac{1}{2} ∑_{j} |a_{j}|^{2}. Let M = ∑_{j} |a_{j}| and v = \frac{1}{2} ∑_{j} |a_{j}|^{2}.

Suppose that 0 ≤ lim inf_{n} u_{n}. For any ε > 0, there is an integer N such that u_{n} ≥ −ε for n ≥ N. We have:

0 ≤ \frac{1}{n} ∑_{N≤k≤n}(u_{k} + ε)^{2} ≤ \frac{1}{n}(M + ε) ∑_{N≤k≤n}(u_{k} + ε).

Making n tend to ∞, we deduce:

0 ≤ v ≤ εM

which contradicts a ≠ (0).

Replacing a by −a, we obtain the desired inequality.

Now, we are in position to prove :

Theorem 2. For a given integer k, there are a_{0} > 0, a_{j} ∈ C, θ_{j} ∈]0, π] such that the moment

of order k of ∥\overrightarrow{Ow(O)}∥^{2} over \{w ∈ W_{a}, l(w) = l\} is

(a_{0} + ℜ(∑_{j} a_{j} exp(ilθ_{j})))l^{2k} + O(l^{2k−1}).

Proof. In order to get (12) below, we will follow some arguments of [6]. For w ∈ W_{a}, there are unique u, v ∈ W_{O} such that w = uw′v where w′ is the word of minimum length in the double coset W_{O}wW_{O}. The word w′v is in W^{O} = {w ∈ W_{a}, l(s_{i}w) = l(w) − 1 for all i ≠ n} which is also the set of the w such that w(A) ⊂ C_{O}. We denote the translation of vector d ∈ L by t_{d}. For any d in L ∩ C_{O}, the unicity of v implies that inf {l(t_{d}w), w ∈ W_{O}} is attained by a unique element w_{d} ∈ W_{O}. Moreover, for all v ∈ W_{O}, we have:

l(t_{d}w_{d}v) = l(t_{d}w_{d}) + l(v).  (8)

For a direction given by a positive root α, the number of hyperplanes H_{α,k} which separate A and w(A) is 0 or (d, α) − 1 according to whether (d, α) = 0 or (d, α) ≠ 0. Hence, we get that

l(t_{d}w_{d}) = (d, ∑_{α∈R_{+}} α) − |{α ∈ R_{+}, (d, α) > 0}| .  (9)

For d = ∑_{i} a_{i}α_{i}^{∨}, we have (see the tables of [2]):

(d, ∑_{α∈R_{+}} α) = 2 ∑_{i} a_{i}.  (10)

Now, let w ∈ W^{O} be of length l. We have ∥\overrightarrow{Ow(O)}∥ = ∥d∥ ≤ (∑_{i} a_{i}) sup_{i} ∥α_{i}^{∨}∥. From (8), (9) and (10):

\{ \begin{matrix}2 ∑_{i} a_{i} − |R_{+}| ≤ l(t_{d}w_{d}) ≤ l \\ l ≤ (d, ∑_{α∈R_{+}} α) + |R_{+}| ≤ ∥d∥ ∥∑_{α∈R_{+}} α∥ + |R_{+}| . \end{matrix}

We deduce:

\frac{l − |R_{+}|}{∥∑_{α∈R_{+}} α∥} ≤ ∥\overrightarrow{Ow(O)}∥ ≤ \frac{1}{2} sup_{i} ∥α_{i}^{∨}∥ (l + |R_{+}|).  (11)

Let w ∈ W be of length l. Then ∥\overrightarrow{Ow(O)}∥ = ∥\overrightarrow{Ow′v(O)}∥ with l − |R_{+}| ≤ l(w′v) ≤ l and w′v ∈ W^{O}. From (11):

\frac{l − |R|}{∥∑_{α∈R_{+}} α∥} ≤ ∥\overrightarrow{Ow(O)}∥ ≤ \frac{1}{2} sup_{i} ∥α_{i}^{∨}∥ (l + |R_{+}|).  (12)

For all x in the affine space R^{p−1} and w in W_{a}, we have w(x) = w(O) + w̃(\overrightarrow{Ox}) where w̃ ∈ W_{O} denotes the linear part of w. We consider a linear representation ρ : W_{a} → GL_{p}(Z) defined by

ρ(w) = ( \begin{matrix}M & d \\ 0 & 1 \end{matrix} )

where M ∈ GL_{p−1}(Z) is the matrix of w̃ in the basis (α_{i}^{∨}). Viewing ∥ ∥^{2k} as a homogeneous polynomial of degree 2k in p variables over R and applying Theorem 1 with ρ_{k}^{∗}, we see that

l ↦ ∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2k} is a linear recurrence sequence. We fix k. From general theory of the linear recurrence sequences in R, we have:

∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2k} = ∑_{0≤p≤m}(a_{0}^{(p)} + ℜ(∑_{j} a_{j}^{(p)} exp(ilθ_{j}^{(p)})))l^{p} with a^{(m)} ≠ (0).

From Lemma 1, we get that:

lim inf_{l} \frac{∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2k}}{l^{m}} < a_{0}^{(m)} < lim sup_{l} \frac{∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2k}}{l^{m}} ,

and from (7) and (11):

0 < lim inf_{l} \frac{∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2k}}{l^{p−2+2k}} ≤ lim sup_{l} \frac{∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2k}}{l^{p−2+2k}} < ∞.

This implies that m = p − 2 + 2k and a_{0}^{(m)} > 0.

Example 2. We consider the affine Weyl group of type B̃_{2}, i.e. with the defining relations:

s_{1}^{2} = s_{2}^{2} = s_{3}^{2} = (s_{1}s_{2})^{4} = (s_{2}s_{3})^{4} = (s_{1}s_{3})^{2} = 1.

[figure: A; C; α_{1}^{∨}; α̃^{∨}; α_{2}^{∨}; O]

Figure 1: The affine Weyl group of type B̃_{2}

The geometric elements are illustrated at figure 1. We have L = Zα_{1}^{∨} ⊕ Zα_{2}^{∨} and α̃^{∨} = α_{1}^{∨} + α_{2}^{∨}. The ρ(s_{i}) ∈ GL_{3}(Z) are given by

ρ(s_{1}) = ( \begin{matrix}−1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ρ(s_{2}) = ( \begin{matrix}1 & 0 & 0 \\ 1 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ρ(s_{3}) = ( \begin{matrix}1 & −2 & 1 \\ 0 & −1 & 1 \\ 0 & 0 & 1 \end{matrix} ) .

The last reflection corresponds to α̃^{∨}. Using ρ_{k}^{∗} for k = 2 and 4:

\{ \begin{matrix}\frac{1}{N_{l}} ∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{2} = \frac{46}{108}l^{2} + O(l) \\ \frac{1}{N_{l}} ∑_{l(w)=l} ∥\overrightarrow{Ow(O)}∥^{4} = \frac{59}{5184}l^{4} + O(l^{3}) \end{matrix}

Remark 2. As the representation ρ can also be used to get ∑_{l(w)=l} \overrightarrow{Ow(O)}, it is not surprising that one can prove:

\frac{1}{N_{l}} ∥ ∑_{l(w)=l} \overrightarrow{Ow(O)}∥ ∈ O(1).

6. Hyperbolic reflection groups

A Coxeter group which admits an hyperbolic form B is isomorphic to a discrete subgroup W_{h} of the full Möbius group Möb(H̃^{m}) acting on

H̃^{m} = {x ∈ R^{m}, x_{m} ≥ 0} ∪ {∞}

where m = dim(E/E^{⊥}) − 1. We say that W_{h} is a hyperbolic reflection group [9]. We get this representation by mapping the s_{i} on reflections in half-planes or half-spheres orthogonal to R^{m−1} = {x ∈ R^{m}, x_{m} = 0}. The mirrors of these reflections are the walls of a fundamental domain C for the action of W_{S} on the Poincaré half-space H^{m} = {x ∈ R^{m}, x_{m} > 0}. From now on, we suppose that ∞ is the only point at infinity of C. In general there is no formula as (3) for the number N_{l} of elements with length l. But in this case, one can show that

0 < lim inf_{l} \frac{ln(N_{l})}{l} ≤ lim sup_{l} \frac{ln(N_{l})}{l} < ∞.

Recall that, for a given z ∈ R^{m−1}, the Hermite algorithm produces at the step n a word w_{n} ∈ W_{h} of length n and we have:

|z − p_{n}q_{n}^{−1}| ≤ \frac{1}{γ |q_{n}|^{2}} .

where p_{n}q_{n}^{−1} = w_{n}(∞), and γ is a constant depending only on W_{h}.

Theorem 3. For any integer k, the sequence l ↦ ∑_{l(w)=l} |q(w)|^{2k} is a linear recurrence sequence.

Proof. We use the geometric representation ρ : W_{S} → O_{B}(E) because (see [8]) |q_{n}|^{2} = −\frac{1}{2}B(w_{n}(v), v) for v (corresponding to ∞) in F = E/E^{⊥}.

For u, v ∈ F, we consider the values of the form B^{k}(w(u), v). By identifying F and its dual F^{∗}, we have

B^{k}(w(u), v) = ⟨w(u), v⟩^{k} = (ρ_{k}^{∗}(w)(u^{k}))(v) = V^{t} · M_{w}^{(k)} · U

where the U_{i} are the coordinate of u^{k} in a basis (E_{i}) of S_{k}(F^{∗}), V_{i} = E_{i}(v) and M_{w}^{(k)} is the matrix of ρ_{k}^{∗}(w) respect to (E_{i}). Hence we have

∑_{w∈W_{S}} |q(w)|^{2k} X^{l(w)} = (−\frac{1}{2})^{k}(E_{i}(e))^{t} · ρ_{k}^{∗}(W_{S})(X) · (e^{k}) in R[[X]] ∩ R(X).

Example 3. From (4), we get

χ(P GL_{2}(Z))(X) = \frac{(1 + X + X^{2})(1 + X)^{2}}{1 − X^{2} − X^{3}} in Q(X) ∩ Z[[X]]  (13)

hence

lim_{l} \frac{ln(N_{l})}{l} ≈ 0.28.

From v = e_{1}^{∗} = −2(e_{2} + e_{3}), we get:

∑_{w} |q(w)|^{2}X^{l(w)} = \frac{X(1 − X)(1 + X)(1 + 2X + X^{2} − 2X^{3} + X^{5} + X^{6})}{1 − X − 2X^{2} + 3X^{4} − 2X^{5} − X^{6} + X^{8}}

and we deduce:

lim_{l} \frac{1}{l} ln (\frac{1}{N_{l}} ∑_{l(w)=l} |q(w)|^{2}) ≈ 0.317.

Similarly, we get

lim_{l} \frac{1}{l} ln (\frac{1}{N_{l}} ∑_{l(w)=l} |q(w)|^{4}) ≈ 0.642.

Remark 3. The same kind of results for the norm of the Möbius transformations of W_{h} are also valid.

References

[1] Bott, R.: An application of the Morse theory to the topology of Lie-groups. Bull. Soc. Math. France 84 (1956), 251–281.

[2] Bourbaki, N.: Groupes et algèbres de Lie, Ch. 4–6, Hermann, Paris, 1968.

[3] Chevalley, C.: Invariants of the finite groups generated by reflections. Amer. J. Math. 77 (1955), 778–782.

[4] Humbert, G.: Sur la méthode d’approximation d’Hermite. Journal de Maths. 2 (1916), 79–103.

[5] Humphreys, J. E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge 1990.

[6] Iwahori, N.; Matsumoto, H.: On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Inst. Hautes Etudes Sci. Publ. Math. 25 (1965), 5–48.

[7] Meignen, P.: Fractions continues cliffordiennes. C. R. Acad. Sci. Paris, 322 (1995), 207–210.

[8] Meignen, P.: Orthochronous Lorentz Group from the Cliffordian point of view. Advances in Applied Clifford Algebras 6 (1996), to appear.

[9] Vinberg, E. B.: Geometry II. Encyclopedia of Maths. Sciences, 29, Springer-Verlag, Berlin Heidelberg, 1993.

Received November 8, 1996