ROOT SYSTEMS AND LENGTH FUNCTIONS

Kirsten Bremke and Gunter Malle

Universität Heidelberg

Abstract. We introduce root systems for those imprimitive complex reflection groups which are generated by involutory reflections, and study the associated length functions. These have many properties in common with the usual length functions for finite Weyl groups.

0. Introduction

In [1] we introduced a root system and a length function for the complex reflection groups W_{n} := G(e, 1, n). This note is motivated by the attempt to generalize this construction to the (imprimitive) complex reflection groups of type W′_{n} := G(e, e, n). These groups can be thought of as being generalizations of the Weyl groups of type D_{n}. They are best studied as reflection subgroups of the wreath products W_{n} = Z_{e} ≀ 𝔖_{n}. Thus we first define a root system for W_{n} (which is different from the one investigated in [1]) and then study the properties of the root system and length function for W′_{n} obtained by restriction. It turns out that this also gives rise to a length function on W_{n}^{∘} := G(e, e/2, n) if e is even.

The general setup is the following. We let Φ be a set on which W_{n} acts, which is partitioned into two subsets Φ = Φ^{+} ∪ Φ^{−}. In our case Φ is a set of eigenvectors for reflections of W_{n} in its reflection representation. Further inside Φ^{+} we choose a subset Ω which is in bijection with the set of reflections of W_{n}. Then we call (Φ, Φ^{−}, Ω) a root system for W_{n}. The associated length function ℓ : W_{n} → ℕ_{0} is defined by ℓ(w) := |w.Ω ∩ Φ^{−}|. The main result can be formulated as follows:

Theorem. There exist a root system (Φ, Φ^{−}, Ω) for W_{n} with associated length function ℓ with the following properties:

(a) ℓ(s) = 1 for all s in a certain set S of standard generators of W_{n}.

(b) The restriction of ℓ to the standard parabolic subgroups W_{J} = ⟨s ∈ J⟩ (for J ⊂ S) agrees with the corresponding length functions defined there.

(c) The polynomial ∑_{w∈W_{n}} q^{ℓ(w)} agrees with the Poincaré polynomial of W_{n}.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 51F15, 20H15.

Key words and phrases. complex reflection groups, root system, length function.

The second author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

(d) In the case e = 2, when W_{n} is the real reflection group of type B_{n}, ℓ agrees with the usual length function on this Coxeter group.

(e) The set Φ consists of eigenvectors of reflections of W_{n} in its reflection representation.

(f) The negative roots are precisely those lying on one side of a certain 2n − 1dimensional real hyperplane in the underlying complex vector space.

(g) The ‘short roots’ in the root system Φ define a root system for the reflection subgroup W′_{n} = G(e, e, n) of W_{n}, with corresponding length function ℓ_{2}, which satisfies the analogues of (a), (b) and (c) above.

(h) In the cases e = 2 or n = 2, when W′_{n} is a real reflection group, ℓ_{2} agrees with the usual length function on these Coxeter groups

(i) If e is even Φ contains a subroot system giving rise to a length function ℓ^{∘} on the reflection subgroup W_{n}^{∘} := G(e, e/2, n) of W_{n}, which satisfies the analogue of (c) above.

1. The Root System

In this section we introduce a root system for W = G(e, 1, n) which consists of eigenvectors of reflections in W and derive some properties of the associated length function ℓ.

Here a root system for a complex reflection group W is a triple (Φ, Φ^{−}, Ω), where Φ is a set with W -action, Φ^{−} is a subset of Φ of cardinality |Φ^{−}| = |Φ|/2, and Ω is a subset of Φ \ Φ^{−} in bijection with the set of reflections of W. The length function ℓ associated with the root system (Φ, Φ^{−}, Ω) is the function W → ℕ_{0} defined by ℓ(w) := |w.Ω ∩ Φ^{−}|.

1A. Definition.

Let V be the unitary space ℂ^{n} with the standard unitary inner product. Denote the standard basis vectors of V by e_{1}, …, e_{n}.

Fix e > 1, and let W_{n} = G(e, 1, n) ⊂ GL(V) be the imprimitive complex reflection group generated by the reflections s_{2}, …, s_{n} of order 2 with roots e_{2} − e_{1}, …, e_{n} − e_{n−1}, respectively, and a (complex) reflection t of order e with root e_{1}. On these generators, W_{n} has a presentation given by the diagram

(1.1)  B_{n}^{(e)} :  ◯_{t}_{e}  ◯_{s_{2}}  ◯_{s_{3}} ⋯ ◯_{s_{n}}

Note that the elements s_{2}, …, s_{n} generate a subgroup of W_{n} isomorphic to the symmetric group 𝔖_{n}, and can be identified with a set of standard generators of this Coxeter group. We write S := {t, s_{2}, …, s_{n}} for the set of fundamental reflections of W_{n}.

We set

δ := \{ \begin{matrix}1 & if e is even, \\ 2 & if e is odd, \end{matrix}

and let ζ ∈ ℂ be a primitive δe-th root of unity such that ζ^{−δ} is the non-trivial eigenvalue of t. We define a a set of roots Φ for W_{n} to be

Φ := {ζ^{i}e_{j} −ζ^{k}e_{l} | 0 ≤ i, k < δe; 1 ≤ l ≤ j ≤ n; i+k ≡ 0 (mod δ); (i, j) ≠ (k, l)} ⊂ V.

Note that Φ is stabilized by the action of W_{n} on V and each φ ∈ Φ is an eigenvector for some reflection of W_{n}. We subdivide Φ into the set of short, respectively long roots

Φ_{1} := {ζ^{i}e_{j} − ζ^{k}e_{j} ∈ Φ}, Φ_{2} := Φ \ Φ_{1} = {ζ^{i}e_{j} − ζ^{k}e_{l} ∈ Φ | j > l}.

Clearly both Φ_{1}, Φ_{2} are stabilized individually by W_{n}. The roots in Φ_{2} belong to the involutory reflections R′ of W_{n}, thus any such root is an eigenvector for precisely one reflection. We write r : Φ_{2} → R′ for the corresponding map associating to φ ∈ Φ_{2} the reflection r(φ) ∈ R′ with non-trivial eigenvector φ. We let

Φ_{1}^{−} := {ζ^{i}e_{j} − ζ^{δk+i}e_{j} ∈ Φ_{1} |0 < i < δe/2 and δe − 2i ≤ δk < δe, or
δe/2 ≤ i < δe and 0 < δk < 2δe − 2i},
Φ_{2}^{−} := {ζ^{i}e_{j} − ζ^{δk+i}e_{l} |δe/2 ≤ i < δe; 0 ≤ k < e; j > l}

be the negative long and short roots, Φ^{−} := Φ_{1}^{−} ∪ Φ_{2}^{−} and Φ_{i}^{+} := Φ_{i} \ Φ_{i}^{−}. Further we define

\begin{matrix}Ω_{1} := & \{e_{j} − ζ^{δk}e_{j} ∈ Φ_{1}\}, \\ Ω_{2} := & \{ζ^{i}e_{j} − ζ^{−i−ϵ}e_{l} | 0 ≤ i < δe/2; ϵ ∈ \{0, 2 − δ\}; j > l\}, \end{matrix}

and Ω := Ω_{1} ∪ Ω_{2}. It is easily seen that Ω_{2} is a section for r, i.e., it contains precisely one eigenvector for each reflection s ∈ R′. We will then also write φ = φ_{s} for φ ∈ Ω_{2} with r(φ) = s. In particular we have

(1.2)  φ_{s_{1}} = ζ^{δe/2−1}e_{2} − ζ^{−δe/2−1+δ}e_{1}, φ_{s_{j}} = e_{j} − e_{j−1} for 2 ≤ j ≤ n,

where s_{1} := t^{−1}s_{2}t. As indicated in the introduction this setup allows to define a length function ℓ on W_{n}:

ℓ : G(e, 1, n) → ℕ_{0}, ℓ(w) := |w.Ω ∩ Φ^{−}| .

Further, we set

ℓ_{2} : G(e, 1, n) → ℕ_{0}, ℓ_{2}(w) := |w.Ω_{2} ∩ Φ_{2}^{−}|.

Remark 1.3. For the real reflection groups G(2, 1, n) the function ℓ agrees with the usual length function on those Coxeter groups.

Proof. Suppose e = 2 so that W_{n} is isomorphic to the Weyl group of type C_{n}. Then

Φ_{1} = {±2e_{j} | 1 ≤ j ≤ n}, Φ_{2} = {±e_{j} ± e_{l} | 1 ≤ l < j ≤ n},

and

Ω_{1} = {2e_{j} | 1 ≤ j ≤ n}, Ω_{2} = {e_{j} ± e_{l} | l < j},

up to a scalar multiple coincide with the sets of long and short roots, respectively long and short positive roots, for C_{n} (compare [3, 3.6(ii)]) and hence ℓ is the usual length function on C_{n}. □

Remark 1.4. The subdivision of Φ into positive and negative roots can be described in an alternative way. Choose t, ζ so that ζ = exp(\frac{2πi}{δe}) is the δe-th root of unity with minimal positive argument. Fix e. Then for ε > 0 sufficiently small we have

(1.5)  Φ_{2}^{−} = Φ_{2} ∩ {(z_{1}, …, z_{n}) ∈ ℂ^{n} | ∑_{j=2}^{n} ε^{2(n−j)}(ℑ(z_{j}) + εℜ(z_{j})) < 0} ,

i.e., Φ_{2}^{−} consists of those roots in Φ lying on a fixed side of a certain real hyperplane in ℂ^{n} considered as a 2n-dimensional real space. Indeed, if φ = ζ^{i}e_{j} − ζ^{δk+i}e_{l} ∈ Φ_{2}^{−}, where δe/2 ≤ i < δe and j > l, then either ℑ(ζ^{i}) < 0, and then the above sum is negative for small enough ε > 0 since j > l, or ℑ(ζ^{i}) = 0, ℜ(ζ^{i}) = −1, and again the sum is negative since j > l.

Similarly we have

Φ_{1}^{−} = {ze_{j} ∈ Φ_{1} | ℜ(z) − εℑ(z) < 0}.

Since the length function ℓ does not change when every root in Φ_{1} is multiplied by the same non-zero complex number and the definition of Φ_{1}^{−}, Ω_{1} is adapted accordingly, we could replace Φ_{1} by iΦ_{1} to obtain a set Φ^{−} with the following defining property: the roots in Φ^{−} are precisely those roots in Φ lying on a fixed side of the hyperplane described by (1.5).

1B. Restriction to standard parabolic subgroups.

For future use let us first record those positive roots made negative by the generating reflections t, s_{2}, …, s_{n}. For t the definitions show that t.Φ_{2}^{+} = Φ_{2}^{+},

(1.6) Φ_{1}^{+} ∩ t^{−1}.Φ_{1}^{−} = {ζ^{i}e_{1} − ζ^{δ−i}e_{1} | δe/2 < i ≤ δe} ∪ {ζ^{i}e_{1} − ζ^{−i}e_{1} | δe/2 < i < δe}

while for s_{j} with 2 ≤ j ≤ n we have

(1.7)  s_{j}.Φ_{1}^{+} = Φ_{1}^{+}, Φ_{2}^{+} ∩ s_{j}.Φ_{2}^{−} = {ζ^{i}e_{j} − ζ^{δk+i}e_{j−1} | 0 ≤ i, δk + i < δe/2}.

Since all these sets intersect Ω in a single element, this shows in particular that all generators s ∈ S = {t, s_{2}, …, s_{n}} have length ℓ(s) = 1. For the powers of t one finds ℓ(t^{−1}) = 2, ℓ(t^{2}) = 3, ℓ(t^{−2}) = 4, ...

Proposition 1.8. Let s ∈ {s_{2}, …, s_{n}}.

(a) For w ∈ W_{n} we have

ℓ(ws) = \{ \begin{matrix}ℓ(w) + 1 & if w.φ_{s} ∈ Φ^{+}, \\ ℓ(w) − 1 & if w.φ_{s} ∈ Φ^{−}, \end{matrix}

(b) For w ∈ W_{n}, we have

ℓ_{2}(ws) = \{ \begin{matrix}ℓ_{2}(w) + 1 & if w.φ_{s} ∈ Φ_{2}^{+}, \\ ℓ_{2}(w) − 1 & if w.φ_{s} ∈ Φ_{2}^{−}, \end{matrix}

Proof. A reflection s = s_{j} with 2 ≤ j ≤ n stabilizes Ω_{1} as well as Ω_{2} \ {φ_{s}} and sends φ_{s} to −φ_{s}. Thus

|ws.Ω ∩ Φ^{−}| = |(ws.(Ω \ {φ_{s}}) ∪ ws.φ_{s}) ∩ Φ^{−}| = |(w.(Ω \ {φ_{s}}) ∪ w. − φ_{s}) ∩ Φ^{−}|

and the result follows. □

We can now show the compatibility with restriction to parabolic subgroups. For a subset J ⊂ S we let W_{J} be the group generated by the elements of J. This will be a product of complex reflection groups of types G(e, 1, m) and 𝔖_{m} for suitable m. In particular, there is a well-defined length function ℓ_{J} on W_{J}.

Proposition 1.9. Let J ⊆ S. Then we have ℓ_{J}(w) = ℓ(w) for all w ∈ W_{J}.

Proof. First assume that t ∉ J. Then W_{J} is a direct product of symmetric groups and Ω_{J} = {e_{j} − e_{l} | s_{k} ∈ J for all l < k ≤ j}. It follows from our definitions that any s ∈ J stabilizes Ω \ Ω_{J} and also Ω_{J} ∪ −Ω_{J} = Ω_{J} ∪ Φ_{J}^{−}. Thus

ℓ(w) = |w.Ω ∩ Φ^{−}| = |w.Ω_{J} ∩ −Ω_{J}| = |w.Ω_{J} ∩ Φ_{J}^{−}|

which is the usual length function on the product of symmetric groups W_{J}, since Ω_{J} and Φ_{J}^{−} consist precisely of the positive respectively negative roots for this Coxeter group.

Now assume that t ∈ J. Write W_{J} = W_{1} × W_{2} with W_{1} ≅ G(e, 1, m) for some m and W_{2} a product of symmetric groups, and let J = J_{1} ∪^{·} J_{2} be the corresponding decomposition of J. It is clear from (1.6) and (1.7) that if s ∈ J_{1}, φ ∈ Φ^{+} with s.φ ∈ Φ^{−} then φ ∈ Φ_{J_{1}}^{+}. Thus W_{1} stabilizes Φ^{+} \ Φ_{J_{1}}^{+} and we have

ℓ(w) = |w.Ω ∩ Φ^{−}| = |w.Ω_{J_{1}} ∩ Φ_{J_{1}}^{−}| = ℓ_{J_{1}}(w) for w ∈ W_{1}.

Now if w = w_{1}w_{2} ∈ W_{J} is the decomposition of a general element of W_{J} we know by the previous case that w_{2} stabilizes Ω \ Ω_{J_{2}}. Moreover it even fixes Ω_{J_{1}} pointwise. Further w_{1} stabilizes Φ^{+} \ Φ_{J_{1}}^{+} and fixes −Ω_{J_{2}} pointwise. Thus the length ℓ(w) can be computed as the sum of the lengths ℓ_{J_{1}}(w_{1}) and ℓ_{J_{2}}(w_{2}), as required. □

1C. The Poincaré polynomial.

In order to be able to determine the Poincaré polynomial of W_{n} with respect to ℓ we first study a particular type of element. For 0 < p ≤ e, 1 ≤ m ≤ n we define

w(p, m) := \{ \begin{matrix}s_{m} ⋯ s_{2}t^{p} & if 0 < p ≤ e/2, \\ s_{m} ⋯ s_{2}t^{p}s_{2} ⋯ s_{m} & if e/2 < p ≤ e. \end{matrix}

Lemma 1.10. We have (a)

ℓ_{2}(w(p, m)) = \{ \begin{matrix}(m − 1)(2p − 1) & for 0 < p ≤ e/2, \\ (m − 1)(2e − 2p) & for e/2 < p ≤ e, \end{matrix}

(b)

ℓ(w(p, m)) = \{ \begin{matrix}m(2p − 1) & for 0 < p ≤ e/2, \\ m(2e − 2p) & for e/2 < p ≤ e, \end{matrix}

(c) ℓ_{2}(w(p_{1}, 1) ⋯ w(p_{n}, n)) = ∑_{i=1}^{n} ℓ_{2}(w(p_{i}, i)),

(d) ℓ(w(p_{1}, 1) ⋯ w(p_{n}, n)) = ∑_{i=1}^{n} ℓ(w(p_{i}, i)).

Proof. From the definitions we see that

(1.11)  w(p, m).e_{k} = \{ \begin{matrix}ζ^{−δp}e_{k} & for e/2 < p ≤ e, k = m, \\ ζ^{−δp}e_{m} & for 0 < p ≤ e/2, k = 1, \\ e_{k−1} & for 0 < p ≤ e/2, 1 < k ≤ m, \\ e_{k} & else. \end{matrix}

In particular, if we denote by Φ′, Ω′ the root system for W_{n−1}, then

(1.12)  w(p, n).Ω_{2} = \{ \begin{matrix}Ω′_{2} ∪ \{ζ^{i}e_{j} − ζ^{−i−ϵ−δp}e_{n}\} & for 0 < p ≤ e/2, \\ Ω′_{2} ∪ \{ζ^{i−δp}e_{n} − ζ^{−i−ϵ}e_{l}\} & for e/2 < p ≤ e. \end{matrix}

Hence the negative roots in w(p, m).Ω_{2} are given by

\begin{matrix}\{ζ^{i}e_{j} − ζ^{−i−ϵ−δp}e_{m} | δe/2 − δp − ϵ < i < δe; j < m\}  for 0 < p ≤ e/2, \\ \{ζ^{i−δp}e_{m} − ζ^{−i−ϵ}e_{l} | δp − δe/2 ≤ i < δe/2; l < m\}  for e/2 < p ≤ e, \end{matrix}

and the length ℓ_{2}(w(p, m)) is as stated in (a). Further, it is immediate from (1.12) that

w′.(w(p, n).Ω_{2} ∩ Φ_{2}^{−}) ⊆ Φ_{2}^{−} for all w′ ∈ W_{n−1},

hence we obtain (c) by induction using Proposition 1.9.

Again from (1.11) we see that

w(p, n).Ω_{1} = Ω′_{1} ∪ {ζ^{−δp}ze_{n} | ze_{n} ∈ Ω_{1}}

so that

w′.(w(p, n).Ω_{1} ∩ Φ_{1}^{−}) ⊆ Φ_{1}^{−} for all w′ ∈ W_{n−1},

and (b), (d) follow as above. □

Proposition 1.13. We have

∑_{w∈W_{n}} q^{ℓ(w)} = P (W_{n})

with the Poincaré polynomial P (W_{n}) of W_{n}.

Proof. Let J := {s_{2}, …, s_{n}}. We claim that

(1.14)  N := {w ∈ W_{n} | w.Ω_{J} ⊆ Φ^{+}}

is a system of left coset representatives of W_{J} ≅ 𝔖_{n} in W_{n}. Indeed, let w_{1}, w_{2} ∈ N with w_{1} = w_{2}w′, where w′ ∈ W_{J}, lie in the same coset. Then by definition w_{2}w′.Ω_{J} ⊆ Φ^{+}. If w′ ≠ 1 then by what we saw in the proof of Proposition 1.9 there exist φ ∈ Ω_{J} with w′.φ ∈ −Ω_{J}. But this is a contradiction to w_{2}.Ω_{J} ⊆ Φ^{+}. Hence different elements of N lie in different cosets.

By (1.11) we have

w(p, n).(e_{k} − e_{k−1}) = \{ \begin{matrix}e_{k−1} − e_{k−2} & for 0 < p ≤ e/2, k > 2, \\ ζ^{δe/2−δp}e_{n} − ζ^{δe/2}e_{1} & for 0 < p ≤ e/2, k = 2, \\ ζ^{−δp}e_{n} − e_{n−1} & for e/2 < p ≤ e, k = n, \\ e_{k} − e_{k−1} & else, \end{matrix}

so it follows by induction that

(1.15)  N_{1} := {w(p_{1}, 1) ⋯ w(p_{n}, n) | 0 < p_{i} ≤ e}

is a subset of N. Since |N_{1}| = e^{n} = (W_{n} : W_{J}) this shows that N_{1} = N is a system of left coset representatives.

By Lemma 1.10(b) and (d) the Poincaré polynomial of N with respect to ℓ is equal to

∑_{w∈N} q^{ℓ(w)} = ∏_{i=1}^{n} \frac{q^{ei} − 1}{q^{i} − 1}.

The Poincaré polynomial of W_{J} with respect to ℓ is equal to ∏_{i=2}^{n}(q^{i} − 1)/(q − 1) by Proposition 1.9. By Proposition 1.8(a) and using induction we see that for w ∈ N, w′ ∈ W_{J} we have ℓ(ww′) = ℓ(w) + ℓ(w′). Hence

∑_{w∈W_{n}} q^{ℓ(w)} = ∑_{w∈N} q^{ℓ(w)} ∑_{w′∈W_{J}} q^{ℓ(w′)} = ∏_{i=1}^{n} \frac{q^{ei} − 1}{q^{i} − 1} · ∏_{i=2}^{n} \frac{q^{i} − 1}{q − 1} = P (W_{n})

as required. □

In the proof of the preceding result we also obtained the following normal form for elements of W_{n}:

Corollary 1.16. Every element w ∈ W_{n} can be written in a unique way as w = w(p_{1}, 1) ⋯ w(p_{n}, n)w′ with w′ ∈ ⟨s_{2}, …, s_{n}⟩, 1 ≤ p_{i} ≤ e, and both ℓ and ℓ_{2} are additive on this decomposition.

1D. Further properties of ℓ.

Proposition 1.17. (a) For all 1 ≠ w ∈ W_{n} there exists s ∈ S ∪ S^{−1} with ℓ(sw) < ℓ(w).

(b) For all w ∈ W_{n} with w ≠ w(p_{2}, 2) ⋯ w(p_{n}, n) where e/2 < p_{i} ≤ e there exists s ∈ S ∪ S^{−1} with ℓ(ws) < ℓ(w).

Proof. Write w = w(p_{1}, 1) ⋯ w(p_{n}, n)w′ as in Corollary 1.16. In (a), if p_{1} ≠ e then either ℓ(tw) < ℓ(w) or ℓ(t^{−1}w) < ℓ(w) by Lemma 1.10(b) and (d). On the other hand, if w = w′ then (a) is true by the corresponding statement for the Coxeter group W_{J}. In the remaining cases let m > 1 be the smallest index such that p_{m} < e. If 0 < p_{m} ≤ e/2 then s_{m}w(p_{m}, m) = w(p_{m}, m − 1) and thus ℓ(s_{m}w) < ℓ(w). If e/2 < p_{m} < e then s_{m}w(p_{m}, m) = w(p_{m}, m − 1)s_{m}. But s_{k}w(p_{l}, l) ∈ {w(p_{l}, l)s_{k}, w(p_{l}, l)s_{k+1}} for k < l, and again we obtain ℓ(s_{m}w) < ℓ(w). This proves (a).

For (b) clearly the result is true if w′ ≠ 1. If w = w(p_{1}, 1) ⋯ w(p_{n}, n) and there exists m with 0 < p_{m} ≤ e/2 then ℓ(wt^{−1}) is smaller than ℓ(w). □

Note that the excluded cases in part (b) are in fact exceptions. For example, let w = s_{2}t^{−1}s_{2} ∈ W_{2} for e > 2. Then ℓ(w) = 4 but ℓ(wt) = ℓ(ws_{2}) = 5, ℓ(wt^{−1}) = 6.

We next compare ℓ to the length function l̃ on W_{n} defined as follows: for w ∈ W_{n}, l̃(w) is the minimum over all expressions w = r_{1}^{a_{1}} ⋯ r_{k}^{a_{k}} of ∑_{j=1}^{k} a_{j}, where r_{j} ∈ S ∪ S^{−1} and a_{j} ≥ 0. Hence this is the standard length function l with respect to the set of generators S ∪ {t^{−1}}. In particular we have l̃(w) = l(w) = ℓ(w) for all w ∈ ⟨s_{2}, …, s_{n}⟩ by Lemma 1.5 in [1] and the subsequent remark.

Proposition 1.18. For all w ∈ W_{n} we have l̃(w) ≤ ℓ(w). Equality occurs precisely for w of the form

w = w(p_{1}, 1) ⋯ w(p_{n}, n)w′ with p_{i} ∈ {1, e} for 1 ≤ i ≤ n.

Proof. By Corollary 1.16 and the definition of l̃ it is sufficient to prove this for w = w(p_{i}, i). In this case it follows from Lemma 1.10(b) that l̃(w(p_{i}, i)) < ℓ(w(p_{i}, i)) unless p_{i} ∈ {1, e}. But in the latter cases, equality holds by Lemma 1.5 in [1] and the following remark. □

Proposition 1.19. Let e be even. Then the longest element w_{0} of W_{n} (with respect to ℓ) satisfies ℓ(w_{0}w) = ℓ(ww_{0}) = ℓ(w_{0}) − ℓ(w) for all w ∈ W_{n}.

Proof. From the explicit matrices for the generators we see that

w_{0} := t^{e/2}(s_{2}t^{e/2}s_{2}) ⋯ (s_{n} ⋯ s_{2}t^{e/2}s_{2} ⋯ s_{n})

acts as −1 on V, hence ℓ(w_{0}) = | − Ω ∩ Φ^{−}| = |Ω|, so in fact w_{0} is the longest element (with respect to ℓ). Further we have

ℓ(w_{0}w) = |w_{0}w.Ω ∩ Φ^{−}| = |w.Ω ∩ w_{0}.Φ^{−}| = |w.Ω ∩ Φ^{+}| = |Ω| − ℓ(w) = ℓ(w_{0}) − ℓ(w)

since Φ^{+} = −Φ^{−}. □

As in the case of the length function studied in [1] we have a connection with basis elements in the cyclotomic algebra H(W_{n}, u) (see [2]):

Corollary 1.20. If x and x′ are reduced expressions for an element w ∈ W_{n} with ℓ(w) = l(w) then T_{x} = T_{x′} in H(W_{n}, u).

2. A Length Function on G(e, e, n)

Restriction to the roots belonging to the reflections lying in the reflection subgroup W′ := G(e, e, n) of W induces a length function on this complex reflection group.

2A. Restriction to short roots.

For e ≥ 2 let W′_{n} = G(e, e, n) be the reflection subgroup of W_{n} of index e generated by the reflections in S′ := {s_{1} := t^{−1}s_{2}t, s_{2}, …, s_{n}}. Thus by our definitions s_{1} acts as

( \begin{matrix}0 & ζ^{−δ} \\ ζ^{δ} & 0 \end{matrix} )

on ⟨e_{1}, e_{2}⟩ and fixes e_{3}, …, e_{n}. It is known (see [2]) that W′_{n} has a presentation with respect to these generators given by the diagram

(2.1)  I_{n}^{(e)} :  e^{^{s_{1}}◯}⧹_{_{s_{2}}◯}⧸◯_{s_{3}}  ◯_{s_{4}} ⋯ ◯_{s_{n}}

The set Φ_{2} only contains roots belonging to reflections in W′_{n}, so (Φ_{2}, Φ_{2}^{+}, Ω_{2}) defines a root system for W′_{n}. The associated length function is the restriction to W′_{n} of ℓ_{2}, which will again be denoted by ℓ_{2}.

Remark 2.2. For the real reflection groups G(2, 2, n) and G(e, e, 2) the function ℓ_{2} agrees with the usual length function on those Coxeter groups.

Proof. If e = 2 then

Φ_{2} = {±e_{j} ± e_{l} | 1 ≤ l < j ≤ n}, Ω_{2} = {e_{j} ± e_{l} | l < j},

up to a scalar multiple coincide with the sets of short, respectively short positive roots, for B_{n} and hence ℓ_{2} is the usual length function on D_{n} (compare [3, 3.6(iv)]).

For n = 2 the group W′_{2} is the complexification of the Coxeter group of type I_{2}(e) (the dihedral group of order 2e). Then

Ω_{2} := {ζ^{i}e_{2} − ζ^{−i−ϵ}e_{1} | 0 ≤ i < δe/2; ϵ ∈ {0, 2 − δ}},

and it is easily checked that ℓ_{2} coincides with the usual length function on this Coxeter group. □

Remark 2.3. By Remark 1.4 the positive roots Φ_{2}^{+} all lie on one side of a certain real hyperplane. But even more is true. The roots in Ω_{2} can be written as linear combinations of the simple roots {φ_{j} := φ_{s_{j}} | 1 ≤ j ≤ n} as follows:

ζ^{i}e_{j} − ζ^{−i−ϵ}e_{l} = ∑_{k=l+1}^{j} ζ^{i}φ_{k} + ∑_{k=3}^{l}(ζ^{i} − ζ^{−i−ϵ})φ_{k} + \frac{ζ^{i} − ζ^{−i−ϵ}}{1 − ζ^{−δ}} (φ_{2} + ζ^{1−δ}φ_{1}),

for j > l > 1,

ζ^{i}e_{j} − ζ^{−i−ϵ}e_{1} = ∑_{k=3}^{j} ζ^{i}φ_{k} + ζ^{1−δ} \frac{ζ^{δ+i−1} − ζ^{−i−1−ϵ}}{1 − ζ^{−δ}} φ_{2} + ζ^{1−δ} \frac{ζ^{i} − ζ^{−i−ϵ}}{1 − ζ^{−δ}} φ_{1}

for j > 1. Since 0 ≤ i < δe/2 all the coefficients occurring in the above decompositions lie in one half plane of the complex plane. If we call the complex numbers in this half plane positive, then in this sense the roots in Ω_{2} are positive linear combinations of the simple roots.

2B. Standard parabolic subgroups, Poincaré polynomial.

The positive roots made negative by s_{1} are given by

(2.4)  Φ_{2}^{+} ∩ s_{1}.Φ_{2}^{−} = {ζ^{i}e_{2} − ζ^{δk+i}e_{1} | 0 ≤ i, δk + i − δ < δe/2},

so it also follows that ℓ_{2}(s) = 1 for s ∈ S′ = {s_{1}, …, s_{n}}.

For a subset J ⊆ S′ of the set of generating reflections of W′_{n} we denote by W′_{J} = ⟨s ∈ J⟩ the corresponding parabolic subgroup. Then W′_{J} splits into a direct product W′_{J} = W′_{1} × ⋯ × W′_{k} with W′_{1} ≅ G(e, e, m) for some m ≥ 0, and W′_{2}, …, W′_{k} are symmetric groups, corresponding to the connected components of the full subdiagram of (2.1) formed by the nodes in J. Thus any element w ∈ W′_{J} can be written as a commuting product w = w_{1} ⋯ w_{k} of elements w_{i} ∈ W′_{i}. Each factor has a length function, so there is a length function ℓ_{J} on W′_{J} defined as the sum of the length functions of the factors w_{i} of an element w. We want to compare this length function ℓ_{J} on W′_{J} with ℓ_{2} on W′_{n}. For this let Φ_{J} := {φ ∈ Φ | r(φ) ∈ W′_{J}}, Φ_{J}^{−} := Φ^{−} ∩ Φ_{J} and Ω_{J} := Ω ∩ Φ_{J}.

Proposition 2.5. Let J ⊆ S′. Then we have ℓ_{J}(w) = ℓ_{2}(w) for all w ∈ W′_{J}.

Proof. If s_{1} ∉ J then W′_{J} is also a parabolic subgroup of W_{n} and the assertion follows from Proposition 1.9 since ℓ(w) = ℓ_{2}(w) for w ∈ W_{J}.

If {s_{1}, s_{2}} ⊆ J then W_{I} with I := {t} ∪ J \ {s_{1}} is a parabolic subgroup of W_{n} and the assertion again follows from the corresponding result (for ℓ_{2}) in Proposition 1.9. So finally assume that s_{1} ∈ J, s_{2} ∉ J. We write J = J_{1} ∪^{·} J_{2} with s_{1} ∈ J_{1}, so that J_{1} contains the nodes in the connected component of the full subdiagram for J containing s_{1}, and let W′_{J} = W′_{1} × W′_{2} be the corresponding direct product decomposition. Here W′_{2} is a direct product of symmetric groups. But W′_{1} is also isomorphic to the symmetric group on the set of standard generators J_{1}. Thus by a well known property of the Coxeter group 𝔖_{n} (see Lemma 1.2 in [1]) any w ∈ W′_{J} can be written w = s_{k}s_{k−1} ⋯ s_{3}s_{1}w′ with w′ ∈ ⟨s_{3}, …, s_{n}⟩. No root of Ω made negative by elements of the form s_{k} ⋯ s_{3}s_{1} lies inside Ω_{J′}. So by Proposition 1.8(b) and induction over the length of w′ we see that ℓ_{2}(w) = ℓ_{2}(s_{k}s_{k−1} ⋯ s_{3}s_{1}) + ℓ_{2}(w′), and the lengths on the right side agree with the lengths in the respective parabolic subgroups. This completes the proof. □

Proposition 2.6. We have

∑_{w∈W′_{n}} q^{ℓ_{2}(w)} = P (W′_{n}),

where

P (W′_{n}) = (1 + q + … + q^{n−1}) ∏_{i=1}^{n−1}(1 + q + … + q^{ie−1})

denotes the Poincaré polynomial of W′_{n}.

Proof. Let J := {s_{2}, …, s_{n}} and let N be defined as in the proof of Proposition 1.13. Then

N ∩ W′_{n} = {w(p_{1}, 1) ⋯ w(p_{n}, n) ∈ W′_{n} | 0 < p_{i} ≤ e; p_{1} + … + p_{n} ≡ 0 (mod e)} has cardinality e^{n−1}. By what we showed in the proof of Proposition 1.13

N′ := {w ∈ W′_{n} | w.Ω_{J} ⊆ Φ^{+}} = N ∩ W′_{n}

is a system of left coset representatives of W′_{J} in W′_{n}.

But since ℓ_{2}(w(p_{1}, 1)) = 0 for all 0 < p_{1} ≤ e we conclude from Lemma 1.10(a) and (c) that the Poincaré polynomial of N′ equals

∑_{w∈N′} q^{ℓ_{2}(w)} = ∏_{i=1}^{n−1} \frac{q^{ei} − 1}{q^{i} − 1}.

Hence by the same argument as in the proof of Proposition 1.13 using Proposition 1.8(b) we have

∑_{w∈W′_{n}} q^{ℓ_{2}(w)} = ∑_{w∈N′} q^{ℓ_{2}(w)} ∑_{w′∈W′_{J}} q^{ℓ_{2}(w′)} = ∏_{i=1}^{n−1} \frac{q^{ei} − 1}{q^{i} − 1} · ∏_{i=2}^{n} \frac{q^{i} − 1}{q − 1} = P (W′_{n}).  □

Corollary 2.7. We have l(w) ≡ ℓ_{2}(w) (mod 2) for all w ∈ W′_{n}.

Proof. Let κ : W_{n} → {±1}, t ↦ 1, s_{i} ↦ −1, be the trivial extension of the sign character of 𝔖_{n} to W_{n}. Then we have to prove: an element w ∈ W′_{n} lies in ker(κ) if and only if ℓ_{2}(w) is even.

By Corollary 1.16 it suffices to prove this for elements w(p_{i}, i) or w′ ∈ W_{J}, J := {s_{2}, …, s_{n}}. But by Lemma 1.10(a) the length ℓ_{2}(w(p_{i}, i)) is odd precisely if κ(w(p_{i}, i)) = −1, and the same is true for the elements w′ ∈ W_{J} by Proposition 2.5. □

According to Proposition 2.6 there exists a unique element w′_{0} ∈ W′_{n} of maximal ℓ_{2}-length. It can easily be checked that w′_{0} acts as −1 on V if both e and n are even. From the definition of ℓ we obtain:

Corollary 2.8. Assume that both e, n are even. Then the longest element w′_{0} ∈ W′_{n} with respect to ℓ_{2} satisfies

ℓ_{2}(w′_{0}w) = ℓ_{2}(w′_{0}) − ℓ_{2}(w) = ℓ_{2}(ww′_{0}) for all w ∈ W′_{n}.

2C. The reflection group G(e, e/2, n).

In this section let e be even and n ≥ 2. Then the root system (Φ, Φ^{−}, Ω) for W_{n} introduced above also gives rise to a length function on the imprimitive reflection subgroup W_{n}^{∘} := G(e, e/2, n) of W_{n} of index e/2. It can be defined as the extension of order 2 of W′_{n} = G(e, e, n) generated by the reflections S^{∘} := S′ ∪ {t^{e/2}}. It has a presentation on S^{∘} symbolized by the diagram

(2.9)  G(e, e/2, n) :  t^{e/2}◯_{e/2+1} ^{◯^{s_{1}}} ⧹_{◯_{s_{2}}} ⧸◯_{s_{3}}  ◯_{s_{4}} ⋯ ◯_{s_{n}}

(see [2]). Any reflection in W_{n}^{∘} has order 2. We define

Ω^{∘} := Ω_{2} ∪ {e_{j} − ζe_{j} | 1 ≤ j ≤ n}

and

ℓ^{∘} : W_{n} → ℕ_{0}, w ↦ ℓ^{∘}(w) := |w.Ω^{∘} ∩ Φ^{−}|,

and we also denote the restriction of ℓ^{∘} to W_{n}^{∘} by ℓ^{∘}. Any generating set of reflections for W_{n}^{∘} has cardinality at least n + 1, so there does not exist a length on W_{n}^{∘} whose generating function is the Poincaré polynomial of W_{n}^{∘} and which takes value 1 on a set of generating reflections. But ℓ^{∘} comes as close to this as possible: we have ℓ^{∘}(t^{e/2}) = ℓ^{∘}(s_{i}) = 1 for 2 ≤ i ≤ n, ℓ^{∘}(s_{1}) = 2 (for e > 2).

For this reason it will also not be true in general that the restriction of ℓ^{∘} to a standard parabolic subgroup agrees with the length function for this subgroup. But it follows easily from Propositions 1.9 and 2.5 that this is true if either s_{1} does not lie in the subgroup or otherwise all of t^{e/2}, s_{1}, s_{2} are contained in it.

In order to determine the generating function of ℓ^{∘} we first evaluate it on the standard elements w(p, m):

Lemma 2.10. We have

ℓ^{∘}(w(p, m)) = \{ \begin{matrix}(m − 1)(2p − 1) + 1 & for 0 < p ≤ e/2, \\ (m − 1)(2e − 2p) & for e/2 < p ≤ e. \end{matrix}

Proof. Taking into account the value of ℓ_{2} from Lemma 1.10(a) it suffices to show

|w(p, m).\{e_{j} − ζe_{j} | 1 ≤ j ≤ n\} ∩ Φ^{−}| = \{ \begin{matrix}1 & for 0 < p ≤ e/2, \\ 0 & for e/2 < p ≤ e, \end{matrix}

which is clear. □

Proposition 2.11. We have

∑_{w∈W_{n}^{∘}} q^{ℓ^{∘}(w)} = P (W_{n}^{∘}),

where

P (W_{n}^{∘}) = (1 + q + … + q^{2n−1}) ∏_{i=1}^{n−1}(1 + q + … + q^{ie−1})

denotes the Poincaré polynomial of W_{n}^{∘}.

Proof. Let J := {s_{2}, …, s_{n}} and

N^{∘} := {w ∈ W_{n}^{∘} | w.Ω_{J} ⊆ Φ^{+}} = N ∩ W_{n}^{∘}

(see (1.14)). From the explicit description of elements in N = N_{1} in (1.15) we have that

N^{∘} = {w(p_{1}, 1) ⋯ w(p_{n}, n) | p_{1} ≡ − ∑_{i=2}^{n} p_{i} (mod e/2)}

so by Lemma 2.10 the Poincaré polynomial of N^{∘} equals

∑_{w∈N^{∘}} q^{ℓ^{∘}(w)} = (1 + q) ∏_{i=2}^{n}(1 + q^{i})(1 + q^{2(i−1)} + … + q^{(e−2)(i−1)}) = (1 + q^{n}) ∏_{i=1}^{n−1} \frac{q^{ie} − 1}{q^{i} − 1}.

The result follows by application of Corollary 1.16 and the additivity of ℓ. □

The longest element w_{0} defined in Proposition 1.19 lies inside W_{n}^{∘}, so also is the longest element of this group (with respect to ℓ^{∘}). Hence as an immediate consequence we obtain:

Corollary 2.12. Let w_{0} be the longest element of W_{n}^{∘} (with respect to ℓ^{∘}). Then ℓ^{∘}(w_{0}w) = ℓ^{∘}(ww_{0}) = ℓ^{∘}(w_{0}) − ℓ^{∘}(w) for all w ∈ W_{n}^{∘}.

References

[1] K. Bremke and G. Malle, Reduced words and a length function for G(e, 1, n), Indag. Mathem. 8 (1997), 453–469.

[2] M. Broué, G. Malle, and R. Rouquier, On complex reflection groups and their associated braid groups, Representations of groups (B.N. Allison and G.H. Cliff, eds.), CMS Conference Proceedings, vol. 16, AMS, Providence, 1995, pp. 1–13.

[3] R. W. Carter, Simple groups of Lie type, John Wiley, London, 1972.

K.B.: Deutsche Bank, Credit Risk Management, Taunusanlage 12, D–60325 Frankfurt, Germany

   E-mail address: kirsten.bremke@zentrale.deuba.com,

G.M.: I W R, Im Neuenheimer Feld 368, D–69120 Heidelberg, Germany

   E-mail address: malle@urania.iwr.uni-heidelberg.de