ON THE RATIONALITY AND FAKE DEGREES

OF CHARACTERS OF CYCLOTOMIC ALGEBRAS

Gunter Malle

Universität Kassel

1. Introduction

Let W be a finite complex reflection group, and H = H(W, u) the corresponding generic (cyclotomic) Hecke algebra as introduced in [8] and [9]. In this paper we study the character fields of the irreducible characters of H. In the case that W is a Weyl group, hence a reflection group over the field of rational numbers, it is well known that all other complex irreducible representations of W can also be realized over ℚ. The corresponding situation for the associated Iwahori-Hecke algebras was investigated by Benson and Curtis [5] and Alvis and Lusztig [1]. They determined the character fields of all absolutely irreducible representations of Iwahori-Hecke algebras. It turned out that sometimes certain square roots of monomials in the parameters have to be adjoined to the ground field. Furthermore they showed that all absolutely irreducible representations can actually be realized over such an extension of the ground field [5,13].

For a complex reflection group W let k denote the character field of the reflection representation of W. It is a result of Benard [4] and Bessis [6] that again all absolutely irreducible complex representations of W can be realized over k and in particular have their character field contained in k. Here we determine the character fields of all generic cyclotomic Hecke algebras associated to complex reflection groups (under a certain assumption known to hold for all but finitely many irreducible types and conjectured to be always true). Partial results in this direction were already obtained in [8] for those complex reflection groups occurring as cyclotomic Weyl groups. Our results show that again the character fields and a splitting field for H can be obtained by adjoining roots of certain monomials in the parameters to the ground field (Corollary 4.8 and Theorem 5.2).

Our methods are similar to those employed in [5, 6, 8]. As in all of the above references, we use a case-by-case analysis, handling each isomorphism type and each irreducible character in turn. As a consequence of our classification we deduce that an irreducible complex reflection group W on ℂ^{n} can be generated by n reflections if and only if the reflection character of the associated 1-parameter cyclotomic algebra H(W, x) has values in k(x) (Corollary 4.9).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 16A64.

The fake degree of an irreducible character χ ∈ Irr(W) is the graded multiplicity of χ in the graded regular representation. It can be defined entirely in terms of W. In Section 6 we prove a remarkable connection between the rationality of characters of cyclotomic algebras and semi-palindromicity of fake degrees, which generalizes a result of Opdam [16] for the real case (Theorem 6.5).

In the last section we take a closer look at the subclass of n-dimensional irreducible reflection groups which can be generated by n of their reflections. These seem to behave nicer than the general reflection groups in several aspects. As an example we give a closed formula for the field of definition k of such a group (Theorem 7.1), and then sharpen some of the results obtained in the previous sections. Acknowledgement. I would like to thank Meinolf Geck for a helpful conversation on splitting fields and Eric Opdam for making available his preprint [17].

2. Some prerequisites

We recall the definition of cyclotomic Hecke algebras and some facts about character values.

2A. Cyclotomic Hecke algebras.

Let W be a finite irreducible complex reflection group and let k denote the character field of this reflection representation. It is known (see [4, 6]) that then k is a splitting field for W (see also Section 7A for a description of k in an important case). Let D be the diagram associated to W in [9]. This defines a presentation of W on a set of generators S subject to the order relations s^{d_{s}} = 1 for s ∈ S, together with certain homogeneous relations, the so-called braid relations. The braid group B = B(W) associated to W is by definition the group generated by a set {s | s ∈ S} in bijection with S, subject to the braid relations of D. Let u = (u_{s,j} | s ∈ S, 0 ≤ j ≤ d_{s} − 1) be transcendentals over ℤ, such that u_{s,j} = u_{t,j} whenever s and t are conjugate in W. The generic cyclotomic Hecke algebra H(W, u) of W with parameter set u is defined to be the quotient

H(W, u) := ℤ[u, u^{−1}]B/I, with I = ( ∏_{j=0}^{d_{s}−1}(s − u_{s,j}) | s ∈ S),

of the group algebra of B over A := ℤ[u, u^{−1}] by the ideal I generated by certain deformed order relations. Thus H := H(W, u) is a finitely generated algebra over A. We will write T_{w} for the image in H of an element w = s_{1} … s_{k} ∈ B. Any ring homomorphism f : ℤ[u, u^{−1}] → R endows R with an A-module structure, and we write

H_{R}(W, u) = H(W, u) ⊗_{A} R

for the corresponding specialization of H. Note that such a homomorphism is uniquely determined by the images f(u_{s,j}), s ∈ S, 0 ≤ j ≤ d_{s} − 1. Also note that under the specialization defined by

(2.1)  u_{s,j} ↦ exp(2πij/d_{s})  for s ∈ S, 0 ≤ j ≤ d_{s} − 1,

H maps to the group algebra of the complex reflection group W. Any specialization of H through which (2.1) factors will be called admissible. One particularly important example is the 1-parameter specialization H(W, x) of H(W, u) induced by the map

(2.2)  u_{s,j} ↦ \{ \begin{matrix}x & j = 0, \\ exp(2πij/d_{s}) & j > 0, \end{matrix}

where x is an indeterminate. This is the analogue of the 1-parameter Iwahori-Hecke algebra for real W.

Henceforth we will make the following assumption:

Assumption 2.3. The generic cyclotomic algebra H(W, u) defined above is free over A = ℤ[u, u^{−1}] of rank |W |.

This assumption is known to hold for all infinite families of irreducible reflection groups by [3,8,2], and for about half of the remaining 34 exceptional groups (see [8]). We conjecture it to be true in all cases (see [9, Sect. 4]).

Let L ≥ A be a splitting field of H. It follows from Assumption 2.3, the fact that k is a splitting field for W, and Tits’ deformation theorem that H_{L} is isomorphic to the group algebra LW. Thus any extension to L of the specialization (2.1) defines a bijection between Irr(H_{L}) and Irr(W).

2B. Character values.

Let W, k, H be as above. For s ∈ S, 0 ≤ j ≤ d_{s}−1, let v_{s,j} be such that v_{s,j}^{|W |} = u_{s,j}. Let L ≥ k(v) be a splitting field for H. We will see later on (Theorem 5.2) that we may take L = k(v). We extend the specialization (2.1) to the integral closure O of A in L, such that on ℤ[v, v^{−1}] we have

(2.4)  v_{s,j} ↦ exp(2πij/(d_{s}|W |))  for s ∈ S, 0 ≤ j ≤ d_{s} − 1 .

By Tits’ deformation theorem this defines a bijection

(2.5)  Irr(W) ⟶^{∼} Irr(H_{L}),  χ ↦ χ_{v} ,

which furthermore carries over to any admissible specialization of H_{O}.

In [9] we defined a certain central element β ∈ Z(B) of the braid group whose image β under the canonical epimorphism B → W generates Z(W). Since we assumed W to be irreducible, Z(W) is cyclic by Schur’s lemma. So π := β^{|Z(W)|} ∈ Z(B) is the smallest power of β which maps to 1 in W. Clearly, the images T_{β}, T_{π} in H are also central. Thus, they act as scalars in any (absolutely) irreducible representation of H_{L}.

Let S′ be a system of representatives of the generators in S up to conjugation in W. Write π = s_{1} … s_{l} in B. For s ∈ S′ let N_{s} = |{j | s_{j} ∼ s}| denote the number of factors in the decomposition of π conjugate to s. By evaluating linear characters of H it is easily seen that N_{s} does not depend on the chosen expression for π. For an irreducible character χ of W let m_{s,j}(χ) denote the multiplicity of the eigenvalue exp(2πij/d_{s}) of s in a representation affording χ. Springer [5] observed that it is possible to compute character values on central elements (of Iwahori-Hecke algebras) without actually constructing the representations; this was already used in [8] to deal with some of the complex reflection groups. Let z = |Z(W)|, so that β^{z} = π. Then the value of the irreducible character χ_{v} ∈ Irr(H_{L}) on T_{β} is given by

(2.6)  χ_{v}(T_{β}) = χ(β) ∏_{s∈S′} ∏_{j=0}^{d_{s}−1} (exp(−2πij/d_{s}|W |) v_{s,j})^{m_{s,j}N_{s}|W |/zχ(1)}.

Indeed, let det_{χ} denote the determinant on a representation affording χ_{v}. Then det_{χ}(T_{β})^{z} = det_{χ}(T_{π}) = ∏_{s∈S′} det_{χ}(T_{s}) = ∏_{s∈S′} ∏_{j=0}^{d_{s}−1} u_{s,j}^{m_{s,j}}. But T_{β} is central, so it acts by a scalar c. This determines c up to a root of unity, which can be computed by specializing to W via (2.4).

2C. Rationality and Schur index.

Let H_{L} be a cyclotomic algebra as above and χ ∈ Irr(H_{L}). The character field of χ is the field generated over k(u) by the values of χ on an A-basis of H_{L} as provided by Assumption 2.3. It is clear that this does not depend on the choice of basis.

Assume that L is Galois over K := Quot(A). Then Gal(L/K) acts on the set Irr(H_{L}) via its action on the character values. We will make use of the following obvious fact. Assume that χ, ψ ∈ Irr(H_{L}) have different multiplicity in a character of H_{L} whose values are invariant under a subgroup H ≤ Gal(L/K). Then χ, ψ do not lie in the same H-orbit. In particular, if there exists a set of K-rational characters of H_{L} such that χ is distinguished from all other irreducible characters by its multiplicities in these characters, then the character field of χ is equal to K.

The field of rationality of a representation of H_{L} will be determined by the following:

Lemma 2.7. Let H be a finite dimensional algebra over a field K_{0} and let B be a basis of H. Let K/K_{0} be an extension field such that H_{K} := H ⊗_{K_{0}} K is split semisimple. Let V be a simple H_{K}-module with character χ such that χ(b) ∈ K_{0} for all b ∈ B and assume that there exists an H-module V′ such that V has multiplicity 1 in V′ ⊗_{K_{0}} K. Then V is realizable over K_{0}.

This follows from standard properties of the Schur index (see for example [10, §74]).

3. Irreducible characters of imprimitive groups

In this section we recall the definition of the imprimitive complex reflection groups and the construction of their complex irreducible characters (see for example [14, Sect. 2A and 5A] or [6, Sect. 1]). From this we deduce some elementary results on multiplicities in induced characters. They are well-known in the case of the real imprimitive reflection groups W (B_{n}) and W (D_{n}).

3A. The groups W_{n}.

Throughout this section we fix an integer m ≥ 2 and let ζ_{m} := exp(2πi/m). For any n ≥ 1 let W_{n} := G(m, 1, n) be the complex linear group on ⊕_{j=1}^{n}ℂe_{j} consisting of all monomial matrices whose non-zero entries lie in {ζ_{m}^{j} | 0 ≤ j ≤ m−1}. Then W_{n} is the semidirect product of its subgroup of diagonal matrices with the subgroup of permutation matrices, that is, W_{n} = Z_{m}^{n}.𝔖_{n} = Z_{m} ≀ 𝔖_{n}. In its above representation W_{n} is generated by the complex reflection t which sends e_{1} to ζ_{m}e_{1} and fixes e_{2}, …, e_{n}, and the permutation matrices s_{l}, 1 ≤ l ≤ n − 1, corresponding to the transpositions (l, l + 1). In particular, W_{n} is an irreducible imprimitive complex reflection group. The generators {t, s_{1}, …, s_{n−1}} together with the relations implied by the diagram

(3.1)  B_{n}^{(m)} :  ◯^{m}  ◯  ◯ … ◯  ◯

yield a presentation for W_{n}. The elements {t, s_{1}, …, s_{n−2}} generate a natural subgroup W_{n−1}, the elements {s_{1}, …, s_{n−1}} a natural subgroup 𝔖_{n}.

We define the following linear character of W_{n}:

γ : W_{n} → ℂ, t ↦ ζ_{m}, s_{l} ↦ 1 for 1 ≤ l ≤ n − 1,

For any m-tuple of partitions α = (α_{0}, α_{1}, …, α_{m−1})⊢_{m}n of n we denote by W_{α} the natural subgroup W_{n_{0}} × … × W_{n_{m−1}} of W_{n}, where α_{j} ⊢ n_{j}, corresponding to the Young subgroup 𝔖_{n_{0}} ×…×𝔖_{n_{m−1}} of 𝔖_{n}. Via the natural projection W_{n_{j}} → 𝔖_{n_{j}} the characters of 𝔖_{n_{j}} may be regarded as characters of W_{n_{j}}. The irreducible characters of 𝔖_{n_{j}} are indexed by partitions α_{j} ⊢ n_{j}. For any multi-partition α⊢_{m}n as above we can thus define a character χ_{α} of W_{n} as the induction of the exterior product

(3.2)  χ_{α} := Ind_{W_{α}}^{W_{n}} (χ_{α_{0}}#(χ_{α_{1}} ⊗ γ)# … #(χ_{α_{m−1}} ⊗ γ^{m−1})) .

It turns out that χ_{α} is irreducible, χ_{α} ≠ χ_{β} if α ≠ β, and that all irreducible characters of W_{n} arise in this way, so Irr(W_{n}) = {χ_{α} | α = (α_{0}, …, α_{m−1})⊢_{m}n}.

Lemma 3.3. (a) Let α = (α_{0}, …, α_{m−1})⊢_{m}n − 1. Then

Ind_{W_{n−1}}^{W_{n}}(χ_{α}) = ∑ χ_{β}

where the sum is over all partitions β = (β_{0}, …, β_{m−1})⊢_{m}n which can be obtained by adding a single 1-hook to α.

(b) For n ≥ 3 the characters of W_{n} are distinguished by their restrictions to W_{n−1}. The characters of W_{2} are distinguished by their restrictions to W_{1} and 𝔖_{2}.

Proof. By construction we have χ_{α} = Ind_{W_{α}}^{W_{n−1}}(ψ) where ψ = (ψ_{0}#ψ_{1}# … #ψ_{m−1}) with ψ_{j} = χ_{α_{j}} ⊗ γ^{j}, thus Ind_{W_{n−1}}^{W_{n}}(χ_{α}) = Ind_{W_{α}}^{W_{n}}(ψ). Now

Ind_{W_{α}}^{W_{α}×W_{1}}(ψ) = ∑_{j=0}^{m−1} ψ#γ^{j}.

It is well-known that

Ind_{𝔖_{n_{j}}}^{𝔖_{n_{j} +1}}(χ_{α_{j}}) = ∑ χ_{β_{j}}

where the sum runs over the partitions β_{j} ⊢ n_{j} + 1 which can be obtained from α_{j} by adding a single 1-hook. We thus obtain that the induced of

ψ#γ^{j} = ψ_{0}# … #(ψ_{j}#γ^{j})# … #ψ_{m−1}

from W_{α} × W_{1} = W_{n_{0}} × … × (W_{n_{j}} × W_{1}) × … × W_{n_{m−1}} to W_{n_{0}} × … × W_{n_{j}+1} × … × W_{n_{m−1}} decomposes into the sum of all ψ_{0}# … #(χ_{β_{j}} ⊗ γ^{j})# … #ψ_{m−1} with β_{j} as before. Finally, the induced of

ψ_{0}# … #(χ_{β_{j}} ⊗ γ^{j})# … #ψ_{m−1}

to W_{n} is by definition the irreducible character χ_{β} with β = (α_{0}, …, β_{j}, …, α_{m−1}). This completes the proof of (a).

In (b), the assertion for n = 2 is easily verified. So now let n ≥ 3 and χ_{α}, χ_{β} ∈ Irr(W_{n}) with α ≠ β. If the distance d := ∑_{j=0}^{m−1} ||α_{j}|−|β_{j}|| of α and β is larger than 2, then there exists a multi-partition ν ⊢_{m} n − 1 such that α but not β can be obtained by adding a single 1-hook to ν, so by (a) the two characters χ_{α}, χ_{β} do not have equal multiplicities in the induced of all irreducible characters of W_{n−1}. Now assume that |α_{j}| and |β_{j}| only differ for j = j_{1}, j_{2}, say |α_{j_{1}}| = |β_{j_{1}}| − 1 and |α_{j_{2}}| = |β_{j_{2}}| + 1. If |α_{j_{1}}| ≠ 0 there exists a multi-partition ν⊢_{m}n − 1 such that α, but not β, is obtained from ν by adding a 1-hook (and similarly if |β_{j_{2}}| ≠ 0). If |α_{j_{1}}| = |β_{j_{2}}| = 0 then since n > 1 there exists j_{3} ≠ j_{1}, j_{2} with |α_{j_{3}}| = |β_{j_{3}}| > 0, and by diminishing this part we again find a partition ν⊢_{m}n − 1 as above. Thus, if the multiplicities of χ_{α}, χ_{β} are the same in the induced of each irreducible character of W_{n−1}, then |α_{j}| = |β_{j}| for all j. Now choose j such that α_{j} ≠ β_{j}. Since n ≥ 3 it follows by elementary combinatorics that there exists a partition ν_{j} ⊢ |α_{j}| − 1 such that α_{j} but not β_{j} can be obtained from ν_{j} by adding a 1-hook (or vice versa). Hence the characters of W_{n} are distinguished by their multiplicities in characters induced from W_{n−1}, and the claim follows by Frobenius reciprocity. □

3B. The infinite series G(m, p, n).

Note that any element of W_{n} can be written as the product of a diagonal element diag(ζ_{m}^{a_{1}}, …, ζ_{m}^{a_{n}}) with a permutation matrix corresponding to some w ∈ 𝔖_{n}. We denote this element by (a_{1}, …, a_{n}; w) for short. Now fix a divisor p|m and let G(m, p, n) be the kernel of the linear character γ^{m/p}, of index p in W_{n}:

G(m, p, n) = {(a_{1}, …, a_{n}; w) | ∑ a_{j} ≡ 0 (mod p), w ∈ 𝔖_{n}} .

Then G(m, p, n) is again an imprimitive complex reflection group, generated by the reflections s_{1}, s̃_{1} := t^{−1}s_{1}t, s_{2}, …, s_{n−1}, and t^{p} (if p ≠ m). Clearly, the reflections s_{1}, s_{2}, …, s_{n−1} again generate a natural reflection subgroup 𝔖_{n} consisting of all permutation matrices.

We describe the irreducible characters of G(m, p, n) in terms of those of W_{n}. Denote by τ the cyclic shift on m-tuples of partitions of n, i.e.,

τ(α_{0}, …, α_{m−1}) = (α_{1}, …, α_{m−1}, α_{0}) .

By definition we then have χ_{τ(α)} ⊗ γ = χ_{α}. Let s_{p}(α) denote the order of the stabilizer of α in the cyclic group ⟨τ^{m/p}⟩. Then upon restriction to G(m, p, n) the irreducible character χ_{α} of W_{n} splits into s_{p}(α) different irreducible constituents, and this exhausts the set of irreducible characters of G(m, p, n). More precisely, let α⊢_{m}n with p̃ := s_{p}(α), W_{α,p} := W_{α} ∩ G(m, p, n), and ψ_{α} the restriction of χ_{α_{0}}#(χ_{α_{1}} ⊗ γ)# … #(χ_{α_{m−1}} ⊗ γ^{m−1}) to W_{α,p}. Then ψ_{α} is invariant under the element σ := (s_{1} ⋯ s_{n−1})^{n/p̃} (note that p̃ = s_{p}(α) divides n), and it extends to the semidirect product W_{α,p}.⟨σ⟩. The induced of the different extensions of ψ_{α} then exhaust the irreducible constituents of the restriction of χ_{α} to G(m, p, n). Thus, we may parametrize Irr(G(m, p, n)) by m-tuples of partitions of n up to cyclic shift by τ^{m/p} in such a way that any α stands for s_{p}(α) different characters.

We will need the following consequence of the Littlewood-Richardson rule for the decomposition of characters induced from Young subgroups. Let α ⊢_{m} n with α_{j} = (α_{j1} ≥ α_{j2} ≥ …), and let β = (β_{1} ≥ β_{2} ≥ …) be defined by β_{l} = ∑_{j} α_{jl}. Let 𝔖_{α} denote the Young subgroup of 𝔖_{n} corresponding to (|α_{0}|, …, |α_{m−1}|). Then we have

(3.4)  ⟨χ_{β}, Ind_{𝔖_{α}}^{𝔖_{n}}(χ_{α_{0}}# … #χ_{α_{m−1}})⟩ = 1

for the character χ_{β} ∈ Irr(𝔖_{n}), and all other constituents of the induction are parametrized by partitions strictly smaller than β in the dominance order. This fact can be proved from [12, Cor. 2.8.14] by induction on m.

Lemma 3.5. (a) Let α ⊢_{m} n and χ_{α} ∈ Irr(G(m, p, n)). Then there exists β ⊢ n such that χ_{α} has multiplicity 1 in the induced of χ_{β} ∈ Irr(𝔖_{n}) from t^{−j}𝔖_{n}t^{j} to G(m, p, n), for some 0 ≤ j ≤ p − 1.

(b) The irreducible constituents of the restriction of χ_{α} ∈ Irr(W_{n}) to G(m, p, n) are distinguished by their restrictions to t^{−j}𝔖_{n}t^{j}, 0 ≤ j ≤ p − 1.

Proof. By our above considerations, χ_{α} is the induced of an irreducible character ψ of H := W_{α,p}.⟨σ⟩ with σ := (s_{1} ⋯ s_{n−1})^{n/p̃}, p̃ = s_{p}(α). By the Mackey theorem, the multiplicity of the induced of χ_{β} in χ_{α} is the same as the scalar product of the restrictions of χ_{β} and of ψ to H ∩ 𝔖_{n}. By Frobenius reciprocity, this is the same as the multiplicity of χ_{β} in (ψ|_{H∩𝔖_{n}})^{𝔖_{n}}.

\begin{matrix}W_{α,p} & H & G(m, p, n) \\ W_{α,p} ∩ 𝔖_{n} & H ∩ 𝔖_{n} & 𝔖_{n} \end{matrix}

But W_{α,p} ∩ 𝔖_{n} is a Young subgroup of 𝔖_{n}, and by the above consequence (3.4) of the Littlewood-Richardson rule, there exists χ_{β} which has multiplicity 1 in the induced of ψ̃ := ψ|_{W_{α,p}∩𝔖_{n}} to 𝔖_{n}. Thus χ_{β} occurs with multiplicity 1 in the induced of exactly one of the extensions of ψ̃ to H ∩ 𝔖_{n}. Hence precisely one of the characters of G(m, p, n) parametrized by α occurs with multiplicity 1 in the induced of χ_{β}. But all characters parametrized by α are conjugate under t, hence both assertions of the lemma follow. □

4. Determination of the character fields

With the preparations in the preceding section we can now determine the character fields of cyclotomic algebras attached to irreducible complex reflection groups. According to the classification by Shephard and Todd [19], the latter fall into an infinite series G(m, p, n) of imprimitive groups, the symmetric groups 𝔖_{n}, and another 34 primitive groups.

4A. The imprimitive groups.

We first consider the imprimitive groups. It is known that the character field k of the reflection representation of G(m, p, n) is equal to ℚ(ζ_{m}), except if n = 2, m = p, when k is the maximal totally real subfield of ℚ(ζ_{m}) (see for example [6]).

Let H(W_{n}; u) with u = (u_{0}, …, u_{m−1}, x_{1}, x_{2}) be the generic cyclotomic algebra for W_{n}. We first determine the splitting field for H(W_{n}; u); this result also follows from the explicit construction of all irreducible representations in [3].

Proposition 4.1 (Ariki and Koike). The field ℚ(u) is a splitting field for H(W_{n}; u).

Proof. We proceed by induction on n. If n = 1 then

H(W_{1}; u) = ℤ[u, u^{−1}, T]/((T − u_{0}) … (T − u_{m−1}))

is commutative, and the irreducible representations are given by T ↦ u_{j}, 0 ≤ j ≤ m− 1. Hence they are all ℚ(u)-rational. Now assume that n > 1. By Lemma 3.3(b) the characters of W_{n} are distinguished by their restrictions to W_{n−1} (and to 𝔖_{n} if n = 2). Thus the characters of H(W_{n}; u) are distinguished by their restrictions to H(W_{n−1}; u) (and to H(𝔖_{n}) for n = 2). Since the irreducible characters of the latter are ℚ(u)- rational by induction, the character field is contained in ℚ(u). By Lemma 3.3(a) and Lemma 2.7 it now follows again by induction on n that all representations can be realized over ℚ(u). □

Let k := ℚ(ζ_{m}). The symmetric group 𝔖_{m} on letters {0, …, m − 1} acts on N := k(u) by permutation of the variables (u_{0}, …, u_{m−1}). Let K := k(u)^{𝔖_{m}} be the fixed field, generated over k(x_{1}, x_{2}) by the elementary symmetric polynomials f_{j}(u) in u. Since H(W_{n}; u) is already defined over K, each automorphism of N/K permutes the set of representations of H(W_{n}; u) over N, hence by Proposition 4.1 in particular it acts on Irr(H(W_{n}; u)). The group 𝔖_{m} also acts in a natural way on the set {α ⊢_{m} n} of m-part partitions of n by permuting the m parts. We have the following compatibility between these two actions:

Lemma 4.2. Let σ ∈ 𝔖_{m} and α ⊢_{m} n. Then the character χ_{α} ∈ Irr(H(W_{n}; u)) satisfies χ_{α} = σ(χ_{σ(α)}).

Proof. For n = 1 the (linear) characters of H(W_{1}; u) are given by χ_{α}(T) = u_{l} if |α_{j}| = δ_{j,l} (see the proof of Proposition 4.1). Let σ ∈ 𝔖_{m} and assume that l = σ(l′). Then σ(α) = (α_{σ(j)}), so |σ(α)_{j}| = δ_{j,l′}, and χ_{σ(α)}(T) = u_{l′}. Thus, σ(χ_{σ(α)})(T) = σ(u_{l′}) = u_{l} as claimed. Now assume by induction that the assertion is true for n − 1. Then it also holds for n by Lemma 3.3(b). □

Note that for any admissible specialization u_{j} ↦ u′_{j} (0 ≤ j ≤ m − 1) the Galois group of k(u′)/k(f_{j}(u′) | 1 ≤ j ≤ m) is in a natural way a subgroup of 𝔖_{m} = Gal(k(u)/k(f_{j}(u) | 1 ≤ j ≤ m)) and the assertion of Lemma 4.2 remains correct for all elements σ of this Galois group.

It follows from a result of Ariki [2, Prop. 1.6] that the cyclotomic algebra for G(m, p, n) may be obtained as a subalgebra of a specialization of the one for W_{n}. More precisely, let now p|m, m = pq, u := (u_{0}, …, u_{q−1}, x_{1}, x_{2}) and

(4.3)  ũ := (ũ_{0}, …, ũ_{q−1}, ζ_{p}ũ_{0}, …, ζ_{p}ũ_{q−1}, …, ζ_{p}^{p−1}ũ_{q−1}, x_{1}, x_{2}) ,

where ζ_{p} := exp(2πi/p) and ũ_{j}^{p} = u_{j}. Then H(G(m, p, n), u) is the subalgebra of H(W_{n}, ũ) generated by T_{t}^{p}, T_{t}^{−1}T_{s_{1}}T_{t}, T_{s_{1}}, …, T_{s_{n−1}}. Note that in the case n = 2, p even, this is not the most general generic algebra for W. The latter case will be treated in the next subsection.

In accordance with our previous notation, let τ = (0, 1, …, m − 1) ∈ 𝔖_{m}, η_{p} = (0, q, 2q, …, (p − 1)q), and S_{p} := ⟨η_{p}^{τ^{j}} | 0 ≤ j ≤ q − 1⟩ ≤ 𝔖_{m} the Galois group of k(ũ)/k(u). For a multi-partition α ⊢_{m} n we define its p-symmetry group

S_{p}(α) := ⟨C_{S_{p}}(α), τ^{q}⟩ ≤ S_{p}.

Note that S_{p}(α) acts naturally on k(ũ) as a subgroup of S_{p} = Gal(k(ũ)/k(u)). We can now describe the character fields of the irreducible characters of cyclotomic algebras of imprimitive complex reflection groups. Note that Assumption 2.3 is satisfied in the case of imprimitive groups by [3,8,2].

Theorem 4.4. Let n ≥ 3, W = G(m, p, n) and α⊢_{m}n. Then the character field of χ_{α} ∈ H(W, u) is the fixed field k(ũ)^{S_{p}(α)} of the p-symmetry group of α.

Proof. We split the proof into two steps.

In the first step we assume that s_{p}(α) = p. By Proposition 4.1 the character field of the irreducible character χ_{α} of H(W_{n}, ũ) is contained in k(ũ). But Lemma 4.2 shows that χ_{α} is even k(u)-rational since S_{p}(α) = S_{p} = Gal(k(ũ)/k(u)). Thus the irreducible constituents of the restriction of χ_{α} to H(G(m, p, n), u) can only be algebraically conjugate among each other. Now Lemma 3.5(b) forces them to be k(u)-rational themselves.

In the second step we let p̃:= s_{p}(α) be arbitrary. By the first step, the constituents of the restriction of χ_{α} to H(G(m, p̃, n), ũp̃) are k(ũp̃)-rational since sp̃(α) = p̃. Moreover, by Lemma 3.5(b), they cannot be conjugate among each other. Now by Lemma 4.2 an element σ ∈ S_{p} = Gal(k(ũ)/k(u)) fixes the character χ_{α} of H(G(m, p, n), u) precisely if σ ∈ S_{p}(α) (remember that τ^{q} fixes χ_{α}). □

Theorem 4.4 shows that, in contrast to the case of real reflection groups, the character field can become an arbitrarily large extension of the ground field:

Example 4.5. Let α = (1, 2, …, q, 0, 0 …, 0) ⊢_{m} n where n = q(q + 1)/2 and m = pq, and χ_{α} the corresponding irreducible character of H(G(m, p, n), u). Then S_{p}(α) = ⟨τ^{q}⟩ and hence the character field of χ_{α} is generated over k(u) by

\{\sqrt[p]{u_{0}/u_{j}} | 1 ≤ j ≤ q − 1\} .

In particular it is not a cyclic extension if q ≥ 3.

4B. Two-dimensional imprimitive groups.

The two-dimensional imprimitive groups differ from the other imprimitive reflection groups in that the generic cyclotomic algebra of G(2m, 2p, 2) has one additional exceptional parameter, thus cannot be obtained as a subalgebra of H(W_{2}, u), and secondly the field of rationality for G(m, m, 2) is of index 2 in ℚ(ζ_{m}). Thus the above determination of the character fields of generic cyclotomic algebras does not apply immediately to this case. On the other hand, the irreducible representations have dimension at most 2, so may easily be constructed explicitly. The linear characters are always rational, and the 2-dimensional characters are indexed by multi-partitions α⊢_{m}2 with α_{j} = 0 unless j = j_{1} or j = j_{2} ≠ j_{1}, and α_{j_{1}} = α_{j_{2}} = 1.

Theorem 4.6. Let W = G(m, p, 2) and α⊢_{m}2 with α_{j_{1}} = α_{j_{2}} = 1. Then the character field of χ_{α} ∈ Irr(H, (u_{0}, …, u_{q−1}, x_{1}, x_{2}, y_{1}, y_{2})) is k(u, \sqrt{x_{1}x_{2}y_{1}y_{2}} \sqrt[p]{u_{j_{1}}/u_{j_{2}}}), where x_{1} = y_{1}, x_{2} = y_{2} if p is odd.

Proof. If p = m then G(m, m, 2) = W (I_{2}(m)) is the dihedral Coxeter group, with generators s_{1}, s_{2} say. The 2-dimensional representations of H(W, (x_{1}, x_{2}, y_{1}, y_{2})) (with y_{j} = x_{j} if m is odd) are given by

T_{s_{1}} ↦ ( \begin{matrix}x_{1} & 0 \\ 1 & x_{2} \end{matrix} ) , T_{s_{2}} ↦ ( \begin{matrix}y_{1} & b_{l} \\ 0 & y_{2} \end{matrix} ) , with 1 ≤ l ≤ \frac{m}{2},

where b_{l} = −x_{1}y_{1} − x_{2}y_{2} + (ζ_{m}^{l} + ζ_{m}^{−l})\sqrt{x_{1}x_{2}y_{1}y_{2}}, thus the result follows. If p < m is odd, no additional parameter occurs and the arguments in the proof of Theorem 4.4 apply unchanged. So it remains to consider G(m, p, 2) with m, p even and p < m. By [15, Prop. 3.8] the corresponding generic cyclotomic algebras can all be realized as subalgebras of suitable specializations of H(G(m, 2, 2), u). The irreducible representations of this algebra were determined in loc. cit., 3B, and the claim follows from these explicit matrices. □

4C. The primitive groups.

For the primitive Coxeter groups 𝔖_{n}, W (E_{6}), W (E_{7}), W (E_{8}) and W (F_{4}) the character fields are known by [5], for W (H_{3}) and W (H_{4}) they can be deduced from [1]. In the case of so-called cyclotomic Weyl groups the following result had already been obtained in [8, Satz 6.3].

Theorem 4.7. Let W be a finite primitive complex reflection group satisfying Assumption 2.3, and χ an irreducible character of H_{L}(W, u). Then the character field of χ is contained in k(u) unless (W, χ(1)) are as in Tables 8.1 or 8.2.

Proof. We discuss the different possibilities case by case. First, it is well known that the assertion holds with k = ℚ for the case W = 𝔖_{n} of symmetric groups. So we may assume that W is one of the 34 exceptional primitive reflection groups G_{4}, …, G_{37} (in the notation of Shephard and Todd [19]). First we consider the twodimensional groups, so W = G_{j} for some 4 ≤ j ≤ 22. The proper parabolic subgroups of W are 1-dimensional and the irreducible characters of cyclotomic algebras for 1dimensional reflection groups are all rational. Thus, by our discussion in Section 2C, only characters with equal restriction to all proper parabolic subgroups can possibly not be k(u)-rational. On the other hand, we may compute certain character values on a central element by (2.6) (this was already done for G_{7}, G_{11}, G_{19} in Tables 3,5,7 of [15], and the values in the other cases can be obtained by specialization according to Tables 4,6,8 of loc. cit.). With these two observations, the result already follows for j ≠ 12, 13, 14, 15, 20, 21, 22.

For j = 12, 13, 14, 15 only the character fields of 2-dimensional irreducible characters cannot be deduced by the previous arguments, but the corresponding representations can easily be determined explicitly. In fact, it suffices to do this for W = G_{15} since the other three groups are subgroups of G_{15}, their cyclotomic algebras are subalgebras of suitable specializations of H(G_{15}, u), and their irreducible representations can be obtained by specialization and restriction of those of H(G_{15}, u) (see Table 6 in [15]). For H(G_{15}, u), six 2-dimensional irreducible representations are rational (they factor over the cyclotomic algebra of the imprimitive group G(2, 1, 2)), while the other twelve require the adjunction of a 4th root as in Table 8.1.

For W = G_{21} the 2- and 3-dimensional representations have to be constructed; they involve the irrationalities stated in the table. For j = 20, 22 we may again descend from the cyclotomic algebra for G_{21}.

Finally let W be one of G_{23}, …, G_{37}. By induction and the results of Section 4A we may further assume that the rationality properties of all irreducible characters of proper parabolic subalgebras of H(W, u) are known. By comparing multiplicities in induced rational characters of parabolic subalgebras we obtain the desired rationality statement for most irreducible characters. For the remaining ones, we at least obtain that certain sums of two or three characters of the same degree must be rational over k(u). For these pairs or triples of characters, we can compute the values on T_{β} by the method of Springer (2.6). It turns out that in all cases the values on T_{β} are either in k(u) and different for the pair of characters in question, or they are algebraically conjugate over k(u), involving the irrationalities stated in the theorem. We give details of the argument in the case W = G_{34}, H = H(W, x), the other ones being similar and easier. Here k = ℚ(ζ_{3}). By using the character tables provided by the computer algebra system Chevie [11] we first check that all irreducible characters of W are uniquely determined by their restrictions to the parabolic subalgebras of types G(3, 3, 5) and W (A_{5}) = 𝔖_{6}, except for certain m_{i}-tuples of characters of degree d_{i} with

(m_{i}, d_{i}) ∈ {(2, 896) (2 pairs), (2, 384) (4 pairs), (3, 729) (2 triples)}.

Since the characters of these subalgebras are k(x)-rational, it follows that at most the above characters can possibly have a character field not contained in k(x). Since the characters of degrees 896 and 384 may be distinguished by their multiplicities in the induced irrational characters of the subalgebra of type G_{33}, their character field over k is equal to k(\sqrt{x}). Next we compute the values of the remaining characters on the central element T_{β} by (2.6). Here β = (s_{1} … s_{6})^{7} by [9, Table 4]. The values of the characters of degree 729 turn out to involve \sqrt[3]{x}. This completes the proof for W = G_{34}. □

4D. Some consequences.

The results in Theorems 4.4, 4.6 and 4.7 show that in all cases the character fields can be generated by roots of suitable monomials in the parameters. Moreover, the order of the root is always bounded by the order of the group of roots of unity in the field of definition k of W; it would be nice to understand this fact by a general argument:

Corollary 4.8. Let W be a finite complex reflection group with character field k. Let μ(k) be the group of roots of unity in k. Then the character field of H(W, u) is contained in

k((ζ_{d_{s}}^{−j}u_{s,j})^{1/|μ(k)|} | s ∈ S, 0 ≤ j ≤ d_{s} − 1) ,

where ζ_{d_{s}} = exp(2πi/d_{s}).

In the recent preprint [17] Opdam has given a general proof that under Assumption 2.3 the character field of H(W, u) over ℚ̄(u) can be generated by suitable roots of monomials in the parameters.

Another consequence of our results is a characterization of those cases where the reflection representation of the 1-parameter cyclotomic algebra is rational:

Corollary 4.9. Let W be an n-dimensional irreducible finite complex reflection group with character field k. Then the reflection representation of H(W, x) is rational over k(x) if and only if W can be generated by n of its reflections.

Proof. The group G(m, p, n) is generated by n of its reflections if and only if p = 1 or p = m. The reflection character of the group G(m, p, n) is parametrized by the multi-partition α = ((n − 1), (1), −, …, −). Since S_{p}(α) = ⟨τ^{q}⟩ is strictly contained in S_{p} if p ≠ 1, m the assertion follows from Theorems 4.4 and 4.6. For the primitive groups, the explicit results show that the reflection character of H(W, x) is nonrational precisely for

(4.10)  W ∈ {G_{7}, G_{11}, G_{12}, G_{13}, G_{15}, G_{19}, G_{22}, G_{31}}.

But these are the primitive groups not generated by n of their reflections, where n denotes the dimension of the irreducible reflection representation of W. □

We will come back to the reflection groups generated by n of their reflections in Section 7.

Remark 4.11. Our results also show that the character field of H(W, u) is contained in k(u) precisely for the irreducible groups

\begin{matrix}𝔖_{n}, G(m, 1, n), G(m, m, n) (n ≥ 3), G(m, m, 2) (m odd), \\ G_{4}, G_{25}, G_{28} = W (F_{4}), G_{35} = W (E_{6}). \end{matrix}

5. Determination of splitting fields

The results of the previous section also allow to find the splitting fields of generic cyclotomic algebras.

Proposition 5.1. Let W be a finite irreducible complex reflection group different from G_{12}, G_{13}, G_{20}, G_{22}, G_{31} and satisfying Assumption 2.3. Let χ ∈ Irr(H(W, u)) with character field N ≥ k(u). Then there exists a representation over N affording χ.

For W of type G_{12}, G_{13}, G_{20}, G_{22}, G_{31} there exists such a representation over an extension of degree at most 2 of N.

Proof. The main ingredient will be Lemma 2.7. Let first G = 𝔖_{n}. Then all character fields are equal to ℚ(u) and any non-linear irreducible character occurs with multiplicity 1 in the permutation character of some proper Young subgroup. Thus the result follows by induction from Lemma 2.7. Let now G = G(m, p, n). The assumptions of Lemma 2.7 are satisfied by Lemma 3.5 since by the previous case k(u) is a splitting field for the representations of the Hecke algebra of 𝔖_{n}. Thus we are left with the primitive exceptional groups. For these it can be checked from the explicit tables that Lemma 2.7 applies except possibly if

(W, χ(1)) ∈ {(G_{12}, 4), (G_{13}, 4), (G_{20}, 6), (G_{22}, 4), (G_{22}, 6), (G_{31}, 36)},

when all multiplicities in induced rational representations from proper parabolic subalgebras are divisible by 2,2,2,2,3,2 respectively. The characters of degree 6 of G_{22} can be obtained by specializing y_{j} ↦ ζ_{3}^{j} in a 6-dimensional representation of H(G_{21}, u), hence can be written over an extension of degree 2 of the character field. Thus they can be realized over the character field. Since each of the remaining exceptional characters χ occurs with multiplicity 2 in some representation which can be written over the character field of χ, the last statement of the proposition follows. □

This allows to obtain the following description of splitting fields:

Theorem 5.2. Let W be a finite complex reflection group satisfying Assumption 2.3 with character field k. Then a splitting field L of H(W, u) is contained in

K_{W} := k( (exp(−2πij/d_{s})u_{s,j})^{1/|μ(k)|} | s ∈ S, 0 ≤ j ≤ d_{s} − 1) .

Proof. By Proposition 5.1 and Corollary 4.8 we only have to consider the exceptional characters of the irreducible groups G_{12}, G_{13}, G_{20}, G_{22}, G_{31} listed in the previous proof. For W = G_{13} the representations of H(W) of degree 4 over K_{W} can be constructed explicitly. The representation of degree 36 of W = G_{31} has multiplicity 1 in an induced representation of the reflection subgroup G(4, 2, 3), thus the corresponding representation of H(W) can be written over K_{W}.

The representation of H(G_{12}) of degree 4 can be obtained by specializing y_{j} ↦ ζ_{3}^{j} in a rational 4-dimensional representation of H(G_{14}) as follows. Let

H = ⟨T_{1}, T_{2} | (T_{1}T_{2})^{4} = (T_{2}T_{1})^{4}, (T_{1} + 1)(T_{1} − x) = (T_{2} − y_{1})(T_{2} − y_{2})(T_{2} − y_{3}) = 0⟩ be the cyclotomic algebra for the complex reflection group G_{14} and let H̃ denote its image under the specialization y_{j} ↦ ζ_{3}^{j}. Then H(G_{12}, (x, −1)) is the subalgebra of H̃ generated by the elements T_{1}, T_{2}T_{1}T_{2}^{−1}, T_{2}^{−1}T_{1}T_{2}. Now

T_{1} ↦ ( \begin{matrix}x & 0 & 0 & 0 \\ 0 & x & 0 & 0 \\ 1 & 0 & −1 & 0 \\ 0 & 1 & 0 & −1 \end{matrix} ) , T_{2} ↦ ( \begin{matrix}y_{1} & 0 & 0 & y_{1}^{2}x^{2} + y_{2}y_{3} \\ 0 & y_{1} & −1 & y_{2} − 2xy_{1} + y_{3} \\ 0 & 0 & y_{2} & (y_{2} − xy_{1})(y_{3} − y_{2}) \\ 0 & 0 & 0 & y_{3} \end{matrix} )

defines a 4-dimensional irreducible representation of H. Under the above specialization this yields the irreducible 4-dimensional representation of H(G_{12}, (x, −1)), over the field ℚ(x, \sqrt{−3}). This gives rise to an 8-dimensional representation over the field ℚ(x). The endomorphism ring of this representation may easily be computed from the explicit matrices given above. The constant coefficient of the characteristic polynomial of an element of this (4-dimensional) endomorphism ring is of the form

(z_{1}^{2} + 3z_{2}^{2} − (z_{3}^{2} + 3z_{4}^{2})x)^{4} ,

with z_{j} in the ground field. Clearly, with z_{j} = \sqrt{x}z_{j+2} for j = 1, 2, this quadratic form does represent zero nontrivially. Thus ℚ(\sqrt{−2}, \sqrt{x}) is a splitting field for H(G_{12}, (x, −1)).

Similarly, the cyclotomic algebra for G_{22} may be obtained by specializing y_{j} ↦ ζ_{3}^{j} in the one for G_{21}. A computation as above shows that the 4-dimensional irreducible representations of H(G_{22}, (x, −1)) can be realized over ℚ(\sqrt{x}).

Finally, the representations of H(G_{20}) of degree 6 can be obtained by specializing x_{j} ↦ (−1)^{j} in representations of H(G_{21}, u) involving the fourth roots of unity. Calculations as above lead to the constant coefficient

(z_{1}^{2} + z_{2}^{2} + 2z_{3}^{2} + 2z_{4}^{2})^{6}

of the characteristic polynomial of an element of the endomorphism ring, which represents zero nontrivially over ℚ(\sqrt{−3}). Thus K_{W} is a splitting field in this case as well. □

6. An observation on fake degrees

6A. Fake degrees.

Let W be a finite complex reflection group on the complex vector space V. The ring of invariants of W in the symmetric algebra S(V) of V is a polynomial ring, generated by homogeneous invariants of degrees d_{1}, …, d_{n}, with n = dim(V). It follows from Molien’s formula for S(V)^{W} that the Poincaré polynomial P_{W} of W is given by

(x − 1)^{n}P_{W} := (\frac{1}{|W |} ∑_{w∈W} \frac{det_{V} (w)}{det_{V} (x − w)})^{−1} = ∏_{j=1}^{n}(x^{d_{j}} − 1) ,

where det_{V} denotes the determinant on V. In particular, P_{W} is a polynomial in x of degree N^{∗} := ∑_{j=1}^{n}(d_{j} − 1). For an irreducible character χ ∈ Irr(W) the fake degree is defined as

(6.1)  R_{χ} := (x − 1)^{n}P_{W} \frac{1}{|W |} ∑_{w∈W} \frac{det_{V} (w)χ(w)}{det_{V} (x − w)} ∈ ℤ[x] .

The fake degree has the following symmetry property:

(6.2)  R_{χ}(x) = x^{N^{∗}}R\overline{det ⊗χ}(x^{−1}),

where ¯ denotes complex conjugation. Indeed, P_{W} (x^{−1}) = x^{−N^{∗}}P_{W} (x), so

R_{χ}(x^{−1}) =x^{−N^{∗}}(x − 1)^{n}(−x)^{−n}P_{W} \frac{1}{|W |} ∑_{w∈W} \frac{det_{V} (w)χ(w)}{det_{V} (x^{−1} − w)}

=x^{−N^{∗}}(x − 1)^{n}P_{W} \frac{1}{|W |} ∑_{w∈W} \frac{det_{V} (w)χ(w)}{det_{V} (wx − 1)} = x^{−N^{∗}}R\overline{det ⊗χ}(x).

For example we have 1 = R_{1}(x^{−1}) = x^{−N^{∗}}R\overline{det}(x), so R\overline{det}(x) = x^{N^{∗}}.

It is possible to express the sum of exponents of the fake degree as follows. By looking at eigenvalues of elements it is easy to see that modulo (x − 1)^{2} the fake degree (6.1) becomes

R_{χ}(x) = \frac{P_{W}}{|W |} (χ(1) + ∑_{r∈R} \frac{det(r)χ(r)}{(x − det(r))}(x − 1)) mod (x − 1)^{2} .

Differentiation yields

R′_{χ}(x) = \frac{P′_{W}}{|W |}χ(1) + \frac{P_{W}}{|W |} ∑_{r∈R} \frac{det(r)χ(r)}{x − det(r)} mod (x − 1).

Since P_{W} = ∏_{j=1}^{n}(x^{d_{j}−1} + x^{d_{j}−2} + … + 1) we obtain by evaluating at x = 1 and using |W | = ∏_{j} d_{j} that

(6.3)  R′_{χ}(1) = χ(1)N^{∗}/2 + ∑_{r∈R} \frac{det(r)χ(r)}{1 − det(r)}.

6B. Semi-palindromicity.

In the case of real reflection groups W, Beynon and Lusztig [7] observed from their tables that most (but not all) fake degrees have an additional symmetry: they are palindromic. Moreover, the ones which are not palindromic are precisely those for which the corresponding character of the 1-parameter Hecke algebra H(W, x) is not rational over ℚ(x). An a priori explanation of this connection was later given by Opdam [16]. A similar phenomenon can be observed for complex reflection groups.

Let us say that χ ∈ Irr(W) has a semi-palindromic fake degree, or by abuse of notation that R_{χ} is semi-palindromic, if there exists a c = c_{χ} ∈ ℕ such that

R_{χ}(x) = x^{c}R\overline{χ}(x^{−1}) .

Again, it turns out that many, but not all irreducible characters of complex reflection groups have semi-palindromic fake degree. It is possible to extend this to a statement about all irreducible characters. But, although the fake degrees can be defined entirely in terms of the group W it seems necessary to appeal to the cyclotomic Hecke algebra attached to W to phrase the result.

Let us first make the following observation on the exponent c above; here R denotes the set of all reflections in W:

Proposition 6.4. Let χ, ψ ∈ Irr(W) be such that χ(1) = ψ(1), χ(r) = ψ(r^{−1}) for all reflections r ∈ R, and R_{χ}(x) = x^{c}R_{ψ}(x^{−1}). Then c = N^{∗} − ∑_{r∈R} χ(r)/χ(1). If all reflections in W have order 2 then we also have 2c = R′_{χ}(1)/χ(1).

Proof. By differentiating the equality R_{χ}(x) = x^{c}R_{ψ}(x^{−1}) with respect to x and evaluating at x = 1 we get

R′_{χ}(1) = cψ(1) − R′_{ψ}(1).

On the other hand, adding (6.3) for χ and ψ and inserting the assumptions gives

R′_{χ}(1) + R′_{ψ}(1) = χ(1)N^{∗} + ∑_{r∈R} χ(r) (\frac{det(r)}{1 − det(r)} + \frac{det(r^{−1})}{1 − det(r^{−1})}) .

The formula for c follows by using that det is a homomorphism and comparing with the first equation.

Now assume that all reflections in W have order 2. Then (6.3) becomes

R′_{χ}(1) = \frac{1}{2} (χ(1)N^{∗} − ∑_{r∈R} χ(r)) ,

proving the second statement. □

Since N^{∗} = |R|, the value of c above can also be written ∑_{r∈R}(1 − χ(r)/χ(1)).

6C. A symmetry property.

Corollary 4.8 implies that the character field of any irreducible character of the 1-parameter cyclotomic algebra H(W, x) is contained in k(y), where y^{|μ(k)|} = x. Let δ be the automorphism of k(y)/k(x) induced by y ↦ exp(2πi/|μ(k)|)y. Then δ also acts on Irr(H(W, x)), so via specialization induces a permutation of the set Irr(W). The element δ together with complex conjugation generate a dihedral group, so the product of δ with complex conjugation is an involution on Irr(W). We have the following extension of the previously mentioned observation on semi-palindromicity:

Theorem 6.5. Let W be a finite complex reflection group satisfying Assumption 2.3 with set of reflections R and let χ ∈ Irr(W). Then

(6.6)  R_{χ}(x) = x^{c}R_{δ(χ̄)}(x^{−1})  with c = N^{∗} − ∑_{r∈R} χ(r)/χ(1),

where δ is the permutation of the set Irr(W) defined above.

Proof. If W is of type A_{n} then all characters of H(W, x) are ℚ(x)-rational and all fake degrees are palindromic, so the assertion holds. If W is of exceptional type G_{j}, 4 ≤ j ≤ 37, the fake degrees can be computed explicitly from (6.1) and the result follows by comparison with the character fields obtained in Theorem 4.7. (Note that for the evaluation of (6.1) it is only necessary to sum over representatives of the conjugacy classes, weighted by the class lengths.)

So we may assume that W = G(m, p, n). In [14, Bem. 2.10] we computed the fake degree of χ_{α} ∈ Irr(W_{n}) as

R_{α}(x) = Γ(α) ∏_{j=0}^{m−1} x^{j|α_{j}|}

where Γ(α) is a palindromic polynomial in x depending only on the set {α_{j}} of components of α. Furthermore, by [14, 5B] the fake degree of χ_{α} ∈ Irr(G(m, p, n)) is given by

R_{χ_{α}}(x) = \frac{x^{nq} − 1}{x^{nm} − 1} \frac{1}{s_{p}(α)} ∑_{j=0}^{p−1} R_{τ^{jq}(α)}(x)

where m = pq. To prove the assertion it hence suffices to show that

(6.7)  ∑_{j=0}^{m−1} j|α_{j+lq}| + ∑_{j=0}^{m−1} j|(δ(\overline{α}))_{m−lq+j}|

is a constant independent of l (here and later on the indices have to be taken modulo m). By our above description of Irr(G(m, p, n)) complex conjugation acts on characters by χ_{α} ↦ χ\overline{α} where \overline{α} = (α_{0}, α_{m−1}, …, α_{1}). Also, δ acts via α ↦ δ(α) = (α′_{j}) with α′_{j} = α_{j−q} if q|j and α′_{j} = α_{j} otherwise. Thus, δ(\overline{α}) = \overline{α}′ with \overline{α}′_{j} = α_{m−j−q} if q|j and \overline{α}′_{j} = α_{m−j} otherwise. We write n_{j} := |α_{j}|. Now let 0 ≤ l ≤ p − 1 be fixed. Then (6.7) becomes

∑_{j=0}^{m−1} jn_{j+lq} + ∑_{j≢0 (mod q)} jn_{m−j−(p−l)q} + ∑_{j≡0 (mod q)} jn_{m−q−j−(p−l)q}

= ∑_{j=0}^{m−1} jn_{j+lq} + ∑_{j=1}^{m} jn_{m−j+lq} + ∑_{j≡0 (mod q)}(j − (q + j))n_{m−q−j+lq}

= m ∑_{j=0}^{m−1} n_{j} − q ∑_{j≡0 (mod q)} n_{j} ,

so it is indeed independent of l, proving the identity for type G(m, p, n).

The value given for c follows from Proposition 6.4. Indeed, since the character δ(\overline{χ}) of H(W, x) is algebraically conjugate to χ̄, we have χ(1) = δ(\overline{χ})(1). Moreover, by the defining relations for H(W, x) the value of any character on an element T_{s} with s a generating reflection is k(x)-rational, hence δ(\overline{χ})(T_{s}) = \overline{χ}(T_{s}). Specialization to W shows that δ(\overline{χ})(s) = \overline{χ}(s) for all generating reflections, and hence for all reflections of W. Thus the assumptions of Proposition 6.4 are satisfied. □

Remark 6.8. It would be nice to have an a priori proof of this theorem, not relying on the classification of the character fields. After reading a preliminary version of this paper, E. Opdam informed the author that he can define (in the spirit of [16]) a map δ^{∗} on the character group of an arbitrary finite complex reflection group W such that the formula in Theorem 6.5 holds with δ replaced by δ^{∗}, without any assumption on the cyclotomic Hecke algebra. However, at the moment there is no a priori proof that δ^{∗} respects irreducibility. Under an additional assumption on the topological braid group of W (which is stronger than our Assumption 2.3) Opdam has a general proof of Theorem 6.5 (see [17, Prop. 7.4]).

Example 6.9. Let W = G_{32} be the primitive 4-dimensional complex reflection group generated by reflections of order 3. Let χ ∈ Irr(W) have degree 64 and ψ := δ(χ̄). Then R_{χ}(x) = x^{c}R_{ψ}(x^{−1}) by Theorem 6.5. But one finds {R′_{χ}(1), R′_{ψ}(1)} = {2240, 2560}, so 2c = (R′_{χ}(1) + R′_{ψ}(1))/64 = 75 ≠ R′_{χ}(1)/χ(1). Hence the assumption in the second part of Proposition 6.4 cannot be dropped.

Clearly, if an irreducible character of the cyclotomic algebra H specialized according to (2.2) has character field not contained in K, the same is true for the corresponding character of the generic cyclotomic algebra. Hence, as an immediate consequence of the preceding results we obtain:

Corollary 6.10. Let W be a finite irreducible complex reflection group with field of definition k. Let χ be an irreducible complex character of W such that the corresponding character of the generic cyclotomic algebra H(W, u) has character field contained in k(u). Then R_{χ} is semi-palindromic.

Example 6.11. By Theorem 6.5 a rational character of the 1-parameter cyclotomic algebra has semi-palindromic fake degree. However, the converse is not always true. Let α = ((2), (1), (1)^{2}, −) ⊢_{4} 5 and χ_{α} ∈ Irr(G(4, 2, 5)) a corresponding irreducible character. Then by what we remarked above about the form of the fake degree, R_{χ_{α}} is semi-palindromic but the corresponding character of H(G(4, 2, 5), x) is not rational by Theorem 4.4.

Nevertheless the following can be checked (see Corollary 7.2 for an extension):

Proposition 6.12. Let W be a finite irreducible complex reflection group with field of definition k and let ρ be the reflection character of W. Then R_{ρ} is semi-palindromic if and only if the corresponding character of the 1-parameter cyclotomic Hecke algebra H(W, x) has character field contained in k(x).

Proof. One direction is clear by Theorem 6.5. Thus we may assume that the character field of ρ is not contained in k(x). If W is of exceptional type, the assertion follows by inspection. So now let W = G(m, p, n) with p ≠ 1, m and α = ((n − 1), (1), −, …, −) (see Corollary 4.9). Then up to a palindromic factor R_{χ_{α}} is equal to x + x^{nq+1} + … + x^{(p−1)nq+1}, while the fake degree of the complex conjugate character equals x^{m−1} + x^{qn−1} + … + x^{(p−1)nq−1} times the same factor. This proves the assertion. □

Remark 6.13. Let W be a complex reflection group on V with degrees d_{1} ≤ … ≤ d_{n} and ρ the character of the reflection representation. Write

(6.14)  R_{ρ} = ∑_{j=1}^{n} x^{c_{j}},  R\overline{ρ} = ∑_{j=1}^{n} x^{d_{j}−1},

with 1 = c_{1} ≤ … ≤ c_{n}. The d_{j} − 1 are the so-called exponents of W, the c_{j} are the coexponents. It was observed by Orlik and Solomon [18] (case by case) that (d_{j} − 1) + c_{n−j+1} = d_{n} for j = 1, …, n if and only if W can be generated by n reflections. This also follows from our Corollary 4.9 and Proposition 6.12. Indeed, W can be generated by n reflections if and only if the reflection character ρ of H(W, x) is rational if and only if R_{ρ} is semi-palindromic, which by (6.14) holds precisely if (d_{j} − 1) + c_{n−j+1} = d_{n} for j = 1, …, n. Although our proof is also case by case, it seems to shed some new light on the observation of Orlik and Solomon.

Conversely, Corollary 4.9 follows from Proposition 6.12 when we assume the result of Orlik and Solomon, and the latter together with Corollary 4.9 gives Proposition 6.12.

7. Well-generated reflection groups

Let us say that a finite n-dimensional irreducible reflection group W ≤ GL_{n}(ℂ) is well-generated if it can be generated by n of its reflections. Thus in particular real reflection groups are well-generated. In the notation of Shephard and Todd [19] the well generated groups are the symmetric groups, the monomial groups G(m, 1, n), G(m, m, n), and those primitive groups G_{j}, 4 ≤ j ≤ 37, which are not listed in (4.10). Well-generated reflection groups seem to behave more nicely in several aspects. For example we saw in Corollary 4.9 that the reflection representation of the 1-parameter cyclotomic algebra H(W, x) is k(x)-rational if and only if W is well-generated.

7A. The field of definition of a well-generated reflection group.

First we give a general formula for the splitting field of a well-generated complex reflection group W which seems to have escaped notice until now. As above write k for the character field of its reflection representation, which by [4,6] is a (hence minimal) splitting field for W. It can be checked case by case that in this case k is generated over ℚ by the coefficients of the characteristic polynomial of a Coxeter element in the reflection representation. Since the eigenvalues of Coxeter elements in well-generated reflection groups can be expressed in a uniform fashion we obtain the following description of k:

Theorem 7.1. Let W be a well-generated reflection group with degrees d_{1} ≤ … ≤ d_{n}. For a primitive d_{n}-th root of unity ζ let G be the setwise stabilizer of (ζ^{d_{j}−1} | 1 ≤ j ≤ n) in the Galois group Gal(ℚ(ζ)/ℚ). Then k is the fixed field k = ℚ(ζ)^{G}.

Proof. This can be checked case by case from the known tables for k and the d_{j}. □

Let us give the connection to the properties of the Coxeter element: By a (caseby-case) result of Shephard and Todd [19, 5.4] there exists an element c ∈ W which is the product in some suitable order over the elements of a set of n generating reflections of W such that its eigenvalues in the reflection representation are given by (ζ^{d_{j}−1} | 1 ≤ j ≤ n) in our above notation (see also [18] or Remark 6.13 for a description of the eigenvalues in terms of the coexponents). Such an element is usually called a Coxeter element of W. Clearly the fixed field of G is now precisely the field generated by the coefficients of the characteristic polynomial of c.

The theorem is no longer true for n-dimensional groups generated by n + 1 reflections. As an example take G = G_{13} with degrees d_{1} = 8, d_{2} = 12, but where k = ℚ(ζ_{8}).

7B. Cyclotomic algebras for well-generated groups.

The general results in the previous sections have the following stronger versions in the case of well-generated groups:

Corollary 7.2. Let W be a well-generated (irreducible) complex reflection group. Then we have:

(a) The character field of H(W, u) is contained in k((ζ_{d_{s}}^{−j}u_{s,j})^{1/z} | s ∈ S, 0 ≤ j ≤ d_{s} − 1), where z := |Z(W)|.

(b) The field in (a) is a splitting field for H(W, u).

(c) Let χ ∈ Irr(W). Then R_{χ} is semi-palindromic if and only if the corresponding character of H(W, x) has its character field contained in k(x).

Proof. Part (a) follows by inspection from the results on character fields and Tables 8.1 and 8.2, where the orders of centers have been included. The second part follows from the first part together with Proposition 5.1. In the third part, let χ_{x} denote the corresponding character of the 1-parameter cyclotomic algebra. By Theorem 6.5 we may assume that χ_{x} is not k(x)-rational. But then W is a primitive group by Remark 4.11. For these the assertion follows by explicit computation of the fake degrees. □

Since |Z(W)| divides |μ(k)| for (absolutely) irreducible W by Schur’s Lemma, the first part is indeed a strengthening of Corollary 4.8. Note that z = gcd(d_{1}, …, d_{n}), and by Theorem 7.1 the field k can be described in terms of the degrees of W, so the same is true for the splitting field in (b).

8. Appendix

Here we collect the results on character fields for primitive groups in the form of a table. In the table, we first list the Shephard-Todd name of W, the order of the center |Z(W)| and the splitting field k of W. The next column contains the degrees of irrational characters of the generic cyclotomic algebra H(W, u), together with the number of such characters. The next column gives the irrationality generating the character field over k(u) (it turns out that for primitive groups the character field is always generated by a single root of a monomial in the parameters). In the last column we give the names of the indeterminates used in describing the irrationalities.

Rationality and fake degrees 21

Table 8.1. Primitive cases.

W |Z|  k  χ(1)  irrationality u
G_{5} 6 ℚ(ζ_{3})  (3×) 3  \sqrt[3]{y_{1}y_{2}y_{3}z_{1}z_{2}z_{3}} (y_{1,2,3}; z_{1,2,3})
G_{6} 4 ℚ(ζ_{12})  (6×) 2  \sqrt{x_{1}x_{2}z_{1}z_{2}} (x_{1,2}; z_{1,2,3})
G_{7} 12 ℚ(ζ_{12}) (18×) 2  \sqrt{x_{1}x_{2}y_{1}y_{2}z_{1}z_{2}} (x_{1,2}; y_{1,2,3}; z_{1,2,3})
(6×) 3  \sqrt[3]{x_{1}^{2}x_{2}y_{1}y_{2}y_{3}z_{1}z_{2}z_{3}}
G_{8} 4  ℚ(i)  (2×) 4  \sqrt{z_{1}z_{2}z_{3}z_{4}} (z_{1,2,3,4})
G_{9} 8 ℚ(ζ_{8}) (12×) 2  \sqrt{x_{1}x_{2}z_{1}z_{2}} (x_{1,2}; z_{1,2,3,4})
(4×) 4  \sqrt[4]{x_{1}^{2}x_{2}^{2}z_{1}z_{2}z_{3}z_{4}}
G_{10} 12 ℚ(ζ_{12}) (12×) 3  \sqrt[3]{y_{1}y_{2}y_{3}z_{1}z_{2}z_{3}} (y_{1,2,3}; z_{1,2,3,4})
(6×) 4  \sqrt{y_{2}y_{3}z_{1}z_{2}z_{3}z_{4}}
G_{11} 24 ℚ(ζ_{24}) (36×) 2  \sqrt{x_{1}x_{2}y_{1}y_{2}z_{1}z_{2}} (x_{1,2}; y_{1,2,3}; z_{1,2,3,4})
(24×) 3  \sqrt[3]{x_{1}^{2}x_{2}y_{1}y_{2}y_{3}z_{1}z_{2}z_{3}}
(12×) 4 \sqrt[4]{x_{1}^{2}x_{2}^{2}y_{1}^{2}y_{2}y_{3}z_{1}z_{2}z_{3}z_{4}}
G_{12} 2 ℚ(\sqrt{−2}) (2×) 2  \sqrt{−x_{1}x_{2}} (x_{1,2})
G_{13} 4 ℚ(ζ_{8})  (4×) 2  \sqrt[4]{x_{1}^{2}x_{2}^{2}u_{1}u_{2}^{3}} (x_{1,2}; u_{1,2})
(2×) 4  \sqrt{u_{1}u_{2}}
G_{14} 6 ℚ(ζ_{3}, \sqrt{−2}) (6×) 2  \sqrt{−x_{1}x_{2}y_{1}y_{2}} (x_{1,2}; y_{1,2,3})
(6×) 3  \sqrt[3]{x_{1}^{2}x_{2}y_{1}y_{2}y_{3}}
G_{15} 12 ℚ(ζ_{24}) (12×) 2  \sqrt[4]{x_{1}^{2}x_{2}^{2}y_{1}^{2}y_{2}^{2}u_{1}u_{2}^{3}} (x_{1,2}; y_{1,2,3}; u_{1,2})
(12×) 3  \sqrt[3]{x_{1}^{2}x_{2}y_{1}y_{2}y_{3}u_{1}^{2}u_{2}}
(6×) 4  \sqrt{y_{2}y_{3}u_{1}u_{2}}
G_{16} 10 ℚ(ζ_{5}) (10×) 4  \sqrt{z_{1}z_{2}z_{3}z_{4}} (z_{1,2,3,4,5})
(5×) 5  \sqrt[5]{z_{1}z_{2}z_{3}z_{4}z_{5}}
G_{17} 20 ℚ(ζ_{20}) (20×) 2  \sqrt{x_{1}x_{2}z_{1}z_{2}} (x_{1,2}; z_{1,2,3,4,5})
(20×) 4  \sqrt[4]{x_{1}^{2}x_{2}^{2}z_{1}z_{2}z_{3}z_{4}}
(10×) 5  \sqrt[5]{x_{1}^{3}x_{2}^{2}z_{1}z_{2}z_{3}z_{4}z_{5}}
(10×) 6  \sqrt{x_{1}x_{2}z_{2}z_{3}z_{4}z_{5}}
G_{18} 30 ℚ(ζ_{15}) (30×) 3  \sqrt[3]{y_{1}y_{2}y_{3}z_{1}z_{2}z_{3}} (y_{1,2,3}; z_{1,2,3,4,5})
(30×) 4  \sqrt{y_{2}y_{3}z_{1}z_{2}z_{3}z_{4}}
(15×) 5  \sqrt[5]{y_{1}^{2}y_{2}^{2}y_{3}z_{1}z_{2}z_{3}z_{4}z_{5}}
(15×) 6  \sqrt[3]{y_{1}^{2}y_{2}^{2}y_{3}^{2}z_{1}^{2}z_{2}z_{3}z_{4}z_{5}}
G_{19} 60 ℚ(ζ_{60}) (60×) 2  \sqrt{x_{1}x_{2}y_{1}y_{2}z_{1}z_{2}} (x_{1,2}; y_{1,2,3}; z_{1,2,3,4,5})
(60×) 3  \sqrt[3]{x_{1}^{2}x_{2}y_{1}y_{2}y_{3}z_{1}z_{2}z_{3}}
(60×) 4 \sqrt[4]{x_{1}^{2}x_{2}^{2}y_{1}^{2}y_{2}y_{3}z_{1}z_{2}z_{3}z_{4}}
(30×) 5 \sqrt[5]{x_{1}^{3}x_{2}^{2}y_{1}^{2}y_{2}^{2}y_{3}z_{1}z_{2}z_{3}z_{4}z_{5}}
(30×) 6 \sqrt[6]{x_{1}^{3}x_{2}^{3}y_{1}^{2}y_{2}^{2}y_{3}^{2}z_{1}^{2}z_{2}z_{3}z_{4}z_{5}}

22 Gunter Malle

Table 8.2. Primitive cases, continued.

W  |Z|  k  χ(1) irrationality u
G_{20}  6 ℚ(ζ_{3}, \sqrt{5})  (6×) 3  \sqrt[3]{y_{1}y_{2}y_{3}} (y_{1,2,3})
(6×) 4  \sqrt{y_{2}y_{3}}
(3×) 6  \sqrt[3]{y_{1}y_{2}y_{3}}
G_{21}  12 ℚ(ζ_{12}, \sqrt{5}) (12×) 2 \sqrt{x_{1}x_{2}y_{1}y_{2}} (x_{1,2}; y_{1,2,3})
(12×) 3 \sqrt[3]{x_{1}^{2}x_{2}y_{1}y_{2}y_{3}}
(12×) 4 \sqrt[4]{x_{1}^{2}x_{2}^{2}y_{1}^{2}y_{2}y_{3}}
(6×) 6 \sqrt[6]{x_{1}^{3}x_{2}^{3}y_{1}^{2}y_{2}^{2}y_{3}^{2}}
G_{22}  4 ℚ(i, \sqrt{5})  (4×) 2  \sqrt{x_{1}x_{2}} (x_{1,2})
(4×) 4  \sqrt{x_{1}x_{2}}
(2×) 6  \sqrt{x_{1}x_{2}}
G_{23} = W (H_{3}) 2 ℚ(\sqrt{5})  (2×) 4  \sqrt{x} (x, −1)
G_{24}  2 ℚ(\sqrt{−7})  (2×) 8  \sqrt{x} (x, −1)
G_{26}  6 ℚ(ζ_{3})  (6×) 8 \sqrt{−x_{1}x_{2}y_{1}y_{2}} (x_{1,2}; y_{1,2,3})
G_{27}  6 ℚ(ζ_{3}, \sqrt{5})  (4×) 8  \sqrt{x} (x, −1)
(6×) 9  \sqrt[3]{x}
G_{29}  4  ℚ(i)  (4×) 16  \sqrt{x} (x, −1)
G_{30} = W (H_{4}) 2 ℚ(\sqrt{5})  (4×) 16  \sqrt{x} (x, −1)
G_{31}  4  ℚ(i)  (4×) 4  \sqrt{x} (x, −1)
(2×) 6  \sqrt{x}
(4×) 10  \sqrt{x}
(8×) 20  \sqrt{x}
(2×) 24  \sqrt{x}
(2×) 30  \sqrt{x}
(4×) 36  \sqrt{x}
(6×) 40  \sqrt{x}
G_{32}  6 ℚ(ζ_{3})  (6×) 64  \sqrt{y_{1}y_{2}} (y_{1,2,3})
(3×) 81  \sqrt[3]{y_{1}y_{2}y_{3}}
G_{33}  2 ℚ(ζ_{3})  (2×) 64  \sqrt{x} (x, −1)
G_{34}  6 ℚ(ζ_{3})  (8×) 384  \sqrt{x} (x, −1)
(6×) 729  \sqrt[3]{x}
(4×) 896  \sqrt{x}
G_{36} = W (E_{7}) 2  ℚ  (2×) 512  \sqrt{x} (x, −1)
G_{37} = W (E_{8}) 2  ℚ  (4×) 4096  \sqrt{x} (x, −1)

For the 2-dimensional groups G_{4}, …, G_{22}, we keep the notation used in [15], for the higher-dimensional groups generated by involutions we have specialized one of the parameters to −1. Note that G_{4}, G_{25}, G_{28} and G_{35} do not occur in the tables since their cyclotomic algebras have splitting field k(u).

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Fachbereich Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Str. 40, D–34132 Kassel, Germany

   E-mail address: malle@mathematik.uni-kassel.de