Invariant Fields of Finite Irreducible Reflection Groups

Gregor Kemper and Gunter Malle^{∗}

January 15, 1999

Abstract

We prove the following result: If G is a finite irreducible reflection group defined over a base field k, then the invariant field of G is purely transcendental over k, even if |G| is divisible by the characteristic of k.

It is well known that in the above situation the invariant ring is in general not a polynomial ring. So the question whether at least the invariant field is purely transcendental (Noether’s problem) is quite natural.

Introduction

A linear group G defined over a field k is called a reflection group if it is generated by elements σ with rank(σ − 1) = 1. If the order of a finite reflection group is not divisible by the characteristic of k, then the invariant ring is isomorphic to a polynomial ring (see, for example, Benson [1, Theorem 7.2.1]). This is no longer true in general if |G| is divisible by char(k). In fact, the authors classified the finite irreducible reflection groups whose invariant rings are polynomial rings in [7]. In view of this, it is natural to ask if at least the invariant field of a finite reflection group is purely transcendental over k. In this paper we show that this is true for all finite irreducible reflection groups. In other words, the answer to Noether’s problem is affirmative for all these groups.

We use the following notation: V is a vector space of finite dimension n over a field k of characteristic p (which may be zero), and G ≤ GL(V) is a finite linear group. Then G also acts on the symmetric algebra k[V] := S(V) of V and on its field of fractions k(V) := Quot(k[V]). We denote the invariant ring and invariant field by k[V]^{G} and k(V)^{G}, respectively.

Theorem 1. With the above notation, let G be an irreducible reflection group. Then the invariant field k(V)^{G} is purely transcendental as an extension of k.

Overview of the proof. If n = dim(V) = 1 then k[V]^{G} is polynomial, and the same is true for n = 2 by Nakajima [10, Theorem 5.1]. In fact, the hypothesis that G is an irreducible reflection group is not needed for n ≤ 2, since the subfield k(V)_{0}^{G} consisting of 0 and all rational invariants which can be written as quotients of homogeneous invariants of equal degrees, has transcendence degree 0 or 1 and hence is purely transcendental by Lüroth’s theorem. But k(V)^{G} is purely transcendental over k(V)_{0}^{G} (see Kemper [6, Proposition 1.1]).

If G is imprimitive, then k[V]^{G} is polynomial by Nakajima [10, Theorem 2.4]. Hence from now on we may assume that n ≥ 3 and G is primitive.

We use the classification of finite irreducible primitive reflection groups by Kantor, Wagner, Zalesskiǐ, and Serežkin. For an overview and references, see Kemper and Malle [7]. From these groups, we only have to consider those whose invariant ring is not polynomial. We get the following list, where not all groups with polynomial invariant rings have been excluded and some groups which are not irreducible primitive reflection groups have been added for simplicity:

∗The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft.

(1) SU_{n}(q) ≤ G ≤ GU_{n}(q), G ≠ SU_{3}(2), where p|q,

(2) Ω_{n}^{(±)}(q) < G ≤ GO_{n}^{(±)}(q), G ≠ SO_{n}^{(±)}(q), where p|q, p ≠ 2.

(3) G = SO_{n}^{(±)}(q), where p = 2, n ≥ 4 is even and p|q,

(4) G = Sp_{n}(q), where n is even and p|q,

(5) G = 𝔖_{n+2}, where n ≥ 5, p|(n + 2),

(6) G = W_{3}(G_{30}), W_{3}(G_{31}), W_{5}(G_{32}), W_{2}(G_{34}), W_{3}(G_{36}), W_{3}(G_{37}), or W_{5}(G_{37}).

Here Ω_{n}^{(±)}(q) denotes the commutator subgroup of GO_{n}^{(±)}(q), 𝔖_{n+2} is the symmetric group of order (n + 2)! with the natural irreducible module of dimension n (see the proof of Proposition 7 for details), and W_{p}(G_{i}) is the p-modular reduction of the i-th complex reflection group in the classification of Shephard and Todd [11].

The cases (1) and (3) are treated in Proposition 3 below. In case (2), k(V)^{G} is purely transcendental by Kemper [6, Theorem 2.4]. The invariants of symplectic groups have been studied by Carlisle and Kropholler (see Benson [1, Section 8.3]). The proof of Theorem 8.3.4 in [loc. cit.] shows that the invariant field is purely transcendental in case (4). Case (5) is treated in Proposition 7 below. Finally, all groups from case (6) are treated in Proposition 8, with the exception of W_{3}(G_{37}), which is treated in Proposition 11.

Most of the proofs are based on a criterion given by the first author in [6, Corollary 1.8], which we repeat in a form which is appropriate for the present purpose.

Proposition 2. Let f_{1}, …, f_{n} ∈ k[V]^{G} be homogeneous invariants such that the Jacobian determinant det(∂f_{i}/∂x_{j}) is non-zero and

∏_{i=1}^{n} deg(f_{i}) < 2|G|.

Then k(V)^{G} = k(f_{1}, …, f_{n}) is purely transcendental.

Another essential ingredient of the proof of Theorem 1 is the usage of a new package for invariant theory of finite groups in the computer algebra system Magma (see Bosma et al. [2], Kemper and Steel [8]). This package proved to be very efficient, and most of the calculations involved in the proof for the exceptional cases would have been utterly impossible without it.

1 The Infinite Series

In this section we prove the assertion of Theorem 1 for the remaining infinite series of groups, which are given by the cases (1), (3) and (5) from the overview of the proof.

Proposition 3. Suppose that SU_{n}(q) ≤ G ≤ GU_{n}(q) with n ≥ 3 and G ≠ SU_{3}(2), or G = SO_{2n}^{(±)}(q), where q is a power of 2 and n ≥ 2. Then k(V)^{G} is purely transcendental over k.

Proof. We first treat the case SU_{n}(q) ≤ G ≤ GU_{n}(q). As in the proof of Theorem 2.5 in Kemper [6], we work with invariants s_{1}, …, s_{n} ∈ k[V]^{GU_{n}(q)} derived directly from the sesquilinear form, which were discovered by Carlisle and Kropholler [4], and a further polynomial t, which in [6] was shown to be a relative invariant with respect to some linear character χ of GU_{n}(q). It was also shown that the Jacobian determinant of s_{1}, …, s_{n−1} and t does not vanish and that

\frac{∏_{i=1}^{n−1} deg(s_{i}) · deg(t)}{| SU_{n}(q)|} < 2.

Since the commutator subgroup of GU_{n}(q) is SU_{n}(q) for (n, q) ≠ (3, 2), χ must be a power of the determinant, say χ(σ) = det(σ)^{e}. Let m = (G : SU_{n}(q)) be the index, then G = {σ ∈ GU_{n}(q) | det(σ)^{m} = 1} and hence t^{m} ∈ k[V]^{G} in this case. It follows by Proposition 2 that s_{1}, …, s_{n−1} and t^{m} generate the invariant field k(V)^{G}. If (n, q) = (3, 2), then G must be GU_{3}(2) since SU_{3}(2) was excluded and the index is 3. In this case, k(V)^{G} is generated by s_{1}, s_{2}, s_{3} (see Carlisle and Kropholler [4]).

Now we treat the case G = SO_{2n}^{(±)}(q) with q a power of 2 and n ≥ 2. We may assume that k = 𝔽_{q}. First we take the invariants of Carlisle and Kropholler, which are (with respect to a suitable basis x_{1}, …, x_{2n} of V)

s_{0} = ∑_{j=1}^{n} x_{j}x_{2n−j+1} + \begin{cases}x_{n}^{2} + ϵx_{n+1}^{2} & if G = SO_{2n}^{−}(q) \\ 0 & if G = SO_{2n}^{+}(q) \end{cases} ,

s_{i} = P^{q^{i−1}}(s_{i−1}) = ∑_{j=1}^{2n} x_{j}^{q^{i}}x_{2n−j+1} (i > 0).

Here ϵ ∈ k is such that the polynomial X^{2} + X + ϵ ∈ k[X] is irreducible, and P^{i} denotes the i-th Steenrod operation (see, for example, Smith [12, page 324]). The Jacobian determinant of s_{0}, …, s_{2n−1} is

𝔍 = |\begin{matrix}x_{2n} & x_{2n−1} & ⋯ & x_{1} \\ x_{2n}^{q} & x_{2n−1}^{q} & ⋯ & x_{1}^{q} \\ ⋮ & ⋮ & & ⋮ \\ x_{2n}^{q^{2n−1}} & x_{2n−1}^{q^{2n−1}} & ⋯ & x_{1}^{q^{2n−1}} \end{matrix} | ≠ 0.

Furthermore, we have

\frac{∏_{i=0}^{2n−1} deg(s_{i})}{|G|} ≤ \frac{∏_{i=0}^{2n−1}(q^{i} + 1)}{2q^{n(n−1)} ∏_{i=1}^{n−1}(q^{2i} − 1) · (q^{n} − 1)} = \frac{∏_{i=n}^{2n−1}(q^{i} + 1)}{q^{n(n−1)} ∏_{i=1}^{n}(q^{i} − 1)},

which for q ≥ 4 is smaller than 2 by Kemper [6, Lemma 2.3]. In view of Proposition 2, this already completes the proof for the case q ≥ 4.

Now we consider the case q = 2. We believe that s_{0}, …, s_{2n−1} generate k(V)^{G} in this case, too, but we cannot prove this since the above fraction turns out to be greater than 2. Instead we give another system of generators. This consists of the invariants s_{0}, …, s_{2n−2} and a further invariant f, which we construct now. Let Q be the quadratic form defining G and V_{0} = {v ∈ V | Q(v) = 0 and v ≠ 0} the set of isotropic vectors. A bilinear form on V is given by ⟨v, w⟩ = Q(v + w) − Q(v) − Q(w), which explicitly is

⟨∑_{i=1}^{2n} α_{i}x_{i}, ∑_{i=1}^{2n} β_{i}x_{i}⟩ = ∑_{i=1}^{2n} α_{i}β_{2n−i+1}.

We define

f = ∑_{v∈V_{0}} p(v), where p(v) = ∏_{\begin{matrix}w∈V_{0}, \\ ⟨v,w⟩=1 \end{matrix}} w.

This clearly is an invariant. All p(v) have the same degree since G acts transitively on V_{0}. For v = x_{1}, the condition ⟨v, w⟩ = 1 means that the last coordinate of w is 1. Then the intermediate 2n−2 coordinates of w can vary freely in k = 𝔽_{2}, while the first coordinate must be chosen such that w ∈ V_{0}. Hence deg(f) = 2^{2n−2}. Now Lemma 4 below assures that ∏_{i=0}^{2n−2} deg(s_{i}) · deg(f) < 2|G|, and by Proposition 2 we only have to show that the Jacobian determinant 𝔍 of s_{1}, …, s_{2n−2}, f does not vanish. Calculating this determinant modulo x_{2n}, we see that it is sufficient to prove that

∂f/∂x_{1} ≢ 0 mod x_{2n}.

Here we can first substitute x_{2n} = 0 and then differentiate. We see that p(v) ≡ 0 mod x_{2n} if ⟨v, x_{2n}⟩ = 1, i.e., if the first coordinate v_{1} of v is one. Hence the sum only runs over those v ∈ V_{0} with v_{1} = 0. For such a v, define

p_{0}(v) = ∏_{\begin{matrix}w∈V_{0}, \\ ⟨v,w⟩=1, \\ w_{1}=0 \end{matrix}} w and p_{1}(v) = ∏_{\begin{matrix}w∈V_{0}, \\ ⟨v,w⟩=1, \\ w_{1}=1 \end{matrix}} w,

hence ∂p(v)/∂x_{1} = p_{0}(v) · ∂p_{1}(v)/∂x_{1}. Choosing a w′ ∈ V with ⟨v, w′⟩ = 1 and w′_{1} = 1 we have

p_{1}(v) = ∏_{\begin{matrix}w∈V, \\ w+w′∈V_{0}, \\ ⟨v,w⟩=0,w_{1}=0 \end{matrix}}(w + w′).

The condition “w + w′ ∈ V_{0}” can be achieved by the (unique) choice of the last coordinate w_{2n} of w, and due to v_{1} = 0, the condition ⟨v, w⟩ = 0 is independent of this choice. Setting

W (v) = {w ∈ V | w_{1} = w_{2n} = 0 and ⟨v, w⟩ = 0},

we obtain

p_{1}(v) ≡ ∏_{w∈W (v)}(w + w′) mod x_{2n}.

But W (v) is a subspace of V, thus ∏_{w∈W (v)}(X +w) ∈ k[V][X] only involves monomials of the form

X^{2^{i}} (see, for example, Wilkerson [13, Proposition 1.1]), and

∂p_{1}(v)/∂x_{1} ≡ ∏_{\begin{matrix}w∈W (v), \\ w≠0 \end{matrix}} w mod x_{2n}.

Now suppose that v ≠ x_{2n}. Then also v′ := v + x_{2n} ∈ V_{0} (since v_{1} = 0). We have W (v′) = W (v) and p_{0}(v′) = p_{0}(v), hence v and v′ contribute the same values to the sum ∂f/∂x_{1} mod x_{2n}, which cancel each other. We are left with

∂f/∂x_{1} ≡ p_{0}(x_{2n}) ∏_{\begin{matrix}w∈W (x_{2n}), \\ w≠0 \end{matrix}} w mod x_{2n},

where p_{0}(x_{2n}) = 1. This proves ∂f/∂x_{1} ≢ 0 mod x_{2n}, and we are done.

The following lemma was used in the above proof.

Lemma 4. For n ≥ 2, the inequality

\frac{∏_{i=0}^{2n−2}(2^{i} + 1) · 2^{2n−2}}{2 · 2^{n(n−1)} · ∏_{i=1}^{n−1}(2^{2i} − 1) · (2^{n} − 1)} < 2

holds.

Proof. The left hand side of the inequality is

\frac{∏_{i=1}^{2n−2}(2^{i} + 1)}{(2 − 2^{1−n}) · 2^{(n−1)^{2}} · ∏_{i=1}^{n−1}(2^{2i} − 1)} = \underbrace{\frac{1}{2 − 2^{1−n}}}_{=:ρ_{1}} ·\frac{∏_{i=n}^{2n−2}(2^{i−n+1} + 2^{1−n})}{∏_{i=1}^{n−1}(2^{i} − 1)} = ρ_{1} · ∏_{i=1}^{n−1} (\frac{2^{i} + 2^{1−n}}{2^{i} − 1} ) ,

which for n ≥ 5 is bounded from above by

\frac{1}{2 − 2^{−4}} · ∏_{i=1}^{4} (1 + \frac{1 + 2^{−4}}{2^{i} − 1} ) · \underbrace{∏_{i=5}^{n−1} (1 + \frac{1 + 2^{−4}}{2^{i} − 1} )}_{=:ρ_{2}} = \frac{1 580 293}{888 832} · ρ_{2}.

Using ln(1 + x) ≤ x for x ≥ 0, we obtain

ln(ρ_{2}) ≤ ∑_{i=5}^{∞} \frac{1 + 2^{−4}}{2^{i} − 1} ≤ (1 + 2^{−4}) ∑_{i=5}^{∞} \frac{1}{2^{i} − 2^{i}/2^{5}} = \frac{1 + 2^{−4}}{(1 − 2^{−5}) · 2^{5}} \frac{1}{1 − 1/2} = \frac{17}{248}.

Gathering the results yields the upper bound

\frac{1 580 293}{888 832} · e^{17/248} ≈ 1.904092843.

Finally, the lemma is verified for 2 ≤ n ≤ 4 by direct calculation.

The next goal is the proof for G = 𝔖_{n+2}. For this we have to introduce some tools. If K is a field, G a group of automorphisms of K, and V is a vector space over K, then an action of G on V is called semilinear if σ(αv + βw) = σ(α)σ(v) + σ(β)σ(w) for all σ ∈ G, α, β ∈ K and v, w ∈ V.

Lemma 5. Let a finite group G of automorphisms of a field K act semilinearly on a K-vector space V. Then there exists a basis of V consisting of G-invariant vectors.

Proof. Take a basis α_{1}, …, α_{n} of K as a vector space over K^{G}. Then the determinant of the matrix (σ(α_{i})) indexed by σ ∈ G and i = 1, …, n is non-zero. For v ∈ V we have

∑_{σ∈G} σ(α_{i})σ(v) = ∑_{σ∈G} σ(α_{i}v) ∈ V^{G}.

By matrix inversion we can therefore represent v as a linear combination of vectors in V^{G}. This yields the result.

The above proof was shown to us by Vladimir Popov. The referee of this paper pointed out that the lemma also follows from the fact that the category of all semilinear G-actions is semisimple.

In the general situation of a finite linear group G ≤ GL(V), a rational function f ∈ k(V) is called homogeneous of degree d if it can be written as a fraction g/h of homogeneous polynomials with deg(g) − deg(h) = d. We write

k(V)_{0} = {f ∈ k(V) | f is homogeneous of degree 0} ∪ {0}

and k(V)_{0}^{G} for the invariants contained in k(V)_{0}. This is a subfield of k(V)^{G}, and k(V)^{G} is purely transcendental over k(V)_{0}^{G}, generated by any homogeneous rational invariant g of minimal positive degree (Kemper [6, Proposition 1.1(a)]). The degree of such a generator g is equal to the number of scalar matrices contained in G, and the subgroup of scalar matrices in G is exactly the kernel of the action of G on k(V)_{0} ([loc. cit., Proposition 1.3]). Part (a) of the next result is contained in Miyata [9].

Proposition 6. Let G ≤ GL(V) be a finite linear group and suppose that U ≤ V is a kG-submodule of V. Then

(a) If G acts faithfully on U, then k(V)^{G} is purely transcendental over k(U)^{G}.

(b) If G acts faithfully on k(U)_{0}, then k(V)_{0}^{G} is purely transcendental over k(U)_{0}^{G}.

Proof. Take a minimal system x_{1}, …, x_{m} ∈ V which together with U generates V as a vector space over k. For the statement (a), set K := k(U) and W := Kx_{1} + ⋯ + Kx_{m}. Then G acts semilinearly on W, so by Lemma 5 there exists a basis y_{1}, …, y_{m} ∈ W^{G} of W. Therefore k(V) = K(x_{1}, …, x_{m}) = K(y_{1}, …, y_{m}) and k(V)^{G} = K^{G}(y_{1}, …, y_{m}). This is purely transcendental over K^{G}.

For (b), let g ∈ k(U)^{G} be homogeneous of degree 1, and set K := k(U)_{0} and W := Kx_{1}/g+⋯+ Kx_{m}/g. As above, there exists a basis y_{1}, …, y_{m} ∈ W^{G} of W, and k(V)_{0}^{G} = K^{G}(y_{1}, …, y_{m}).

As an application, we obtain:

Proposition 7. If G = 𝔖_{n+2} with p = char(k) | (n + 2) and n ≥ 3, then k(V)^{G} is purely transcendental over k.

Proof. Take the natural (n+2)-dimensional permutation module V_{n+2} of G with basis x_{1}, …, x_{n+2}, and let V_{n+1} = V_{n+2}/(k · (x_{1} + ⋯ + x_{n+2})). Then k[V_{n+1}]^{G} is generated by the projections of the elementary symmetric polynomials s_{2}, …, s_{n+2} ∈ k[V_{n+2}]^{G} to k[V_{n+1}]^{G}. Indeed, the quotient ring of k[V_{n+1}] modulo the ideal generated by these projections is isomorphic to k[V_{n+2}]/(s_{1}, …, s_{n}), which is of Krull dimension zero, and the product of the degrees of the s_{i} is equal to |G|. From this it follows by Smith [12, Prop. 5.5.5] that the projections of the s_{i} generate k[V_{n+1}]^{G}. Hence k(V_{n+1})^{G} is purely transcendental over k, and so is k(V_{n+1})_{0}^{G} by Kemper [6, Proposition 1.1]. We are interested in the invariant field k(V)^{G}, where V is the submodule of V_{n+1} consisting of the classes modulo k · (x_{1} + ⋯ + x_{n+2}) of vectors of V_{n+2} with coordinate sum zero. Since the action of G on V is faithful, every σ ∈ G which acts on V as a scalar must lie in the center of G, which is trivial. Hence G acts faithfully on k(V)_{0}, and k(V_{n+1})_{0}^{G} is purely transcendental over k(V)_{0}^{G} by Proposition 6(b). Thus both upper extensions in the following diagram are purely transcendental of transcendence degree 1.

[figure: k(V)^{G}; k(V_{n+1})_{0}^{G}; k(V)_{0}^{G}; k]

From this we obtain an isomorphism between k(V)^{G} and k(V_{n+1})_{0}^{G}. But k(V_{n+1})_{0}^{G} is purely transcendental over k and hence so is k(V)^{G}.

We remark that the system of algebraically independent generators for k(V)^{G} obtained by this proof does not consist of homogeneous invariants.

2 The Exceptional Groups

In this section we prove Theorem 1 for the exceptional groups listed in case (6) from the overview of the proof. Much of this depends on extensive computations that were performed on a computer using the system Magma (see Bosma et al. [2]). In order to make all computations transparent and reproducible at any time, we wrote Magma-input files which run the computations. Upon request we can provide these files for readers of the paper.

Proposition 8. If G is one of the groups W_{3}(G_{30}), W_{3}(G_{31}), W_{5}(G_{32}), W_{2}(G_{34}), W_{3}(G_{36}) or W_{5}(G_{37}), then the invariant field k(V)^{G} is purely transcendental over k.

Proof. For each group G = W_{p}(G_{k}), we give the corresponding group Ĝ = W_{0}(G_{k}) in characteristic 0. We then produce invariants of Ĝ by choosing an orbit Ĝ · v of a vector v and forming i-th elementary symmetric polynomials of this orbit, which will always be denoted by s_{i}. This method is only feasible for relatively small degrees, and we construct further invariants of Ĝ by such means as Hessian determinants, bordered Hessians, Laplacians and scalar products of gradients. Details shall be given below. We then obtain invariants of G by forming suitable rational linear combinations of Ĝ-invariants, checking with the computer that their coefficients are in fact integral, and reducing modulo p. Sometimes additional G-invariants are produced by means of Steenrod operations. Using these techniques, we construct n = dim(V) homogeneous invariants g_{i} for each group. We then check that their Jacobian determinant does not vanish (again by using the computer and sometimes suitably specializing some of the variables), and that ∏_{i=1}^{n} deg(g_{i}) < 2|G|. By Proposition 2, this is sufficient. Before addressing the various cases, let us remark that many of the invariants were first obtained by means of linear algebra, i.e., by computing all invariants of some given degrees (see Kemper and Steel [8]). Only after we knew the degrees of the generating invariants did we start to look for the “recipes” to construct them which are given here. This way of double checking adds to the certainty that the results from our computations are correct.

For the group G = W_{3}(G_{30}) of order 14 400 (which is the 3-modular reduction W_{3}(H_{4}) of the Coxeter group of type H_{4}), Ĝ can be generated by the matrices

(\begin{matrix}−1 & 0 & 0 & 0 \\ α & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & α & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 1 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & −1 \end{matrix}) ∈ GL_{4} (ℚ(\sqrt{5})) ,

where α = (1 + \sqrt{5})/2. The orbit of the first basis vector x_{1} has length 120, and we take elementary symmetric polynomials s_{2}, s_{12} and s_{20} of it. Then f_{2} = s_{2}/60 is integral, and we take g_{2} to be its 3-modular reduction, and g_{10} = P^{1}(g_{2}) the first Steenrod operation. We verify that

\frac{3α − 2}{3}s_{12}^{2} − \frac{1}{3}f_{2}^{2}s_{20} + (α − 1)f_{2}^{6}s_{12} + \frac{3α − 1}{3}f_{2}^{12}

is integral and take g_{24} to be its reduction. This turns out to be divisible by g_{2}, and we set g_{22} = g_{24}/g_{2}. Finally, take g_{36} to be the reduction of s_{36}. Then g_{2}, g_{10}, g_{22} and g_{36} generate k(V)^{G}. It would lead too far to explain how the coefficients in the above linear combination were found. For the proof it is sufficient to verify that they fulfill their purpose.

For the group G = W_{3}(G_{31}) of order 46 080, Ĝ can be generated by the matrices

(\begin{matrix}0 & −1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , \frac{1}{4} (\begin{matrix}2α & −2α & 2α & −2α + 2 \\ −2 & −2 & −2 & −2α − 2 \\ 2α & −2α & −2α & 2α − 2 \\ α − 1 & α − 1 & −α + 1 & 2 \end{matrix}) ∈ GL_{4} (ℚ(α))

where α^{2} = −1. A first invariant of G is g_{4} = x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + 2x_{4}^{4}. We construct invariants of Ĝ by taking elementary symmetric polynomials s_{i} of the Ĝ-orbit of 2x_{1}, which has length 240. We find that

\frac{2}{9}s_{20} + \frac{5}{9}s_{8}s_{12} and \frac{1}{27}s_{12}^{2} + \frac{938}{729}s_{8}^{3}

are integral and take their reductions g_{20} and g_{24}. A further G-invariant is g_{28} = P^{1}(g_{20}). It is now verified that g_{4}, g_{20}, g_{24} and g_{28} generate k(V)^{G}.

Next we consider G = W_{5}(G_{32}) of order 155 520, for which Ĝ can be generated by

(\begin{matrix}ω & 0 & 0 & 0 \\ −ω & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 1 & 0 & 0 \\ 0 & ω & 0 & 0 \\ 0 & −ω & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & ω & 0 \\ 0 & 0 & −ω & 1 \end{matrix}) , (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & ω \end{matrix}) ∈ GL_{4} (ℚ(ω)) ,

where ω^{2} + ω + 1 = 0. The orbit of x_{1} has length 240, and we take the elementary symmetric polynomials s_{12}, s_{18} and s_{24}. Then

f_{12} = \frac{s_{12}}{42} , f_{18} = \frac{s_{18}}{2184} and f_{24} = \frac{s_{24}}{21}

are integral, and the reduction of f_{12} is a square g_{6}^{2} with g_{6} ∈ k[V]^{G}. Furthermore, let g_{24} be the reduction of f_{24}. We find that

\frac{9}{125}f_{12}^{3} + \frac{104}{125}f_{18}^{2} − \frac{1}{5}f_{12}f_{24}

is integral and reduce it to obtain g_{36} ∈ k[V]^{G}. It is then verified that g_{6}, g_{24}, g_{36} and g_{48} = P^{1}(g_{24}) generate k(V)^{G}.

For G = W_{2}(G_{34}), the characteristic 0 group Ĝ is generated by the permutations (1,2), (2,3), (3,4), (4,5) and (5,6), together with the matrices

(\begin{matrix}0 & ω & 0 & 0 & 0 & 0 \\ ω^{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) , \frac{1}{3} (\begin{matrix}ω + 2 & ω − 1 & ω + 2 & 0 & 0 & 0 \\ ω − 1 & ω + 2 & ω + 2 & 0 & 0 & 0 \\ −2ω − 1 & −2ω − 1 & ω + 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & −ω + 1 & −ω − 2 & −ω − 2 \\ 0 & 0 & 0 & −ω − 2 & −ω + 1 & −ω − 2 \\ 0 & 0 & 0 & −ω − 2 & −ω − 2 & −ω + 1 \end{matrix}) ,

where ω^{2} + ω + 1 = 0. Here Ĝ has order 39 191 040, and its reduction G has half this order. The orbit M of 3(−ωx_{1} + x_{2}) has length 756. Here s_{i} denotes the i-th power sum, i.e., s_{i} = ∑_{f∈M} f^{i}. We calculate s_{6}, s_{12} and s_{18} (beyond which evaluations of s_{i} are not possible due to storage requirements) and set

f_{6} = \frac{1}{8748}s_{6}, f_{12} = \frac{−1}{177147000}s_{12} + \frac{17}{375}f_{6}^{2},
f_{18} = \frac{−ω}{8930472738660}s_{18} + \frac{911ω}{1400355}f_{6}^{3} − \frac{2702ω}{31119} f_{12}f_{6}

which are all integral. Furthermore, the Hessian determinant h_{24} = det(∂^{2}f_{6}/∂x_{i}∂x_{j}) is an invariant since the matrices from Ĝ all have determinant ±1 (see, for example, Smith [12, p. 20]), and we set

f_{24} = \frac{−ω − 1}{23328000000}h_{24} + \frac{1}{4}f_{12}^{2} − \frac{ω + 1}{4}f_{18}f_{6}.

We shall need one additional invariant of Ĝ, which is obtained from the bordered Hessian of f_{6} and f_{12}, which is by definition the 7 × 7 determinant

h_{42} = det ( \begin{matrix}& & & ∂f_{12}/∂x_{1} \\ & \frac{∂^{2}f_{6}}{∂x_{i}∂x_{j}} & & ⋮ \\ & & & ∂f_{12}/∂x_{6} \\ ∂f_{12}/∂x_{1} & ⋯ & ∂f_{12}/∂x_{6} & 0 \end{matrix} ) .

As for the Hessian determinant, if follows by the chain rule that this is an invariant, and its degree is 42. Set f_{42} = −h_{42}/3499200000, which is integral. We obtain the following invariants of G: The reduction of f_{6} is a square g_{3}^{2} with g_{3} ∈ k[V]^{G}. In fact, we obtain g_{3} = x_{1}^{3} + ⋯ + x_{6}^{3}. Furthermore, we take g_{9} = P^{2}(g_{3}). Now we reduce

(f_{18}f_{6} + ωf_{12}^{2} − 4ωf_{24})/4

and obtain an invariant g′_{24} which is divisible by g_{3}, hence g_{21} = g′_{24}/g_{3} ∈ k[V]^{G}. Now we reduce f_{24} to obtain g_{24}, take g_{33} = P^{8}(g_{9}) and finally reduce

(−f_{12}^{3}f_{6}^{2} + 49f_{12}^{4} + (ω + 1)f_{18}f_{12}f_{6}^{3} + 36(ω + 1)f_{18}f_{12}^{2}f_{6} + 43ωf_{18}^{2}f_{6}^{2} + 4f_{24}f_{12}f_{6}^{2} −

8f_{24}f_{12}^{2} − 16(ω + 1)f_{24}f_{18}f_{6} + 16f_{24}^{2} − 4(ω + 1)f_{42}f_{6})/64

to an invariant g_{48}. Then g_{3}, g_{9}, g_{21}, g_{24}, g_{33} and g_{48} generate k(V)^{G}.

We now turn to the group G = W_{3}(G_{36}) = W_{3}(E_{7}) (in Coxeter notation), whose order is 2 903 040. Here Ĝ can be generated by the permutations (1,2), (2,3), (3,4), (4,5), and (5,6), together with the matrices

(\begin{matrix}0 & −1 & 0 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) , \frac{1}{8} (\begin{matrix}6 & −2 & −2 & −2 & −2 & −2 & −8 \\ −2 & 6 & −2 & −2 & −2 & −2 & −8 \\ −2 & −2 & 6 & −2 & −2 & −2 & −8 \\ −2 & −2 & −2 & 6 & −2 & −2 & −8 \\ −2 & −2 & −2 & −2 & 6 & −2 & −8 \\ −2 & −2 & −2 & −2 & −2 & 6 & −8 \\ −1 & −1 & −1 & −1 & −1 & −1 & 4 \end{matrix}) ∈ GL_{7}(ℚ).

The Ĝ-orbit of 2x_{1} + 4x_{7} has length 56, and we take the elementary symmetric polynomials s_{2}, s_{6}, s_{8}, s_{10} and s_{18} in that orbit. Then s_{2}/6 is integral and its reduction yields an invariant g_{2}. Furthermore, we set g_{4} = P^{1}(g_{2}) and g_{10} = P^{3}(g_{4}). The resulting invariants are g_{i} = x_{1}^{i}+⋯+x_{6}^{i}−x_{7}^{i} (i = 2, 4, 10). A further invariant g′_{10} of degree 10 is obtained by reducing

(−29s_{2}^{5} + 702s_{6}s_{2}^{2} + 810s_{8}s_{2} + 405s_{10})/3^{6},

which is integral. Now we set g_{16} = P^{3}(g′_{10}), g_{22} = P^{3}(g_{16}), and take g_{18} to be the reduction of s_{18}. Then it is verified that g_{2}, g_{4}, g_{10}, g′_{10}, g_{16}, g_{18} and g_{22} generate k(V)^{G}.

Finally, for G = W_{5}(G_{37}) = W_{5}(E_{8}) of order 696 729 600, Ĝ can be generated by the permutations (1,2,3,4,5,6,7), and (7,8), and the matrices

(\begin{matrix}0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ −1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) , \frac{1}{4} (\begin{matrix}3 & −1 & −1 & −1 & −1 & −1 & −1 & −1 \\ −1 & 3 & −1 & −1 & −1 & −1 & −1 & −1 \\ −1 & −1 & 3 & −1 & −1 & −1 & −1 & −1 \\ −1 & −1 & −1 & 3 & −1 & −1 & −1 & −1 \\ −1 & −1 & −1 & −1 & 3 & −1 & −1 & −1 \\ −1 & −1 & −1 & −1 & −1 & 3 & −1 & −1 \\ −1 & −1 & −1 & −1 & −1 & −1 & 3 & −1 \\ −1 & −1 & −1 & −1 & −1 & −1 & −1 & 3 \end{matrix})

The Ĝ-orbit of 2(x_{1} + x_{2}) has length 240, and we take the elementary symmetric polynomials s_{2} and s_{8}. We have f_{2} := −s_{2}/120 = x_{1}^{2} + ⋯ + x_{8}^{2}, which shows that Ĝ lies in the orthogonal group GO_{8}(ℝ). We write ∇f = (∂f/∂x_{1}, …, ∂f/∂x_{8}) for the gradient of f. It follows by application of the chain rule that the scalar product ⟨∇f, ∇g⟩ of the gradients of two invariants f and g is again an invariant. Moreover, the Laplacian Δf = ∂^{2}f/∂x_{1}^{2} + ⋯ + ∂^{2}f/∂x_{8}^{2} of an invariant f is an invariant. We form

f_{14} = ⟨∇s_{8}, ∇s_{8}⟩ and f_{12} = Δf_{14}.

The expressions

(3f_{14} − 2f_{2}f_{12} + 12f_{2}^{3}s_{8} − 8f_{2}^{7})/5 and (22s_{8}^{2} − 52f_{2}f_{14} − 43f_{2}^{2}f_{12} − 3f_{2}^{4}s_{8} + 18f_{2}^{8})/25

are integral and we take their reductions g_{14} and g_{16} ∈ k[V]^{G}. Furthermore, let g_{2} be the reduction of f_{2} and set

g_{6} = P^{1}(g_{2}), g_{20} = P^{1}(g_{16}), g_{24} = P^{1}(g_{20}), g_{26} = P^{5}(g_{6}), and g_{36} = P^{5}(g_{16}).

Then g_{2}, g_{6}, g_{14}, g_{16}, g_{20}, g_{24}, g_{26}, and g_{36} generate k(V)^{G}

The proof for the group W_{3}(G_{37}) = W_{3}(E_{8}) turns out to be harder than the previous ones since there do not exist 8 algebraically independent invariants whose degree product is smaller than twice the group order. Before coming to this proof, we have to make some remarks on Steenrod operations.

If V is a vector space over the finite field 𝔽_{q}, then an action of the Steenrod algebra P^{∗} on 𝔽_{q}(V) can be defined by an obvious generalization of the action on 𝔽_{q}[V] as given by Smith [12, page 324]: Take the algebra 𝔽_{q}(V)[[ξ]] of formal power series in ξ over 𝔽_{q}(V) and define a homomorphism P: 𝔽_{q}(V) → 𝔽_{q}(V)[[ξ]] of 𝔽_{q}-algebras by

P (v) = v + v^{q}ξ for v ∈ V.

Then the i-th Steenrod operations P^{i} are defined by P (f) = ∑_{i=0}^{∞} P^{i}(f) · ξ^{i} for f ∈ 𝔽_{q}(V). The product formulas

P^{k}(f · g) = ∑_{i=0}^{k} P^{i}(f) · P^{k−i}(g)

follow immediately. A subfield L ≤ 𝔽_{q}(V) containing 𝔽_{q} is called a field over the Steenrod algebra P^{∗} if P^{i}(f) ∈ L for all f ∈ L and i ∈ ℕ. As an example, invariant fields of linear groups are fields over the Steenrod algebra since the above homomorphism P commutes with 𝔽_{q}-linear transformations.

Proposition 9. Let V be a vector space over the prime field 𝔽_{p}, and let L = 𝔽_{p}(f_{1}, …, f_{m}) ≤ 𝔽_{p}(V) be a field generated by polynomials f_{1}, …, f_{m} ∈ 𝔽_{p}[V] such that

P^{p^{j}}(f_{i}) ∈ L for 1 ≤ i ≤ m and j ∈ ℕ_{0}.

Then L is a field over the Steenrod algebra P^{∗}.

It was indicated to us by Mara D. Neusel and Larry Smith that related results already appeared at various places in the literature. Nevertheless, we could not find an explicit proof, so for the convenience of the reader we give a short argument here.

Proof. We prove by induction on k that P^{k}(f) ∈ L for all f ∈ L. This is true for k = 0, so we assume that k > 0.

First we show that P^{k}(f_{i}) ∈ L for 1 ≤ i ≤ m. If k is a power of p, then this is true by assumption. Otherwise, take j ∈ ℕ_{0} such that p^{j} < k < p^{j+1} and set l = k − p^{j}. We have

l ≤ (p − 1)p^{j} − 1 = (p − 2) · p^{j} + ∑_{ν=0}^{j−1}(p − 1) · p^{ν}.

This means that every coefficient in the p-adic expansion of l is smaller than or equal to the corresponding coefficient in the expansion of (p−1)p^{j} −1, which implies that the binomial coefficient \binom{(p−1)p^{j}−1}{l} is non-zero in 𝔽_{p}. We consider the Adem relation

P^{l}P^{p^{j}}(f_{i}) = (−1)^{l} · \binom{(p − 1)p^{j} − 1}{l}P^{k}(f_{i}) + ∑_{ν=1}^{[l/p]}(−1)^{l+ν} · \binom{(p − 1)(p^{j} − ν) − 1}{l − pν}P^{k−ν}P^{ν}(f_{i}),

where [l/p] is the greatest integer ≤ l/p (see Bullett and Macdonald [3]). By the induction hypothesis, all terms except for the one containing P^{k}(f_{i}) are contained in L, hence also P^{k}(f_{i}) ∈ L since the equation can be solved for P^{k}(f_{i}).

By the multiplicative property of the Steenrod operations it follows immediately now that P^{k}(f) ∈ L for f ∈ 𝔽_{p}[f_{1}, …, f_{m}]. Also, for f, g ∈ 𝔽_{p}[f_{1}, …, f_{m}] with g ≠ 0 we have

P^{k}(f) = P^{k}(f/g · g) = P^{k}(f/g) · g + ∑_{i=1}^{k} P^{k−i}(f/g)P^{i}(g).

This proves P^{k}(L) ⊂ L.

The next proposition is just a reformulation of the result of Carlisle and Kropholler [4] and a result contained in the proof of Proposition 3.

Proposition 10. Let G = GO_{n}^{(±)}(q) be an orthogonal group acting on a vector space V over 𝔽_{q} with q > 2, defined by an invariant f_{2} ∈ 𝔽_{q}[V]^{G} of degree 2. Then 𝔽_{q}(V)^{G} is generated by f_{2} as a field over the Steenrod algebra P^{∗}.

We can now complete the proof of Theorem 1 by determining the invariant field of W_{3}(E_{8}).

Proposition 11. Let G = W_{3}(G_{37}) = W_{3}(E_{8}). Then the invariant field k(V)^{G} is purely transcendental over k.

Proof. We apply similar techniques as in the proof of Proposition 8 to obtain invariants of G by using the corresponding group Ĝ = W_{0}(E_{8}) in characteristic 0. As in that proof (where generators for Ĝ were given on page 9, we take the Ĝ-orbit of 2(x_{1} + x_{2}) of length 240 and its elementary symmetric polynomials s_{2} and s_{8}. With the notation introduced in the proof of Proposition 8, we form the Ĝ-invariants

f_{2} = \frac{−1}{120}s_{2}, f_{8} = \frac{−1}{384}s_{8} + \frac{159397}{8}f_{2}^{4},

h_{14} = ⟨∇f_{8}, ∇f_{9}⟩, h_{12} = Δh_{14},

f_{12} = \frac{1}{1568}h_{12} − \frac{1}{28}f_{2}^{6} − \frac{15}{14}f_{2}^{2}f_{8}, f_{14} = \frac{13}{420}h_{14} + \frac{13}{105}f_{2}^{3}f_{8} + \frac{2}{15}f_{2}f_{12},

h_{18} = ⟨∇f_{8}, ∇f_{12}⟩, f_{18} = \frac{1}{24}h_{18},

h_{20} = ⟨∇f_{8}, ∇f_{14}⟩, f_{20} = \frac{1}{8}h_{20},

h_{24} = ⟨∇f_{12}, ∇f_{14}⟩, f_{24} = \frac{1}{24}h_{24}.

Now we can construct invariants of G. First, let g_{2} be the reduction of f_{2} and set g_{4} = P^{1}(g_{2}), g_{10} = P^{3}(g_{4}), and g_{28} = P^{9}(g_{10}). Next we convince ourselves that the linear combinations

f′_{24} = (−10f_{12}^{2} + 16f_{8}^{3} + 18f_{2}f_{8}f_{14} + 30f_{2}^{2}f_{8}f_{12} + 12f_{2}^{4}f_{8}^{2} − 9f_{2}^{5}f_{14} + 35f_{2}^{6}f_{12})/3^{3},
f_{26} = (54f_{12}f_{14} − 305f_{8}f_{18} + 270f_{2}f_{12}^{2} + 161f_{2}f_{8}^{3} − 6f_{2}^{2}f_{8}f_{14} − 340f_{2}^{3}f_{8}f_{12} +
154f_{2}^{4}f_{18} − f_{2}^{5}f_{8}^{2} − 222f_{2}^{6}f_{14} − 313f_{2}^{7}f_{12})/3^{5}, and
f_{28} = (81f_{14}^{2} + 189f_{8}f_{20} + 216f_{8}^{2}f_{12} + 27f_{2}f_{12}f_{14} + 233f_{2}f_{8}f_{18} + 324f_{2}^{2}f_{24} +
72f_{2}^{2}f_{12}^{2} + 271f_{2}^{2}f_{8}^{3} − 318f_{2}^{3}f_{8}f_{14} + 189f_{2}^{4}f_{20} + 331f_{2}^{4}f_{8}f_{12} + 269f_{2}^{5}f_{18} +
10f_{2}^{6}f_{8}^{2} − 156f_{2}^{7}f_{14} − 146f_{2}^{8}f_{12})/3^{5}

are integral, and take their reductions g_{24}, g_{26} and g′_{28}. Finally, we set g_{42} = P^{9}(g_{24}). We now claim that

L := k(g_{2}, g_{4}, g_{10}, g_{24}, g_{26}, g_{28}, g′_{28}, g_{42}) = k(V)^{G}.

From now on, we only work in characteristic 3. First, we verify that the Jacobian determinant of the generators of L is non-zero, hence they are algebraically independent. Unfortunately, the degree product of the generators exceeds 2 · |G|, so we are not finished by the usual argument. But we see that, due to g_{2}, G ≤ GO_{8}^{+}(3). If we can show that L is a field over the Steenrod algebra P^{∗}, then k(V)^{GO_{8}^{+}(3)} ≤ L by Proposition 10, and by Galois theory L is the fixed field of an intermediate group G′ with G ≤ G′ ≤ GO_{8}^{+}(3). Moreover, the index (G′ : G) is bounded from above by 2 since the degree product of the generators of L is smaller than 3·|G|. But there is only one such intermediate group G′, namely G′ = G, as can easily be extracted from the Atlas (Conway et al. [5]).

Hence we have to show that L is a field over the Steenrod algebra by verifying the assumptions of Proposition 9. In order to show that P^{3^{j}}(f) ∈ L for a generator f and j ∈ ℕ, we try to solve the equation

h_{d} · P^{3^{j}}(f) = g_{deg(f)+2·3^{j}+d}

for some d ∈ ℕ_{0}, where h_{d} and g_{deg(f)+2·3^{j}+d} are general elements of k[g_{2}, g_{4}, g_{10}, g_{24}, g_{26}, g_{28}, g′_{28}, g_{42}] of degrees d and deg(f)+2·3^{j} +d, respectively. Comparing coefficients in the above equation yields a homogeneous system of linear equations for the coefficients of h_{d} and g_{deg(f)+2·3^{j}+d}, which we try to solve non-trivially. If this is successful, then a representation of P^{3^{j}}(f) as an element of L has been found. Otherwise, we pass to a larger d. This approach was successful for all f and j with 3^{j} ≤ deg(f). Since P^{l}(f) = 0 for l > deg(f), this suffices. The largest P^{3^{j}}(f) that has to be represented is P^{27}(g_{42}) of degree 96. Here the calculation was successful for d = 36, i.e., we could represent P^{27}(g_{42}) as a quotient of an element of degree 132 by one of degree 36. To make the calculations possible, we used the subgroup H ≤ G generated by the first three generators of G as given on page 9. H is the irreducible imprimitive reflection group W_{3}(D_{8}). It has order 5 160 960 and fundamental invariants of degrees 2,4,6,8,8,10,12, and 14. All invariants of G can thus be expressed as polynomials in these fundamental invariants, and it was these representations that we used for the calculations. The largest linear system that we had to solve had 4 150 unknowns. The solution was found by first specializing all but two of the variables (which are, in fact, the invariants of H) to random linear combinations of these two to get a larger solution space and then, repeating this process, “funneling down” the solution space until a space of dimension one was reached. The solution obtained in this way was then verified without any specialization.

The total computation time for this group was about 34 hours on a Sun Ultrasparc workstation. The computation required more than 600 Mbytes of memory.

Remark. We find it remarkable to what extent the degrees of invariants found for the p-modular reductions of complex reflection groups differ from the degrees in characteristic zero. ◁

3 Reducible Groups

If G ≤ GL(V) is a finite linear group and U ≤ V is a faithful submodule of V such that G acts on U as an irreducible reflection group, then k(V)^{G} is purely transcendental over k by Theorem 1 and Proposition 6(a). G need not be a reflection group on V for this. On the other hand, not all reflection groups are covered this way, since the socle of V is not always faithful if G is a reflection group. The most striking examples are reflection groups which are p-groups, where p = char(k). But if G is a p-group, then k(V)^{G} is always purely transcendental (see Miyata [9]). In fact, we see from the proof given in [9] that k(V)^{G} is purely transcendental over k(U)^{G}, whenever U ≤ V is a submodule of codimension 1. Combining this and Proposition 6(a) with Theorem 1, we obtain:

Corollary 12. Let G ≤ GL(V) be a finite linear group and let

U_{0} < U_{1} < … < U_{r} = V

be a chain of kG-submodules such that G acts as an irreducible reflection group on U_{0} (possibly not faithfully), and for each i = 1, …, r one (or both) of the following holds:

(a) dim_{k}(U_{i}/U_{i−1}) = 1, or

(b) the G-actions on U_{i} and on U_{i−1} have the same kernel.

Then k(V)^{G} is purely transcendental over k.

If G ≤ GL(V) is a reflection group and V = V_{1}⊕⋯⊕V_{r} is a decomposition into kG-submodules, then k(V)^{G} ≅ k(V_{1})^{G} ⊗ ⋯ ⊗ k(V_{r})^{G}. Hence if the assumption of the above corollary is true for all V_{i}, then k(V)^{G} is purely transcendental also.

We do not know whether Theorem 1 remains true if the assumption that G be irreducible is dropped. It would be nice to have a proof of Theorem 1 and possible generalizations which does not use the classification of irreducible reflection groups.

References

[1] David J. Benson, Polynomial Invariants of Finite Groups, Lond. Math. Soc. Lecture Note Ser. 190, Cambridge Univ. Press, Cambridge 1993.

[2] Wieb Bosma, John J. Cannon, Catherine Playoust, The Magma Algebra System I: The User Language, J. Symbolic Computation 24 (1997), 235–265.

[3] S. R. Bullett, I. G. Macdonald, On the Adem Relations, Topology 21 (1982), 329–332.

[4] David Carlisle, Peter H. Kropholler, Rational Invariants of Certain Orthogonal and Unitary Groups, Bull. London Math. Soc. 24 (1992), 57–60.

[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford 1985.

[6] Gregor Kemper, A Constructive Approach to Noether’s Problem, Manuscripta Math. 90 (1996), 343–363.

[7] Gregor Kemper, Gunter Malle, The Finite Irreducible Linear Groups with Polynomial Ring of Invariants, Transformation Groups 2 (1997), 57–89.

[8] Gregor Kemper, Allan Steel, Some Algorithms in Invariant Theory of Finite Groups, in: P. Dräxler, G.O. Michler, C. M. Ringel, eds., Computational Methods for Representations of Groups and Algebras, Euroconference in Essen, April 1-5 1997, Progress in Mathematics 173, Birkhäuser, Basel 1999.

[9] Takehiko Miyata, Invariants of Certain Groups I, Nagoya Math. J. 41 (1971), 69–73.

[10] Haruhisa Nakajima, Invariants of Finite Groups Generated by Pseudo-Reflections in Positive Characteristic, Tsukuba J. Math. 3 (1979), 109–122.

[11] G. C. Shephard, J. A. Todd, Finite Unitary Reflection Groups, Canad. J. Math. 6 (1954), 274–304.

[12] Larry Smith, Polynomial Invariants of Finite Groups, A. K. Peters, Wellesley, Mass. 1995.

[13] Clarence Wilkerson, A Primer on the Dickson Invariants, Amer. Math. Soc. Contemp. Math. Series 19 (1983), 421–434.

Gregor Kemper  Gunter Malle
IWR, Mathematisches Institut  FB Mathematik/Informatik
Universität Heidelberg  Universität Gh Kassel
Im Neuenheimer Feld 368  Heinrich-Plett-Str. 40
69 120 Heidelberg  34 132 Kassel
Germany  Germany

Gregor.Kemper@iwr.uni-heidelberg.de  malle@mathematik.uni-kassel.de