Invariant rings and fields of finite groups

Gregor Kemper and Gunter Malle

IWR der Universität Heidelberg Im Neuenheimer Feld 368 69120 Heidelberg Germany

Abstract. We consider polynomials and rational functions which are invariant under the action of a finite linear group. The aim is to give a survey over the knowledge on some structural properties of such rings and fields of invariants. Particular emphasis lies on the modular case, where the characteristic of the ground field divides the group order.

1 Introduction

The study of rings and fields of invariants of linear groups has a long history. In classical invariant theory, which had its golden age before and around the turn of the century, the focus was on continuous groups such as SL_{2}(ℂ) and their representations. Recently invariant theory has undergone a renaissance, which is due to the emergence of better algorithms and machinery for the computation, and a new interest in the modular case, which in some aspects parallels modular representation theory.

In this paper we restrict our attention to invariants of finite groups, but the ground fields will be arbitrary. We look at various structural properties which rings or fields of invariants can have, and ask for connections between these properties and properties of the group or the representation in question. For example, it is a natural question whether the ring of invariants is isomorphic to a polynomial ring, or whether the field of invariants is purely transcendental. Weakening the condition of polynomiality, we get a hierarchy of properties which rings of invariants can have. In particular, we will address the complete intersection property, the Gorenstein property and the Cohen-Macaulay property. When writing the paper we were inspired by the beautiful survey article of Stanley [29], who 20 years ago considered the same properties of rings of invariants but focused on the non-modular case.

The first section of this paper is intended as an account, similar in spirit to Stanley’s paper, of the present-day knowledge on these question, with special emphasis on the peculiarities in the modular case. In the course of the paper we will particularly stress the example of groups generated by pseudo-reflections for which recently a number of new results could be shown. These groups are interesting since in characteristic zero they are exactly the groups whose rings of invariants are polynomial rings. On the other hand, they provide good examples to illustrate the differences between the modular and the characteristic zero theory, since over fields of positive characteristic there exist reflection groups whose invariants are not even Cohen-Macaulay. As general reading on invariant theory we refer to the books [2,27,30] of Benson, Smith and Sturmfels.

Pseudo-reflection groups make another appearance in the second section of this paper, where we consider fields of invariants and ask whether they are purely transcendental (“Noether’s problem”). As it turns out, the fields of invariants of irreducible pseudo-reflection groups are always purely transcendental, even in the modular case. This illustrates how different the behavior of fields of invariants and rings of invariants can be. Noether’s problem is a natural and interesting question on the structure of fields of invariants, but it also has applications in the construction of so-called generic polynomials for certain Galois extensions. We will explain this connection to Galois theory and give a few examples.

2 Rings of invariants

Let k be a field and V a finite dimensional vector space over k. Let G ≤ GL(V) be a finite linear group on V. Then G also acts in a natural way on the symmetric algebra k[V] of V. We are interested in the structure of the ring of invariants

k[V]^{G} = {f ∈ k[V] | σf = f for all σ ∈ G}

of G in this action. By choosing a basis {x_{1}, …, x_{n}} of V we may identify the symmetric algebra k[V] with the polynomial ring k[x_{1}, …, x_{n}]. Putting the generators x_{i} in degree 1 this endows k[V] and hence k[V]^{G} with the structure of a graded k-algebra. Moreover by the theorem of Hilbert-Noether k[V]^{G} is finitely generated as k-algebra. In the following section we recall some general notions and results for finitely generated graded k-algebras (which can be found in [2], for example) before looking more closely at invariant rings.

2.1 Properties of graded algebras

Let A be a finitely generated commutative graded k-algebra, so

A = ⨁_{i=0}^{∞} A_{i} with A_{0} = k .

By the Noether normalization theorem there exist algebraically independent homogeneous elements f_{1}, …, f_{n} ∈ A such that A is a finitely generated module over k[f_{1}, …, f_{n}], where n = dim A is the Krull-dimension of A. Such a set {f_{1}, …, f_{n}} is called a homogeneous system of parameters for A. The algebra A is called Cohen-Macaulay if A is a free module over k[f_{1}, …, f_{n}] for some (and then for all) homogeneous systems of parameters {f_{1}, …, f_{n}}. An equivalent definition is the following: A is Cohen-Macaulay if there exist homogeneous f_{1}, …, f_{n} ∈ A of positive degree such that f_{i} is not a zero divisor in A/(f_{1}, …, f_{i−1}), for i = 1, …, n. Such a sequence f_{1}, …, f_{n} is called a regular sequence, and it turns out that if A is Cohen-Macaulay, then every homogeneous system of parameters is a regular sequence.

We shall now introduce some further properties of graded algebras which are more special than the Cohen-Macaulay property. Consider the Hilbert series of A, which is by definition the formal power series

H(A, t) := ∑_{i=0}^{∞} dim(A_{i}) t^{i} .

By the theorem of Hilbert-Serre, for any choice {f_{1}, …, f_{n}} of a homogeneous system of parameters, this power series can be written in the form

H(A, t) = \frac{g(t)}{∏_{i=1}^{n}(1 − t^{d_{i}})}, with d_{i} = deg(f_{i}),

for some Laurent polynomial g(t) with integer coefficients. It is not surprising that H(A, t) contains a lot of information about the structure of A. For example, one can define the Gorenstein property by saying that A is Gorenstein if it is a Cohen-Macaulay domain and its Hilbert-series as a rational function in t satisfies

H(A, t^{−1}) = (−1)^{dim(A)}t^{l}H(A, t)  (1)

for some l ∈ ℤ. By the “original definition” of the Gorenstein property, a Cohen-Macaulay algebra A is said to be Gorenstein if the canonical module for A is isomorphic to A. It is a theorem of Stanley that this definition coincides with ours for domains, see [28]. We will say that A is strongly Gorenstein if l = dim(A) in (1). One can check that our notion of strongly Gorenstein coincides with Benson’s notion of graded Gorenstein, which imposes a condition on the degree of the isomorphism between A and its canonical module.

A k-algebra A is called a complete intersection if it has a presentation as

A = k[f_{1}, …, f_{n+m}]/(g_{1}, …, g_{m})  (2)

for some m ≥ 0, where n = dim(A). Let us explain why this property implies the Gorenstein property. Indeed, take polynomials g_{m+1}, …, g_{m+n} ∈ R := k[f_{1}, …, f_{n+m}] whose images in A form a homogeneous system of parameters of A. Then R/(g_{1}, …, g_{m+n}) is finitely generated as a k-vector space, which by Nakayama’s lemma (as stated in [27, Cor 5.2.5], for example) implies that R is finitely generated as a module over k[g_{1}, …, g_{m+n}]. In other words, {g_{1}, …, g_{m+n}} form a homogeneous system of parameters for R. But choosing {f_{1}, …, f_{n+m}} as a homogeneous system of parameters shows that R is Cohen-Macaulay, hence g_{1}, …, g_{m+n} is a regular sequence. Therefore, the images of g_{m+1}, …, g_{m+n} are a regular sequence in A, hence A is Cohen-Macaulay. Now we must verify Equation (1) for the Hilbert series of A. A slight generalization of what we said about the Cohen-Macaulay property (see [27, Cor 6.2.8]) allows us to conclude from the regularity of g_{1}, …, g_{m} that R is free as a module over k[g_{1}, …, g_{m}]. If S is a minimal set of homogeneous generators for R over k[g_{1}, …, g_{m}] and H(S, t) is the formal power series whose i-th coefficient is the number of generators in S of degree i, then we obtain the equation

H(S, t) = \frac{H(R, t)}{H(k[g_{1}, …, g_{m}], t)} = \frac{∏_{i=1}^{m}(1 − t^{e_{i}})}{∏_{i=1}^{n+m}(1 − t^{d_{i}})},

where deg(f_{i}) = d_{i}, deg(g_{i}) = e_{i}. On the other hand, the image of S under the canonical map R → A is a basis of A as a vector space over k by Nakayama’s lemma, hence H(A, t) = H(S, t). Together with the above equation we obtain that the Hilbert series of a complete intersection A is of the form

H(A, t) = \frac{∏_{i=1}^{m}(1 − t^{e_{i}})}{∏_{i=1}^{n+m}(1 − t^{d_{i}})},  (3)

so indeed it satisfies Equation (1).

We remark that a complete intersection with m = 1 is also called a hypersurface. The geometric idea behind this term is obvious. Finally, the most special property that we consider is that A is (isomorphic to) a polynomial ring, or, in other words, a complete intersection with m = 0. In summary, we have reached the following hierarchy between the properties:

polynomial ⇒ complete intersection ⇒ Gorenstein ⇒ Cohen-Macaulay.

Invariant rings of finite groups. Much of the theory of graded algebras was prompted by the study of rings of invariants, and indeed it seems that rings of invariants are not very much better behaved than graded algebras which are domains in general. We will however mention a few special features here before coming to the one-by-one investigation of the above properties for rings of invariants.

Most importantly, the study of rings of invariants subdivides naturally into two quite different cases. Either the characteristic of k divides the order of G (the modular case), or it does not (the so-called coprime characteristic case). In the latter case, much more is known about the structure of k[V]^{G}. For example, it is known that then k[V]^{G} is always Cohen-Macaulay (see Th. 2.19). This is no longer true in the modular case. Furthermore, in the coprime characteristic case the Hilbert series of the ring of invariants may be calculated by Molien’s formula: for any linear character χ of G the Hilbert series for the module of relative invariants

k[V]_{χ}^{G} = {f ∈ k[V] | σ(f) = χ(σ)f for all σ ∈ G}

is given by

H(k[V]_{χ}^{G}, t) = \frac{1}{|G|} ∑_{σ∈G} \frac{χ(σ^{−1})}{det_{V} (1 − σt)}.  (4)

In the modular case, there is no such simple formula for H(k[V]^{G}, t) (but see the paper [13] of the first author in this volume).

We note that in the situation where A = k[V]^{G} is a ring of invariants (and G finite), the Krull dimension of A is equal to the k-dimension of V. Furthermore, a homogeneous system of parameters for A is also called a system of primary invariants. Given primary invariants f_{1}, …, f_{n}, a system of generators g_{1}, …, g_{s} of k[V]^{G} as a module over k[f_{1}, …, f_{n}] is called a system of secondary invariants of G.

2.2 Polynomial rings

We first consider the question under which conditions the ring of invariants k[V]^{G} of a finite group is a polynomial ring.

General criteria. In characteristic 0 the question of polynomiality has found a beautiful complete answer. The criterion for polynomial invariant rings can be given in terms of certain types of elements of the linear group G. In order to describe this we introduce the following notation. An element σ ∈ GL(V) is called a (pseudo-)reflection if dim(ker(σ − 1)) = dim(V) − 1, i.e., if σ fixes a hyperplane pointwise and σ ≠ 1. Note that all elements in finite linear groups G over a field of characteristic 0 are diagonalizable. Diagonalizable reflections will also be called homologies, while non-diagonalizable reflections are called transvections.

Theorem 2.1. (Chevalley, Serre, Shephard/Todd) Let G ≤ GL_{n}(k) have order prime to the characteristic of k. Then the ring of invariants of G is polynomial if and only if G is generated by pseudo-reflections.

This theorem was first proved by Shephard/Todd [26] for the field k = ℂ of complex numbers. For one direction of the proof they needed to classify all irreducible finite groups generated by complex pseudo-reflections. (The reduction to the irreducible case is trivial here, since all complex representations of finite groups are completely reducible.) Later, Chevalley [5] gave a conceptual proof avoiding the classification. The statement for arbitrary, but coprime characteristic was given by Serre [25]. In that same paper, he also showed a necessary condition in the case that the characteristic divides the order of G. Here the centralizers

C_{G}(W) = {σ ∈ G | σw = w for all w ∈ W }

of subspaces W ≤ V enter the picture:

Theorem 2.2. (Serre) Let G ≤ GL(V) be a finite group such that k[V]^{G} is polynomial. Then G is generated by pseudo-reflections, and the point-wise stabilizer of any subspace of V has polynomial ring of invariants (and thus is generated by pseudo-reflections).

A necessary and sufficient criterion, which is useful in studying explicitly given examples, was proved by the first author in [11]:

Proposition 2.3. (Kemper) Let G ≤ GL(V) be finite. Then k[V]^{G} is polynomial if and only if there exist n = dim(V) homogeneous invariants f_{1}, …, f_{n} ∈ k[V]^{G} such that the Jacobian determinant det((∂f_{i}/∂x_{j})_{i,j}) is nonzero and |G| = ∏_{i=1}^{n} deg(f_{i}).

Irreducible groups. By the Theorem of Serre (Theorem 2.2) a finite group with polynomial invariants is necessarily generated by reflections. In the case of irreducible linear groups, the authors also found a sufficient criterion. Note that an irreducible linear group generated by reflections is already absolutely irreducible.

Theorem 2.4. (Kemper/Malle) Let V be a finite-dimensional vector space, G ≤ GL(V) a finite irreducible linear group generated by pseudo-reflections such that the pointwise stabilizer in G of any nontrivial subspace of V has a polynomial ring of invariants. Then k[V]^{G} is a polynomial ring.

The proof of Theorem 2.4 in [14] is similar in spirit to the one of Shephard/Todd in the complex case, in that it uses the classification of finite irreducible linear groups generated by pseudo-reflections. This classification was achieved by Kantor, Wagner and Zalesskii/Serezkin (see [14] for a precise statement of this classification). As a by-product of the proof of Theorem 2.4 we obtained a complete list of the irreducible reflection groups with polynomial ring of invariants. This is reproduced in the form of Table 2.5.

As a corollary of the proof we can also verify the following assertion, which is in partial confirmation of a conjecture of Kac [9]:

Corollary 2.6. Let G ≤ GL(V) be a finite irreducible linear group containing no transvections. Then k[V]^{G} is polynomial if and only if the pointwise stabilizer in G of any subspace of V is generated by pseudo-reflections.

The example of the transvection group Sp_{4}(2) in its natural 4-dimensional representation over 𝔽_{2} shows that the assumption that G contains no transvections is necessary in Corollary 2.6 (see [14, Ex. 2.2]).

The following example shows that even when a complex linear group and its p-modular reduction are isomorphic and both have polynomial invariants, the degrees of generating invariants and hence the corresponding Hilbert series need not coincide:

Example 2.7. Let G ≤ GL_{3}(ℂ) be the primitive complex reflection group of order 648 denoted G_{25} by Shephard and Todd [26]. Its ring of invariants is polynomial with algebraically independent generators in degrees 6, 9, 12. Let L be a G-invariant ℤ_{2}[\sqrt{−3}]-lattice in ℂ^{3} and consider its 2-modular reduction. This yields a 3-dimensional reflection representation of G in characteristic 2. According to [14, Table 6.2] this identifies G with the 3-dimensional unitary group GU_{3}(2) ≤ GL_{3}(4). Here, the ring of invariants is also polynomial, but the generators are in degrees 3, 9, 24.

Table 2.5. The irreducible reflection groups

G ≤ GL_{n}(k), char(k) = p  k[V]^{G} polynomial  K[V]^{G} not polynomial
G imprimitive  all  -
n = dim(V) ≤ 2  all  -
SL_{n}(p^{f}) ≤ G ≤ GL_{n}(p^{f})  all  -
Ω_{n}^{(±)}(p^{f}) < G ≤ GO_{n}^{(±)}(p^{f}), GO_{3}(p^{f}), R^{+}O_{3}(p^{f}),  all other
G ≠ SO_{n}^{(±)}(p^{f})  GO_{4}^{−}(p^{f})
SU_{n}(p^{f}) ≤ G ≤ GU_{n}(p^{f})  GU_{3}(p^{f})  all other
G = Sp_{n}(p^{f}), n ≥ 4 even  -  all
G = 𝔖_{n+1}, p |̸(n + 1)  all  -
G = 𝔖_{n+2}, p|(n + 2),  -  all
n ≥ 5
G = W_{p}(G_{i})  W_{7}(G_{24}), W_{3}(G_{28}), W_{3}(G_{30}),
(i = 23, …, 37),  all except →  W_{5}(G_{30}), W_{3}(G_{31}), W_{5}(G_{32}),
G = EJ_{3}(5), G = J_{4}(4)  W_{3}(G_{33}), W_{2}(G_{34}), W_{3}(G_{36}),
W_{3}(G_{37}), W_{5}(G_{37})

Nilpotent groups. The rings of invariants of p-groups generated by reflections where p is the characteristic of k, (i.e., by transvections), have been studied by Nakajima. The question of polynomiality of the ring of invariants could only be settled completely for pseudo-reflection groups over the prime field 𝔽_{p}. In fact, the following result covers a larger class of groups, which includes all nilpotent groups:

Theorem 2.8. (Nakajima) Let G ≤ GL_{n}(𝔽_{p}) be the direct product of a p-group P with a p′-group H. Then k[V]^{G} is polynomial if and only if

i) H is generated by pseudo-reflections, and

ii) there exists a basis {x_{1}, …, x_{n}} of V = 𝔽_{p}^{n} such that

∏_{i=1}^{n} |P.x_{i}| = |P |

and the ⨁_{i=1}^{j} kx_{i} are kP -submodules of V for 1 ≤ j ≤ n.

This was proved in [21, Th. 1.4].

Unique factorization domains. An obvious question to ask about a ring of invariants is whether it is a unique factorization domain. This property does not fit into the hierarchy stated in Section 2.1, but it has a strong connection to pseudo-reflections, as the following theorem by Nakajima shows.

Theorem 2.9. (Nakajima) Let G be a finite group. Then k[V]^{G} is a unique factorization domain if and only if any non-trivial homomorphism G → k^{×} is non-trivial on some reflection of G.

A proof can be found in [2, Cor. 3.9.3]. The theorem implies that rings of invariants of pseudo-reflection groups and of p-groups, where p is the characteristic of k, are always unique factorization domains.

2.3 Complete intersections

Let us say that a linear group G ≤ GL(V) is a CI-group if the ring of invariants k[V]^{G} is a complete intersection. The following example shows that there are CI-groups whose ring of invariants is not polynomial

Example 2.10. Let k be a field of characteristic different from 2 and G = Z_{2} the cyclic group of order two acting by scalar matrices on V = k^{2}. Then

k[V]^{G} = k[f_{1}, f_{2}, h_{1}] / (f_{1}f_{2} − h_{1}^{2})

with f_{1} = x_{1}^{2}, f_{2} = x_{2}^{2}, h_{1} = x_{1}x_{2}, is a complete intersection, but not a polynomial ring by Theorem 2.2. Here, this can also be seen from the Hilbert series

H(k[V]^{G}, t) = \frac{1 + t^{2}}{(1 − t^{2})^{2}} .

In view of the fact that Theorem 2.1 fails in the modular case, it seems interesting to determine which modular reflection groups are at least CI-groups. This question is still open at present, even the irreducible case has not been settled. The following partial results have been obtained:

Theorem 2.11. (Carlisle/Kropholler) The transvection groups G = Sp_{2n}(q) in their natural matrix representation are CI-groups.

A proof can be found in [2, Thm. 8.3.11]. All exceptional irreducible pseudoreflection CI-groups are known (by what we said above, they must be among the groups in the lower right-hand corner of Table 2.5):

Proposition 2.12. (Kemper/Malle) Let G be an exceptional complex reflection group such that its p-modular reduction W_{p}(G) is isomorphic to G, still acts irreducibly, and is not isomorphic to a classical group. Then W_{p}(G) occurs in Table 2.13 and k[V]^{W_{p}(G)} is a complete intersection but not a polynomial ring if and only if the corresponding entry is ’CI’.

Table 2.13. Non-polynomial invariant rings for exceptional reflection groups

G dim  |W_{p}(G)|  2  3  5 7
G_{24} 3  336  –  pol  pol CI
G_{28} 4  1152  –  CI  pol pol
G_{30} 4  14400  –  CI RO_{4}^{+}(5) pol
G_{31} 4  46080  –  CI  pol pol
G_{32} 4  155520  –  –  nCI pol
G_{33} 5  51840  – R^{−}O_{5}(3)  pol pol
G_{34} 6  19595520  CI  –  – –
G_{36} 7  2903040  –  nCI  pol pol
G_{37} 8 696729600  –  nCI  nCI pol

The entries – indicate that either W_{p}(G) is strictly smaller than G, or it acts reducibly. An entry ‘nCI‘ signifies that the ring of invariants is not a complete intersection. In two cases, the p-modular reduction W_{p}(G) in Table 2.13 is isomorphic to a classical group. For these, the question of when they are CI-groups is not yet settled. The proof relies on computer calculations as well as on the subsequent Theorem 2.14 and will be published elsewhere. See also Proposition 2.23 for a related statement.

Kac and Watanabe have proved a necessary condition for CI-groups very similar to Serre’s criterion (Theorem 2.2) for polynomiality. It uses the following generalization of the notion of pseudo-reflection. An element σ ∈ GL(V) is called m-special if it centralizes a subspace of codimension m, i.e., if dim(ker(σ − 1)) ≥ dim(V) − m. Thus pseudo-reflections are 1-special. Elements which are 2-special but not 1-special are also called bi-reflections.

Theorem 2.14. (Kac/Watanabe) Let G ≤ GL(V) be a finite CI-group. Then G is generated by 2-special elements and the point-wise stabilizer of any subspace of V is again a CI-group (and thus is generated by 2-special elements).

This is proved in [9] (see also [8] for fields of characteristic 0).

Theorem 2.14 focuses attention on groups generated by bi-reflections. In characteristic 0 the irreducible ones among these groups were classified around 1980. Up to imprimitive groups and groups of low dimension, these are just the alternating and symmetric groups in their natural representation. Using this classification, Gordeev [8] and independently Nakajima ([22] and the references cited there) determined the CI-groups G ≤ GL_{n}(ℂ). They show that the question for reducible groups can be answered if the irreducible and the abelian CI-groups are known. Their result consists of a list of those linear groups over ℂ which are CI-groups, which we will not reproduce here. (The irreducible imprimitive CI-groups of dimension 4 are not classified completely.) The problem of finding a nice structural criterion for CI-groups is still open even in characteristic 0.

2.4 Gorenstein rings

Recall that a Cohen-Macaulay ring of invariants k[V]^{G} is Gorenstein if and only if for some l ∈ ℤ its Hilbert series satisfies the symmetry property H(k[V]^{G}, t^{−1}) = (−1)^{dim(V)}t^{l}H(k[V]^{G}, t) . The following gives a simple example for a Gorenstein non-CI-group.

Example 2.15. Let k be a field of characteristic different from 3 and G = Z_{3} the cyclic group of order three acting by scalar matrices on V = k^{3}. Then Molien’s formula (4) gives

H(k[V]^{G}, t) = \frac{1 + 7t^{3} + t^{6}}{(1 − t^{3})^{3}} ,

hence the ring of invariants is (even strongly) Gorenstein, but since the zeros of the numerator are not roots of unity, it is not a complete intersection by (3).

Assume that the characteristic of k is prime to |G|. Comparing with Molien’s formula (4) one sees that the Laurent expansion at t = 1 of the Hilbert series starts as follows

H(k[V]^{G}, t) = \frac{1}{|G|}(1 − t)^{− dim(V)} + \frac{r}{2|G|}(1 − t)^{− dim(V)+1} + …,

where r is the number of pseudo-reflections in G (see for example [2, 2.6]). Thus, if k[V]^{G} is Gorenstein then the exponent l above equals l = dim(V) + r. In particular, in coprime characteristic a Gorenstein ring of invariants k[V]^{G} is strongly Gorenstein if and only if G contains no pseudo-reflections. On the other hand, since

det(1 − σt^{−1}) = (−t)^{− dim(V)} det(σ) det(1 − σ^{−1}t)

Molien’s formula (4) gives

H(k[V]^{G}, t^{−1}) = \frac{1}{|G|} ∑_{σ∈G} \frac{1}{det(1 − σt)}

= \frac{(−t)^{dim(V)}}{|G|} ∑_{σ∈G} \frac{det(σ)}{det(1 − σt)} = (−t)^{dim(V)}H(k[V]_{det^{−1}}^{G}, t) .

Hence we have obtained Watanabe’s characterization of groups whose invariants are strongly Gorenstein:

Proposition 2.16. Let G ≤ GL(V) have order prime to the characteristic of k. Then k[V]^{G} is strongly Gorenstein if and only if G ≤ SL(V).

Indeed, G ≤ SL(V) is equivalent to k[V]_{det^{−1}}^{G} = k[V]^{G}, thus by our above calculations to

H(k[V]^{G}, t^{−1}) = (−t)^{dim(V)}H(k[V]_{det}^{G}, t) = (−t)^{dim(V)}H(k[V]^{G}, t) .

Not much is known about the Gorenstein property in the modular case. The following proposition obtained in [3] shows that it is a useful property.

Proposition 2.17. (Campbell/Geramita/Hughes/Shank/Wehlau) If the ring of invariants k[V]^{G} is Gorenstein, then it can be generated by invariants of degrees at most dim(V) · (|G| − 1). Moreover, if k[V]^{G} is a hypersurface, it can be generated by invariants of degrees at most |G|.

2.5 Cohen-Macaulay rings

It is easy to find linear groups whose rings of invariants are Cohen-Macaulay but not Gorenstein:

Example 2.18. Let k be a field of characteristic different from 3 and G = Z_{3} the cyclic group of order three acting by scalar matrices on V = k^{2}. Then Molien’s formula gives

H(k[V]^{G}, t) = \frac{1 + 2t^{3}}{(1 − t^{3})^{2}} ,

hence the ring of invariants is not Gorenstein. On the other hand it is clear that k[V]^{G} is generated by x_{1}^{2}x_{2}, x_{1}x_{2}^{2} as a free module over k[x_{1}^{3}, x_{2}^{3}], hence k[V]^{G} is Cohen-Macaulay.

In the coprime characteristic case, all rings of invariants are Cohen-Macaulay (see for example [2, Th. 4.3.6]):

Theorem 2.19. (Hochster/Eagon) Let G ≤ GL(V) be a finite group and k a field whose characteristic is zero or coprime to the order of G. Then k[V]^{G} is Cohen-Macaulay.

It is perhaps the most outstanding deficiency of modular invariant theory that Theorem 2.19 is not true in the modular case.

Example 2.20. One of the simplest example of an invariant ring which is not Cohen-Macaulay is given by the action of the cyclic group Z_{2} of order 2 on k[V] = 𝔽_{2}[x_{1}, x_{2}, x_{3}, y_{1}, y_{2}, y_{3}] by x_{i} ↦ y_{i} ↦ x_{i}. In fact, if k[V]^{G} were Cohen-Macaulay, then the sequence f_{1} = x_{1} + y_{1}, f_{2} = x_{2} + y_{2}, f_{3} = x_{3} + y_{3} would have to be regular, since it is the beginning of a homogeneous system of parameters. But the relation

0 = | \begin{matrix}f_{1} & f_{2} & f_{3} \\ f_{1} & f_{2} & f_{3} \\ x_{1} & x_{2} & x_{3} \end{matrix} | = u_{23}f_{1} + u_{31}f_{2} + u_{12}f_{3} with u_{ij} = x_{i}y_{j} + x_{j}y_{i}

shows that f_{3} is a zero divisor in k[V]^{G}/(f_{1}, f_{2}).

A characterization of those linear groups whose rings of invariants are Cohen-Macaulay is still missing. The following result was proved in [12].

Proposition 2.21. (Kemper) Suppose that for an integer r > 0 the r-th cohomology group H^{r}(G, k[V]) is nonzero, and that no element in G of order p =char(k) is (r + 1)-special (see page 9). Then k[V]^{G} is not Cohen-Macaulay.

Example 2.20 arises as an application of the theorem, since H^{1}(Z_{2}, 𝔽_{2}) ≠ 0. As a corollary, we obtain the following converse of Theorem 2.19.

Theorem 2.22. If G is a finite group and k a field such that for every finite dimensional kG-module V the invariant ring k[V]^{G} is Cohen-Macaulay, then the characteristic of k is zero or coprime to the order of G.

For further examples, we take another look at the Table 2.5 of irreducible reflection groups and ask if the rings of invariants are at least always Cohen-Macaulay. As it turns out, this is not the case. The groups 𝔖_{n+2} appearing in the table are the symmetric groups on n + 2 symbols with the following representation: Let W = k^{n+2} be the natural permutation representation and suppose that the characteristic p of k divides n + 2. Then the 1-dimensional invariant subspace W_{1} = ⟨x_{1} + … + x_{n+2}⟩ is contained in the n + 1-dimensional invariant subspace

W_{n+1} = {∑ a_{i}x_{i} | ∑ a_{i} = 0} .

Thus we obtain a representation of 𝔖_{n+2} on the n-dimensional space V := W_{n+1}/W_{1} as a group generated by reflections. The following is another application of Theorem 2.21 (see [12, Ex. 2.6]).

Proposition 2.23. (Kemper) Let n > 3, G = 𝔖_{n+2} in its n-dimensional reflection representation over a field k of characteristic p|(n + 2), p ≥ 5, constructed above. Then k[V]^{G} is not Cohen-Macaulay.

Another series of pseudo-reflection groups whose rings of invariants are not Cohen-Macaulay was discovered by Nakajima [20]. This series consists of pgroups (which hence are not irreducible).

The role of reflections and bi-reflections is also emphasized by the following result proved in [12].

Theorem 2.24. (Kemper) Suppose that G is a p-group, with p the characteristic of k, and k[V]^{G} is Cohen-Macaulay. Then G is generated by elements which are 2-special.

3 Fields of invariants

So far we have been looking at properties of polynomials f ∈ k[V] invariant under a finite group G. Instead one could study invariant rational functions, i.e., the field of invariants of G

k(V)^{G} := Quot(k[V]^{G}) = {f ∈ k(V) := Quot(k[V]) | σf = f for all σ ∈ G}

which is equal to the field of fractions of k[V]^{G} since G is finite. This will lead us to Noether’s problem and the notion of generic polynomials.

3.1 Noether’s problem and generic polynomials

Let G ≤ 𝔖_{n} be a permutation group and assume that the field of invariants k(x_{1}, …, x_{n})^{G} is purely transcendental over k. In 1918, Emmy Noether [23] showed that in this case we have a parametrization of the polynomials with Galois group G in the following sense: there exists a polynomial g(t_{1}, …, t_{n}, X) is a polynomial in X whose coefficients are rational functions in k(t_{1}, …, t_{n}) such that the Galois group of g(X) is G, and every polynomial f(X) ∈ k[X] with Galois group G whose coefficients avoid a certain Zariski-closed subset can be obtained by a suitable specialization of g(X): f(X) = g(a_{1}, …, a_{n}, X) with a_{i} ∈ k. This raises the question, now known as Noether’s problem, whether the field of invariants k(V)^{G} is always a rational function field. Of course this question can also be generalized to the situation where G ≤ GL(V) is a (finite) linear group. We explain the significance of this question to Galois theory below.

The property that the field of invariants of a linear group is purely transcendental is the field-theoretic analogue to the property that the ring of invariants is polynomial. If k[V]^{G} is polynomial, then k(V)^{G} clearly is purely transcendental. A look at Example 2.10 shows that the pure transcendence if k(V)^{G} is (strictly) weaker than the polynomiality of k[V]^{G}. Indeed, we have f_{2} = h_{1}^{2}/f_{1} in that example, hence k(V)^{G} = k(f_{1}, h_{1}) is purely transcendental.

Not much is known about Noether’s problem. Already Noether observed that k(V)^{G} is purely transcendental if dim(V) ≤ 2 or if dim(V) = 3 and k is algebraically closed. This follows by theorems of Lüroth and Castelnuovo (see the references in [23]). The field of invariants is also purely transcendental if G is abelian and the ground field k is algebraically closed [7]. It was shown by Swan [31] and Voskresenskii [32] that this need not be true any longer if the condition on k is dropped. Indeed, for G = Z_{47} the cyclic group of order 47 in its faithful permutation representation on ℚ^{47} the field of invariants is not a rational function field, which gives a negative answer to Noether’s problem. More generally Lenstra [16] gave a criterion when the field of invariants of an abelian group is purely transcendental over k by using Galois descent. Later Saltman found counter examples where the ground field is algebraically closed. Among these examples are certain nilpotent groups over k = ℂ. Apart from the case of abelian groups and small dimensions, Noether’s problem remains wide open. Positive answers are known for just a few groups, which include A_{4} and A_{5} in their natural permutation representation over ℚ [17].

One reason for the interest in this question is the connection to generic polynomials, which we now explain. Let k be a field and G a finite group. A polynomial f(t_{1}, …, t_{m}, X) ∈ k(t_{1}, …, t_{m})[X] is said to be a generic polynomial for G over k if Gal(f(X)) ≅ G and secondly for every infinite field K ≥ k and every Galois extension N/K with group G there exist a_{1}, …, a_{m} ∈ K such that N is the splitting field of f(a_{1}, …, a_{m}, X). Notice that only every field extension but not every polynomial with Galois group G can be obtained by specializing g(X). In this sense, generic polynomials are weaker than the parametrizations introduced by Noether, but on the other hand generic polynomials are a more satisfactory notion since no Zariski-closed subset must be excluded. It is also worth noting that if we have a generic polynomial f for G over a Hilbertian field k, then G occurs as a Galois group over k. An obvious example is the generic polynomial f = X^{n} + t_{n−1}X^{n−1} + … + t_{0} for the symmetric group 𝔖_{n}. Other examples are polynomials for Artin-Schreier and Kummer extensions.

Now let G ≤ GL(V) be a finite group such that the field of invariants k(V)^{G} is purely transcendental, generated by s_{1}, …, s_{n}, say. Let B be a finite G-invariant subset of V which generates V and define

f(X) := ∏_{v∈B}(X − v) .

Then the coefficients of f are G-invariant, hence can be expressed in s_{1}, …, s_{n}, so f(X) = f(s_{1}, …, s_{n}, X) ∈ k(s_{1}, …, s_{n})[X]. Clearly, by construction, Gal(f) = G. But we even have (see for example [10, Satz 1.7]):

Proposition 3.1. The polynomial f constructed above is generic for G over k.

The first author has recently shown that the polynomial f has even the following stronger property: if K ≥ k is an infinite field and N/K a Galois extension whose Galois group is a subgroup of G, then N is the splitting field of f(a_{1}, …, a_{n}, X) with suitable a_{1}, …, a_{n} ∈ K.

Example 3.2. Let G = GL_{n}(q) in its natural representation on V = 𝔽_{q}^{n}. Dickson has shown that the ring of invariants is polynomial (see Table 2.5, generated by the non-vanishing coefficients c_{0}, …, c_{n−1} of the polynomial

f(X) = ∏_{0≠v∈V} (X − v) = ∑_{i=0}^{n} c_{i}X^{q^{i}−1} ∈ 𝔽_{q}[c_{0}, …, c_{n−1}](X) .

In particular, f as above is the generic polynomial for GL_{n}(q) over 𝔽_{q} for the subset B = V \ {0}. Note that all exponents of X in f are divisible by q − 1. The polynomial f(X^{1/(q−1)}) now has Galois group PGL_{n}(q) (and is generic for those PGL_{n}(q)-extensions embeddable into a GL_{n}(q)-extension). A simple argument shows that the invariant c_{0} is a (q − 1)st power. Dickson shows that the ring of invariants of SL_{n}(q) is also polynomial, generated by u := c_{0}^{1/(q−1)}, c_{1}, …, c_{n−1}. Thus, considered as a polynomial over 𝔽_{q}[u, c_{1}, …, c_{n−1}], f is generic for SL_{n}(q) over 𝔽_{q} and similarly as above we obtain a polynomial with group PSL_{n}(q). This example also shows why families of polynomials such as the ones found by Abhyankar (see [1] and the papers cited there) have to exist.

Example 3.3. The previous example did not yield generic PSL_{n}(q)-polynomials. At least for n = 2 these can be obtained as follows. Let V = Sym^{2}(W) be the symmetric square of the natural module W for SL_{2}(q). Since the center of SL_{2}(q) acts trivially on V, this defines a faithful representation of G = PSL_{2}(q) on V. It can be shown that G leaves invariant a quadratic form on V, and is even isomorphic to the derived group O_{3}(q) of the special orthogonal group SO_{3}(V) with respect to this form. By a result of the first author [10, Satz 3.14] the field of invariants k(V)^{O_{3}(q)} is purely transcendental, and this yields generic polynomials for G = PSL_{2}(q). In the cases q = 5, 7 one obtains the following two generic polynomials g_{q}(t_{1}, t_{2}, X) ∈ 𝔽_{q}(t_{1}, t_{2})[X] with two parameters :

g_{5}(X) = X^{12} − 2 u^{2} X^{8} + 2 (t_{1} + 1)u^{4} X^{4} + 2 t_{2} u^{5} X^{2} + (t_{1} − 2)^{2}u^{6}

with u = t_{2}^{2} + (t_{1} − 2)^{2} (t_{1} + 2)^{3}, respectively

g_{7}(X) = X^{24} − 3u^{3}X^{18} + (t_{1} + 1)u^{6}X^{12} − 3(t_{1}^{2} + t_{1} − 1)u^{9}X^{6} + u^{11}X^{3} − (t_{1} − 2)^{3}u^{12}

with u = t_{2}^{2} + (t_{1} − 2)^{3} (t_{1} + 2)^{4}.

It should be noted however that not all generic polynomials arise from the construction sketched above. Indeed, Saltman has shown that the cyclic group G = Z_{n} has a generic polynomial over any field k of characteristic prime to n for which k(ζ_{n})/k is cyclic [24]. This cannot be constructed as indicated above since the field of invariants of Z_{47} is not purely transcendental over ℚ by [31].

3.2 Fields of invariants of reflection groups

By the result of Shephard/Todd and Chevalley, the field of invariants of a reflection group is always purely transcendental in characteristic 0 since the ring of invariants is polynomial. This simple argument fails in positive characteristic by Theorem 2.4. The authors recently showed that at least in the irreducible case the result nevertheless remains true [15]:

Theorem 3.4. (Kemper/Malle) Let G be an irreducible pseudo-reflection group. Then the invariant field k(V)^{G} is a purely transcendental extension of k.

Notice that the classical groups GO_{n}^{(±)}(q), GU_{n}(q), SU_{n}(q), and Sp_{n}(q) appear among the irreducible reflection groups (Table 2.5). Thus Theorem 3.4 together with Proposition 3.1 gives generic polynomials for these groups in defining characteristic. For the proof, only those groups occurring in the right-hand column of Table 2.5 have to be considered. Most of them could be handled by the following criterion (see for example [10]):

Proposition 3.5. (Kemper) Let f_{1}, …, f_{n} ∈ k[V]^{G}, n = dim(V), be homogeneous invariants such that ∏_{i=1}^{n} deg(f_{i}) < 2|G| and the Jacobian determinant det(∂f_{i}/∂x_{j}) is non-zero. Then k(V)^{G} = k(f_{1}, …, f_{n}) is purely transcendental.

In the case of classical groups this can be applied to invariants found by Carlisle/Kropholler [4] respectively the first author. No generating set of invariants is known for the symmetric groups 𝔖_{n+2} in their n-dimensional representation over 𝔽_{p} (see page 12). Here, we are able to apply a theorem of Endo and Miyata (see [18]), which is of some interest in itself.

Proposition 3.6. (Endo/Miyata) Suppose G ≤ GL(V) is a finite linear group and U ≤ V is a faithful kG-submodule. Then k(V)^{G} is purely transcendental over k(U)^{G}. In particular, if k(U)^{G} is purely transcendental over k, then so is k(V)^{G}.

No reduction principle of this sort exists for the polynomiality of the ring of invariants. In fact, G does in general not act as a reflection group on V even if it does on U ≤ V. This illustrates how different the behavior of fields and rings of invariants are: while the structure of rings of invariants very heavily depends on the linear representation of the group, the field of invariants is in some sense much less dependent on the representation.

It should be noted that one side effect of our case-by-case approach to the proof of Theorem 3.4 is the fact that minimal bases for the fields of invariants are explicitly known in each case (except for the symmetric groups 𝔖_{n+2}), so that generic polynomials can be computed from this at least in principle. (We have the obvious generic polynomials for the symmetric groups.)

It would be interesting to know whether Theorem 3.4 continues to hold for reducible reflection groups. In the extreme case of p-groups, where p is the characteristic of k, we have the following result which is not restricted to reflection groups:

Theorem 3.7. (Miyata) Let k be of characteristic p and G ≤ GL_{n}(k) be a p-group. Then k(V)^{G} is purely transcendental.

See [18] for a proof. We see how different the behavior of rings and fields of invariants can be by comparing to the examples of Nakajima [20] of p-groups whose ring of invariants is not even Cohen-Macaulay.

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