PSEUDO-REFLECTION GROUPS AND ESSENTIAL DIMENSION

ALEXANDER DUNCAN AND ZINOVY REICHSTEIN

Abstract. Given a pseudo-reflection group, we give a simple formula for the essential dimension at a prime p. Additionally, we determine the absolute essential dimension in most cases. We also study the “poor man’s essential dimension” of an arbitrary finite group, an intermediate notion between the absolute essential dimension and the essential dimension at a prime p.

1. Introduction

Let k be a field and G be a finite group. Throughout, we assume that char(k) does not divide the order of G. Our finite groups will be viewed as split algebraic groups over k. We will denote by k̄ the algebraic closure of k and by ζ_{d} a primitive dth root of unity in k̄ where d is a positive integer coprime to char(k). By a variety we will mean a separated reduced scheme of finite type over k, not necessarily irreducible.

We will be interested in the essential dimension ed(G) and the relative essential dimension ed(G; p), where p is a prime. For the definition of essential dimension, a survey of its properties and further references, see [R10].

A theorem of N. A. Karpenko and A. S. Merkurjev [KM08] asserts that

(1.1)  ed(G; p) = ed(G_{p}) = rdim(G_{p}) .

Here G_{p} is any Sylow p-subgroup of G, and for a finite group H, rdim(H) denotes the minimal dimension of a faithful representation of H defined over k, and we assume that ζ_{p} ∈ k. However, note that, since [k(ζ_{p}) : k] is prime to p, ed(G; p) is the same over k or k(ζ_{p}).

The Karpenko-Merkurjev theorem (1.1) shows that ed(G; p) is a purely representationtheoretic quantity. It can be computed, at least in principle, if one has a description of the irreducible representations of G_{p} over k(ζ_{p}). In contrast, the “absolute” essential dimension ed(G) is more mysterious; in particular, it is unknown for many groups. By definition, ed(G; p) ⩽ ed(G) for every prime p, and the best known lower bound for ed(G) is usually deduced from this inequality.

The case where G = S_{n} is the symmetric group, is of particular interest because it relates to classical questions in the theory of polynomials; see [BR97, BR99]. Here the relative essential dimension is known exactly for every prime p,

(1.2)  ed(S_{n}; p) = ⌊\frac{n}{p}⌋;

2010 Mathematics Subject Classification. 20F55, 20D15.

Key words and phrases. Essential dimension, pseudo-reflection group, p-group.

A. Duncan was partially supported by National Science Foundation RTG grants DMS 0838697 and DMS 0943832.

Z. Reichstein was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 250217-2012.

see [MR09, Corollary 4.2]. The absolute essential dimension ed(S_{n}) is largely unknown. In characteristic zero we know only that

(1.3)  max_{p} ed(S_{n}; p) = ⌊\frac{n}{2}⌋ ⩽ ⌊\frac{n + 1}{2}⌋ ⩽ ed(S_{n}) ⩽ n − 3

for any n ⩾ 6; see [BR97], [Dun10] and [Mac11]. We know even less about ed(S_{n}) in prime characteristic.

M. MacDonald [Mac11, Section 5.1] recently computed ed(G; p), for all irreducible Weyl groups G, extending (1.2). He also computed the absolute essential dimension ed(G) when G is not isomorphic to S_{n} or W (E_{6}), the Weyl group of the root system of type E_{6}. His proofs are based on case-by-case analysis.

The aim of this paper is twofold. First, we will generalize MacDonald’s results to all finite groups generated by pseudo-reflections, with a uniform statement and proof. Second, we will investigate a new intermediate notion between max_{p} ed(G; p) and ed(G), which we call “poor man’s essential dimension.”

We will adopt the following notational conventions inspired by [Spr74]. Let ϕ: G ↪ GL(V) be a faithful representation of G and m be a positive integer prime to the characteristic of k. Choosing a primitive mth root of unity ζ_{m} ∈ k̄, we define V (g, ζ_{m}) := ker(ζ_{m}I − ϕ(g)) as the ζ_{m}-eigenspace of g and let

a_{ϕ}(m) := max_{g∈G} dim V (g, ζ_{m}) .

Note that V (g, ζ_{m}) is defined over k(ζ_{m}) but may not be defined over k. Replacing g by suitable powers, we see that a_{ϕ}(m) depends only on ϕ and m and not on the choice of the primitive mth root of unity ζ_{m}. If the reference to ϕ is clear from the context, we will write g in place of ϕ(g) and a(m) in place of a_{ϕ}(m). By convention, we set a(m) = 0 if m is a multiple of the characteristic of k.

Recall that an element g ∈ GL(V) is a pseudo-reflection if it is conjugate to a diagonal matrix of the form diag(1, …, 1, ζ), where ζ ≠ 1 is a root of unity.

Theorem 1.1. Let G be a finite subgroup of GL(V). Assume that the characteristic of the base field k does not divide |G|. Then

(a) ed(G; p) ⩽ a(p) for every prime p.

(b) Moreover, if G is generated by pseudo-reflections then ed(G; p) = a(p) for every prime p.

Suppose that ϕ: G ↪ GL(V) is generated by pseudo-reflections with n = dim(V). Then k[V]^{G} = k[f_{1}, …, f_{n}] for some homogeneous polynomials f_{1}, …, f_{n} such that d_{i} = deg(f_{i}). The integers d_{1} …, d_{n} are called the degrees of the fundamental invariants of ϕ. These numbers are uniquely determined by ϕ up to reordering. They are independent of the choice of f_{1}, …, f_{n} and can be recovered directly from the Poincaré series of k[V]^{G}; see, e.g., [Kan01] or [LT09]. T. A. Springer [Spr74, Theorem 3.4(i)] showed that

(1.4)  a(m) = |{i | d_{i} is divisible by m}|

Note that while the base field k is assumed to be the field of complex numbers ℂ in [Spr74, Theorem 3.4(i)], the above formula remains valid under our less restrictive assumptions on k; see, e.g., [Kan01, Section 33-1].

Complex groups generated by pseudo-reflections have been classified by G. C. Shephard and J. A. Todd [ST54]. Their classification lists d_{1}, …, d_{n} in every case; Springer’s theorem (1.4) makes it possible to read a(m) directly off their table for every G and every m. The same can be done for other base fields k, as long as char(k) does not divide |G|; for details and further references, see Section 4.

Example 1.2. For G = W (E_{8}) (group number 37 in the Shephard-Todd classification), the values of d_{1}, …, d_{8} are

2, 8, 12, 14, 18, 20, 24 and 30,

respectively; see, e.g., [LT09, Appendix D]. Counting how many of these numbers are divisible by each prime p and applying Theorem 1.1(b) in combination with (1.4), we recover the following values from [Mac11, Table IV].

\begin{matrix}p & 2 & 3 & 5 & 7 & > 7 \\ ed(W (E_{8}); p) & 8 & 4 & 2 & 1 & 0 \end{matrix}

We will leverage Theorem 1.1 to compute the absolute essential dimension for most irreducible pseudo-reflection groups. Recall that, in the Shephard-Todd classification there are three infinite families: the symmetric groups, the family G(m, l, n) depending on three integer parameters (m, l, n), and the cyclic groups. In addition, there are 34 exceptional groups.

Theorem 1.3. Let G ⊂ GL(V) be an irreducible representation of a finite group generated by pseudo-reflections. Suppose G is not isomorphic to a symmetric group S_{n} and char(k) does not divide |G|. Then

(a) ed(G) = dim(V) − 2 = 4, if G = W (E_{6}),

(b) ed(G) = dim(V) − 1 = n − 1, if G ≃ G(m, m, n) with m, n relatively prime,

(c) ed(G) = dim(V) in all other cases.

As we mentioned above, the exact value of ed(S_{n}) is not known; see (1.3). The value of the essential dimensions G = W (E_{6}) is new; see [Mac11, Remark 5.2]. The proof of part (a) relies on a geometric construction suggested to us by I. Dolgachev.

We now recall that ed(G) is the minimal dimension of a versal G-variety and ed(G; p) is the minimal dimension of a p-versal G-variety; see [Ser03, Section 5] and [DR13, Remark 2.5]. Poor man’s essential dimension, denoted pmed(G), is defined as the minimal dimension of a G-variety which is simultaneously p-versal for every prime p. We have

(1.5)  max_{p} ed(G; p) ⩽ pmed(G) ⩽ ed(G) .

The term “poor man’s essential dimension” is meant to suggest that pmed(G) is a more accessible substitute for ed(G). Where exactly it fits between max_{p} ed(G; p) and ed(G), is a key motivating question for this paper.

Theorem 1.4. Let G be a finite subgroup of GL(V). Assume that the base field k is infinite and char(k) does not divide |G|. Then

(a) pmed(G) ⩽ max_{p} a(p).

(b) Moreover, if G is generated by pseudo-reflections then pmed(G) = max_{p} a(p) = max_{p} ed(G; p).

In both parts the maximum is taken over all prime integers p.

In particular, pmed(S_{n}) = ⌊\frac{n}{2}⌋ for every n, assuming char(k) = 0, a result we found

somewhat surprising, considering that ed(S_{n}) > ⌊\frac{n}{2}⌋ for every odd n ≥ 7; see (1.3).

In view of Theorem 1.4(b), it is natural to ask if

(1.6)  pmed(G) = max_{p} ed(G; p)

for every finite group G. In addition to the case of pseudo-reflection groups covered by Theorem 1.4(b), we will also prove that this is the case for alternating groups (Example 10.1) and for groups all of whose Sylow subgroups are abelian (Proposition 11.1). A conjectural approach to proving (1.6) for other finite groups is outlined at the end of Section 11.

2. Proof of Theorem 1.1(a)

Throughout this section we fix a prime p and assume that the base field k is of characteristic ≠ p.

Lemma 2.1. Let V be a finite-dimensional k-vector space, and G_{p} ⊂ GL(V) be a finite p-group. Assume ζ_{p} ∈ k and V′ is a minimal (with respect to inclusion) faithful G_{p}- subrepresentation of V. Then there exists a central element g ∈ G_{p} of order p such that V′ ⊂ V (g, ζ_{p}), where ζ_{p} is a primitive pth root of unity.

Proof. Let C be the socle of G_{p}; i.e., the p-torsion subgroup of the centre Z(G_{p}).

Decompose V′ = V_{1} ⊕ ⋯ ⊕ V_{r} as a direct sum of irreducible G_{p}-representations. Each V_{i} decomposes into a direct sum of character spaces for C. Since C is central, each of these character spaces is G_{p}-invariant. As V_{i} is irreducible as a G_{p}-module, there is only one such component. That is, C acts on each V_{i} by scalar multiplication via a character χ_{i} : C → k^{∗}.

We will view the characters χ_{i} as elements of the dual group C^{∗} = Hom(C, k^{∗}). Note that since C is an elementary abelian p-group, C^{∗} has the natural structure of an 𝔽_{p}- vector space. An easy argument shows that χ_{1}, …, χ_{r} form an 𝔽_{p}-basis of C^{∗}; see [MR10, Lemma 2.3]. Consequently, there is a unique element g ∈ C such that χ_{i}(g) = ζ_{p} for every i = 1, …, r. In other words, V′ ⊂ V (g, ζ_{p}), as desired. □

Proof of Theorem 1.1(a). Neither ed(G; p) nor a(p) will change if we replace k by k(ζ_{p}). Hence, we may assume without loss of generality that k contains ζ_{p}. Let G_{p} be a Sylow p-subgroup of G and define V′ and g as in Lemma 2.1. Then V′ ⊂ V (g, ζ_{p}). Thus

ed(G; p) = ed(G_{p}; p) ⩽ ed(G_{p}) ⩽ dim(V′) ⩽ dim V (g, ζ_{p}) ⩽ a(p) ,

as desired. Note that the inequality ed(G_{p}) ⩽ dim(V′) is an immediate consequence of the definition of essential dimension; see, e.g., [R10, (2.3)]. □

We conclude this section with a refinement of Lemma 2.1 which will be used in the proofs of both Theorem 1.1(b) and Corollary 5.1.

Lemma 2.2. Let V be a finite-dimensional k-vector space, G ⊂ GL(V) be a finite group generated by pseudo-reflections, and G_{p} be a p-Sylow subgroup of G. Assume that ζ_{p} ∈ k and V′, g are as in statement of Lemma 2.1. Then dim V (g, ζ_{p}) = a(p).

Proof. By a theorem of Springer [Spr74, Theorem 3.4(ii)] there exists an h ∈ G such that dim V (h, ζ_{p}) = a(p) and V′ ⊂ V (g, ζ_{p}) ⊂ V (h, ζ_{p}); see [Spr74, Theorem 3.4(ii)]. Springer originally proved this result over ℂ; a proof over an arbitrary base field (containing ζ_{p}) can be found in [Kan01, Chapter 33].

After replacing h by a suitable power, we may assume that the order of h is a power of p. Let N = {x ∈ G | x(V′) = V′} be the stabilizer of V′ in G. Note that G_{p} ⊂ N and thus G_{p} is a p-Sylow subgroup of N. Moreover, we also have h ∈ N. Since the order of h is a power of p, there exists an element n ∈ N such that h′ = nhn^{−1} is in G_{p}. Since h acts on V′ as ζ_{p} id_{V′}, so does h′. Now, h′ and g both lie in G_{p} and have identical actions on V′, which is a faithful representation of G_{p}. Thus h′ = g, and a(p) = dim V (h, ζ_{p}) = dim V (h′, ζ_{p}) = dim V (g, ζ_{p}), as desired. □

3. Proof of Theorem 1.1(b): First reductions

We now turn to the proof of Theorem 1.1(b). In view of part (a), it suffices to show that ed(G; p) ⩾ a(p). Since ed_{k}(G; p) ⩾ ed_{l}(G; p), for any field extension l/k, we may assume without loss of generality that k is algebraically closed, and, in particular, that ζ_{p} ∈ k.

Our proof of Theorem 1.1(b) will proceed by contradiction. We begin by studying a minimal counterexample, with the ultimate goal of showing that it cannot exist.

Proposition 3.1. Let ϕ: G ↪ GL(V) be a counterexample to Theorem 1.1(b) of minimal dimension. That is, V is a vector space of minimal dimension such that there exists a finite group G with an embedding ϕ: G ↪ GL(V), where ϕ(G) is generated by pseudo-reflections, and

(3.1)  ed(G; p) < a_{ϕ}(p)

for some prime p. Then (a) dim(V) ⩾ 2.

(b) ϕ is irreducible.

(c) Some element g ∈ G of order p acts on V as a scalar. In particular, a_{ϕ}(p) = dim(V).

(d) G contains no elements of order p with exactly two eigenvalues.

(e) G contains no pseudo-reflections of order p.

(f) If p = 2 then g = − id_{V} is the unique element of order 2 in G.

(g) G_{p} is contained in the commutator subgroup [G, G]. Here, as usual, G_{p} denotes a p-Sylow subgroup of G.

(h) Let g ∈ G be as in part (c) and ϕ′: G → GL(V′) be an irreducible representation such that ϕ′(g) ≠ 1. Then dim(V′) is a multiple of p. In particular, dim(V) is a multiple of p.

(i) dim(V) ⩾ 2p.

Proof. (a) Assume the contrary: dim(V) = 1. In this case G is a cyclic group. If |G| is divisible by p then ed(G; p) = a(p) = 1; otherwise ed(G; p) = a(p) = 0. In both cases, (3.1) fails, a contradiction.

(b) Assume the contrary: V = V_{1}⊕V_{2}, where V_{1} and V_{2} are proper G-stable subspaces. Each pseudo-reflection g ∈ G acts non-trivially on exactly one summand V_{i}. For i = 1, 2, let G_{i} be the subgroup of G generated by those reflections that act non-trivially on V_{i}. Then G is isomorphic to the direct product G_{1} × G_{2}, and ϕ = ϕ_{1} ⊕ ϕ_{2}, where ϕ restricts to ϕ_{i}: G_{i} → GL(V_{i}), and ϕ_{1}(G_{1}), ϕ_{2}(G_{2}) are generated by pseudo-reflections. Note that a_{ϕ}(p) = a_{ϕ_{1}}(p) + a_{ϕ_{2}}(p). In addition, by [KM08, Theorem 5.1],

ed(G; p) = ed(G_{1}; p) + ed(G_{2}; p) .

By minimality of ϕ, we have that ed(G_{1}; p) ⩾ a_{ϕ_{1}}(p) and ed(G_{2}; p) ⩾ a_{ϕ_{2}}(p). Thus ed(G; p) ⩾ a_{ϕ}(p), a contradiction.

(c) Choose V′ and g as in Lemmas 2.1 and 2.2. Recall that g is a central element of G_{p} of order p and a_{ϕ}(p) = dim V (g, ζ_{p}). Set W := V (g, ζ_{p}). The element g acts on W as a scalar; our goal is to show that W = V.

Let S = {s ∈ G | sW = W } be the stabilizer of W in G and let S_{0} be the subgroup of S consisting of elements that fix W pointwise. Note that since g is central in G_{p}, we have G_{p} ⊂ S. Moreover, since G_{p} acts faithfully on V′ ⊂ W, we have G_{p} ∩ S_{0} = {1}. Restricting the action of S to W, we obtain a faithful representation of H = S/S_{0} on W, which we will denote by ψ. By [LM03, Theorem 1.1], ψ(H) ⊂ GL(W) is generated by pseudo-reflections. (Note that, while [LM03, Theorem 1.1] assumes k = ℂ, its proof goes through under our less restrictive assumptions on k.) By our construction,

a_{ϕ}(p) = dim(W) = a_{ψ}(p) .

Since G_{p} ⊂ S and G_{p} ∩ S_{0} = {1}, the quotient H = S/S_{0} contains an isomorphic image of G_{p}, which is a Sylow p-subgroup of H, so that

ed(G; p) = ed(G_{p}; p) = ed(H; p) .

Thus by (3.1), ed(H; p) = ed(G; p) < a_{ϕ}(p) = a_{ψ}(p). By the minimality of ϕ, we see that dim(V) = dim(W), i.e., V = W = V (g, ζ_{p}). This proves part (c).

(d) Assume the contrary: an element h of G of order p has exactly two distinct eigenvalues, ζ_{p}^{i} and ζ_{p}^{j}. After replacing h by a suitable power of hg^{−i}, where g is the central element we constructed in part (c), we may assume that i = 0 and j = 1. Then V is the direct sum of eigenspaces V_{0} ⊕ V_{1}, where V_{i} = V (h, ζ_{p}^{i}). Let G_{1} (resp. G_{0}) be the subgroup of G consisting of elements which fix V_{0} (resp. V_{1}) pointwise (note the reversed indices).

Given a finite group of order prime to the characteristic of k, any invariant subspace has a unique invariant complement. Note that V_{0} is the unique ⟨gh^{−1}⟩-invariant complement to V_{1}. Since ⟨gh^{−1}⟩ ⊂ G_{0}, we see that V_{0} is, in fact, the G_{0}-invariant complement. Similarly, since h ∈ G_{1}, the G_{1}-invariant complement to V_{0} is V_{1}. We now see that G_{0} and G_{1} commute and G_{0} ∩ G_{1} = {1}. Hence, G_{0} and G_{1} generate a subgroup of G isomorphic to G_{0} × G_{1}. By abuse of notation we shall denote this group by G_{0} × G_{1}.

Note that ϕ restricts to faithful representations ϕ_{0} : G_{0} → GL(V_{0}) and ϕ_{1} : G_{1} → GL(V_{1}). Since ϕ_{0}(gh^{−1}) = ζ_{p} id_{V_{0}} and ϕ_{1}(h) = ζ_{p} id_{V_{1}}, we have

a_{ϕ_{0}}(p) = dim(V_{0}) and a_{ϕ_{1}}(p) = dim(V_{1}).

We now recall that by a theorem of R. Steinberg [Ste64, Theorem 1.5], G_{0} and G_{1} ⊂ GL(V) are both generated by pseudo-reflections. (In positive characteristic this is due to J.-P. Serre [Ser68]; cf. [DK01, Proposition 3.7.8].) Since G_{1} acts trivially on V_{0} and G_{0} acts trivially on V_{1}, we conclude that ϕ_{0}(G_{0}) and ϕ_{1}(G_{1}) are also generated by pseudo-reflections.

By the minimality of ϕ, Theorem 1.1(b) holds for ϕ_{0} and ϕ_{1}. Thus

ed(G; p) ⩾ ed(G_{0} × G_{1}; p) = ed(G_{0}; p) + ed(G_{1}; p) =
a_{ϕ_{0}}(p) + a_{ϕ_{1}}(p) = dim(V_{0}) + dim(V_{1}) = dim(V) = a_{ϕ}(p).

Here the first equality is [KM08, Theorem 5.1], and the second follows from the minimality of ϕ. The resulting inequality contradicts (3.1).

(e) By part (a), dim(V) ⩾ 2. Hence, a pseudo-reflection has exactly two distinct eigenvalues, and (e) follows from (d).

(f) Every element of GL(V) of order 2, other than − id_{V}, has exactly two distinct eigenvalues and thus cannot lie in G by (d).

(g) By (e), G does not have any pseudo-reflections of order p, and hence of any order divisible by p. The finite abelian group G/[G, G] is generated by the images of the pseudo-reflections. All of these images have order prime to p. Hence, the order of G/[G, G] is prime to p. We conclude that G_{p} ⊂ [G, G].

(h) Since g is central, ϕ′(g) = λ id_{V′}, where λ is a primitive pth root of unity. Thus det ϕ′(g) = λ^{dim(V′)}. On the other hand, by part (g), g ∈ G_{p} ⊂ [G, G] and hence, det ϕ′(g) = 1. Thus dim(V′) is divisible by p.

(i) Let C = ⟨g⟩, where g is as in part (c). Applying [R10, Theorem 4.1] (with r = 1) to the central exact sequence 1 → C → G → G/C → 1 we obtain the inequality

(3.2)  ed(G; p) ⩾ gcd_{ϕ′} dim(ϕ′) ,

where ϕ′: G → GL(V′) runs over all irreducible representations of G such that the restriction of ϕ′ to C is non-trivial, or equivalently, ϕ′(g) ≠ 1. Note that the statement of [R10, Theorem 4.1] only gives this inequality for ed(G). However, it remains valid for ed(G; p); see [R10, Section 5] or the proof of [LMMR13, Theorem 3.1].

By part (h), dim(ϕ′) is divisible by p for every such ϕ′. Thus ed(G; p) ⩾ p. Assumption (3.1) now tells us that dim(V) > p. Since dim(V) is divisible by p by (h), we conclude that dim(V) ⩾ 2p. □

4. Conclusion of the proof of Theorem 1.1(b)

The remainder of the proof of Theorem 1.1(b) relies on the classification of irreducible pseudo-reflection groups due to Shephard and Todd [ST54]. Shephard and Todd worked over the field k = ℂ of complex numbers. We are working over a base field k such that char(k) does not divide |G|. As we explained at the beginning of the previous section, we may (and will) assume that k is algebraically closed. Before we proceed with the proof of Theorem 1.1(b), we would like to explain how the Shephard-Todd classification applies in this more general situation.

If k is an algebraically closed field of characteristic zero, then any representation of a finite group over k descends to \overline{ℚ} ⊂ k; see [Ser77, Section 12.3]. Hence, this representation is defined over ℂ, and the entire Shephard-Todd classification remains valid over k.

Now suppose k is an algebraically closed field of positive characteristic. Let A = W (k) be its Witt ring. Recall that A is a complete discrete valuation ring of characteristic zero, whose residue field is k. Denote the fraction field of A by K and the maximal ideal by M. It is well known that if char(k) does not divide |G| (which is our standing assumption) then every n-dimensional k[G]-module V lifts to a unique A[G]-module V_{A}, which is free of rank n over A.

It is shown in [Ser77, Section 15.5] that the lifting operation V ↦ V_{K} := V_{A} ⊗ K and the “reduction mod M” operation V_{K} ↦ V give rise to mutually inverse bijections between the representation rings R_{k}(G) and R_{K}(G) of G. These bijections send irreducible k-representations to irreducible K-representations of the same dimension, and they are functorial in both V and G. In particular, if g ∈ G and ζ_{d} ∈ k is a primitive dth root of unity then the eigenspace V (g, ζ_{d}), viewed as a representation of the cyclic subgroup ⟨g⟩ ⊂ G, lifts to V_{K}(g, η_{d}) for some primitive dth root of unity η_{d} ∈ A such that

(4.1)  ζ_{d} = η_{d} (mod M)

Taking d = 1, we see that if g ∈ G acts on V as a pseudo-reflection if and only if it acts on V_{K} as a pseudo-reflection.

This shows that for every pseudo-reflection group ϕ: G ↪ GL(V) over k there is an abstractly isomorphic pseudo-reflection group ϕ_{K}: G ↪ GL(V_{K}) over K. For each g ∈ G, the eigenvalues of ϕ(g) and ϕ_{K}(g) are the same, modulo M, in the sense that if η_{d} is an eigenvalue of ϕ_{K}(g) then ζ_{d} is an eigenvalue of ϕ(g), as in (4.1). Thus dim_{k} V (g, ζ_{d}) = dim_{K} V (g, η_{d}) and consequently,

a_{ϕ}(d) = max_{g∈G} dim_{k} V (g, ζ_{d}) = max_{g∈G} dim_{K} V_{K}(g, η_{d}) = a_{ϕ_{K}}(d)

for every d ≥ 1. Note also that the degrees of the fundamental invariants are the same since they can be recovered from the a(d)’s as d varies; cf. (1.4).

We conclude that if k is an algebraically closed field satisfying the above assumptions, then many properties of irreducible pseudo-reflection groups, whose orders are prime to char(k), are the same over k as they are over ℂ: their isomorphism types, the numbers a(d) for each d ≥ 1, the numbers of pseudo-reflections of each order, the number of central elements of each order, and the degrees of the fundamental invariants. This allows us to use the Shephard-Todd classification (e.g., from [LT09, Appendix D], where k is assumed to be ℂ) in our setting; cf. [Kan01, Section 15.3].

We now proceed with the proof of Theorem 1.1(b). Let ϕ: G ↪ GL(V) be a minimal counterexample, as in the statement of Proposition 3.1. Then by Proposition 3.1, ϕ is irreducible. By the Shephard-Todd classification, there are three infinite families and 34 exceptional groups. We will denote the infinite families by ST_{1}, ST_{2} and ST_{3}, and the exceptional groups ST_{4} through ST_{37}, following the numbering in [ST54].

The infinite families ST_{1} – ST_{3}.

Case ST_{1}: Here V is the natural (n − 1)-dimensional representation of G := S_{n}. For n ≥ 3, G has trivial center and hence, cannot be minimal by Proposition 3.1(c). For n = 2, dim(V) = 1, contradicting Proposition 3.1(a).

Case ST_{2}: Here G = G(m, l, n) ⊂ GL_{n}, where m, n > 1, l divides m, and (m, l, n) ≠ (2, 2, 2). Here G(m, l, n) is defined as a semidirect product of the diagonal subgroup

A(m, l, n) = {diag(ζ_{m}^{a_{1}}, …, ζ_{m}^{a_{n}}) | a_{1} + ⋯ + a_{n} ≡ 0 (mod l)} ⊂ GL_{n}

and the symmetric group S_{n}, whose elements are viewed as permutation matrices in GL_{n}; see [LT09, Chapter 2]. (Note that [LT09] assumes k = ℂ, but the same construction works in our more general context.) By Proposition 3.1(c), G(m, l, n) contains the scalar matrix ζ_{p} id. This matrix has to be contained in A(m, l, n); hence, p divides m. Moreover by Proposition 3.1(i), we may assume n ⩾ 2p. Consider g = diag(ζ_{m}^{m/p}, …, ζ_{m}^{m/p}, 1, …, 1) ∈ A(m, l, n) ⊂ G(m, l, n), where ζ_{m}^{m/p} occurs p times. This element has order p and exactly two eigenvalues, contradicting Proposition 3.1(d).

Case ST_{3}: Here G is cyclic and V is a 1-dimensional. Once again, this contradicts Proposition 3.1(a).

The exceptional cases ST_{4} – ST_{37}.

All of the exceptional cases satisfy dim(V) ⩽ 8. On the other hand, by Proposition 3.1(i), dim(V) = mp, where m ⩾ 2. We conclude that either (I) p = 2 and dim(V) = 4, 6 or 8, or (II) p = 3 and dim(V) = 6.

Case I: We need to consider the groups ST_{28}–ST_{32}, ST_{34}, ST_{35}, and ST_{37}, with p = 2. With the exception of ST_{32}, each of these groups has a reflection of order 2 and thus is ruled out by Proposition 3.1(e). The group ST_{32} is isomorphic to ℤ/3ℤ × Sp_{4}(𝔽_{3})

(see [LT09, Theorem 8.43]). The group Sp_{4}(𝔽_{3}) has non-central elements of order 2, contradicting Proposition 3.1(f).

Case II: We only need consider two groups, ST_{34} and ST_{35}. The group ST_{35} has trivial centre and thus is ruled out by Proposition 3.1(c). (Recall that the order of the centre is the greatest common divisor of the degrees d_{1}, …, d_{6}. For ST_{35} = W (E_{6}) these are, 2, 5, 6, 8, 9, and 12.) This leaves us with G = ST_{34}, otherwise known as the Mitchell group. The structure of this group was investigated by J. H. Conway and N. J. A. Sloane. In [CS83, Section 2] they constructed four isomorphic lattices, Λ^{(i)}, where i = 2, 3, 4 and 7, whose automorphism group is ST_{34}. In subsection 2.3 they showed that ST_{34} ≃ Aut(Λ^{(3)}) contains the group (2 × 3^{5}) ⋊ S_{6}, which, in turn, contains, the non-exceptional group G(3, 3, 6). Thus

ed(G(3, 3, 6); 3) ⩽ ed(ST_{34}; 3) < a(3) = 6 ,

where the second inequality follows from our assumption that ST_{34}, with its natural 6-dimensional representation, is a counterexample to Theorem 1.1(b) of minimal dimension. (Here p = 3). Then G(3, 3, 6), with its natural 6-dimensional representation and p = 3, is also a counterexample of minimal dimension. Since we have ruled out every non-exceptional group as a counterexample of minimal dimension, this is a contradiction. The proof of Theorem 1.1(b) is now complete. □

5. A representation-theoretic corollary

Before proceeding further we record a representation-theoretic corollary of our proof of Theorem 1.1(b), which, to the best of our knowledge, has not been previously noticed. As before, rdim(H) the minimal dimension of a faithful representation of a finite group H over the base field k.

Corollary 5.1. Suppose ζ_{p} ∈ k. Let G ⊂ GL(V) be a finite subgroup generated by pseudo-reflections, G_{p} be a p-Sylow subgroup of G, and V′ ⊂ V be a minimal (with respect to inclusion) faithful k-subrepresentation of G_{p}. Then dim(V′) = rdim(G_{p}).

Proof. Since ζ_{p} ∈ k, rdim(G_{p}) = ed(G; p) by the Karpenko-Merkurjev theorem (1.1). Choose g as in Lemma 2.1. Then, by Lemma 2.2,

ed(G; p) = rdim(G_{p}) ⩽ dim(V′) ⩽ dim V (g, ζ_{p}) = a(p) .

By Theorem 1.1(b), ed(G; p) = a(p) and thus the above inequalities are all equalities. This completes the proof of Corollary 5.1. □

The following example shows that Corollary 5.1 fails if G ⊂ GL(V) is not assumed to be generated by pseudo-reflections.

Example 5.2. Let p > 2 be a prime, H be a non-abelian group of order p^{3}, and ψ: H ↪ GL(U) be a faithful p-dimensional representation of H. Set G = H × H and

ϕ = ψ_{1} ⊗ ψ_{2} ⊕ ψ_{1} : G → GL(U ⊗ U ⊕ U) ,

where for i = 1, 2, ψ_{i} is the composition of ψ with the projection G → H to the ith factor. Both ψ_{1} ⊗ ψ_{2} and ψ_{1} are irreducible representations of G; the irreducibility of ψ_{1} ⊗ ψ_{2} follows from [Ser77, Theorem 3.2.10(i)]. These irreducible representations are distinct, because dim(ψ_{1} ⊗ ψ_{2}) = p^{2} and dim(ψ_{1}) = p.

Note that G = G_{p} is a group of order p^{6}, and V = U ⊗U ⊕U is a faithful representation of G. Since it is a direct sum of two distinct irreducibles, neither of which is faithful, the only faithful G_{p}-subrepresentation V′ of V is V itself. On the other hand, G has a 2pdimensional faithful representation ψ_{1} ⊕ ψ_{2}; hence, rdim(G) ⩽ 2p. In summary, G = G_{p}, V = V′ and dim(V′) = p^{2} + p > 2p ⩾ rdim(G_{p}). Thus the assertion of Corollary 5.1 fails for ϕ(G) ⊂ GL(V).

6. Proof of Theorem 1.3(a)

The degrees of the fundamental invariants of W (E_{6}) are 2, 5, 6, 8, 9 and 12; see, e. g., [LT09, p. 275]. Thus by Theorem 1.1(b), ed(W (E_{6}); 2) = 4. This shows that ed W (E_{6}) ⩾ 4.

Recall that ed W (E_{6}) is the minimal value of dim(Y) such that there exists a dominant rational W (E_{6})-equivariant map V −−→ Y defined over k, where V is a linear representation of W (E_{6}), and Y is a a k-variety with a faithful action of W (E_{6}); see, e.g., [R10, Section 2]. To prove the opposite inequality, ed W (E_{6}) ⩽ 4, it thus suffices to establish the following lemma suggested to us by I. Dolgachev.

Lemma 6.1. Let k be a field of characteristic ≠ 2, 3. There exists a dominant W (E_{6})- equivariant map

f: 𝔸^{6} −−→ Y

where 𝔸^{6} is a linear representation of W (E_{6}) and Y is a 4-dimensional variety with a faithful action of W (E_{6}).

Proof. First, we construct Y. Consider the space (ℙ^{2})^{6} of ordered 6-tuples of points in the projective plane, and let U ⊂ (ℙ^{2})^{6} be the dense open consisting of 6-tuples (a_{1}, …, a_{6}) such that no two of the points a_{i} lie on the same line, and no six lie on the same conic. This open subset is invariant under the natural (diagonal) PGL_{3}-action on (ℙ^{2})^{6}. Moreover, U is contained in the stable locus of (ℙ^{2})^{6} for this action; see, e.g., [DO88, p. 116]. Thus there exists a geometric quotient q: U → Y := U/ PGL_{3}. The explicit description in [DO88, Example I.3] show that Y and q are defined over k. Note that

dim(Y) = dim(U) − dim(PGL_{3}) = dim (ℙ^{2})^{6} − dim(PGL_{3}) = 12 − 8 = 4,

as desired.

Now, we construct the affine space 𝔸^{6} and its map to Y. Let x, y, z be projective coordinates on ℙ^{2} and C ⊂ ℙ^{2} be the cubic yz^{2} = x^{3}. Note that C has a cusp at (0 : 1 : 0). The smooth locus C_{sm} = C \ {(0 : 1 : 0)} is an algebraic group isomorphic to the additive group 𝔾_{a}. Indeed, we identify 𝔾_{a} ≃ 𝔸^{1} with C_{sm} via t ↦ (t : t^{3} : 1). Thus the space C_{sm}^{6} is isomorphic to affine space 𝔸^{6}.

This yields a rational map

ϕ: C_{sm}^{6} → C^{6} ↪ (ℙ^{2})^{6} .

Three points in t_{1}, t_{2}, t_{3} ∈ C_{sm} lie on a line if and only if t_{1} + t_{2} + t_{3} = 0; six points t_{1}, …, t_{6} ∈ C_{sm} lie on a conic if and only t_{1} + ⋯ + t_{6} = 0. Thus for general (t_{1}, …, t_{6}) ∈ C_{sm}^{6}, we have ϕ(t_{1}, …, t_{6}) ∈ U. In other words, we may view ϕ as a rational map C_{sm}^{6} −−→ U. We now define the map f: C_{sm}^{6} −−→ Y as the composition

f: C_{sm}^{6} −−→^{ϕ} U →^{q} Y .

By [Sh95, Lemma 13], over the algebraic closure, if (t_{1}, …, t_{6}) is a 6-tuple of points in general position in ℙ^{2} then there is a cuspidal cubic C′ ⊂ ℙ^{2} such that t_{1}, …, t_{6} lie in the smooth locus of C′. Since any two cuspidal cubics in ℙ^{2} are projectively equivalent (recall our assumptions on the characteristic), we conclude that f is dominant.

It remains to construct actions of W (E_{6}) on 𝔸^{6} and Y, and to show that f is equivariant. Recall that blowing up 6 points in ℙ^{2} produces a cubic surface X with the 6 exceptional divisors of the blow-up corresponding to a “sixer”: 6 pairwise disjoint lines in X. Conversely, any sixer can be blown down to produce 6 points on ℙ^{2}. Over an algebraically closed field, the elements of W (E_{6}) act freely and transitively on the set of sixers in X (where we keep track of the ordering of the 6 lines). This produces a faithful action of W (E_{6}) on Y which is defined over k. This action of the Weyl group W (E_{6}) on Y is sometimes called the Cremona representation or the Coble representation. For more details, see [Dol83, Section 7], [Dol08, Section 6], and [DO88, Chapter 6].

We recall how W (E_{6}) acts on the Picard group N of a smooth cubic surface X ⊂ ℙ^{3} over an algebraically closed field; see, e.g., [Dol83, Sections 4 and 5] or [Man86, Section IV.26]. The Picard group N ≃ ℤ^{7} with its intersection form is a lattice with a symmetric bilinear form given by diag(1, −1, …, −1) with respect to the basis e_{0}, …, e_{6}, where e_{0} is the hyperplane section of X and e_{1}, …, e_{6} is a collection of 6 mutually disjoint lines on X.

We consider a set of fundamental roots in N given by

α_{1} = e_{0} − e_{1} − e_{2} − e_{3}, α_{2} = e_{2} − e_{1}, … α_{6} = e_{6} − e_{5} .

The reflections associated to these roots generate a group isomorphic to W (E_{6}). Note that the reflections associated to α_{2}, …, α_{6} generate a subgroup isomorphic to S_{6} which permute the basis elements e_{1}, …, e_{6}. The symmetric group S_{6} naturally acts on C_{sm}^{6} and (ℙ^{2})^{6} by permutations; thus f is S_{6}-equivariant. It remains to consider the reflection g ∈ W (E_{6}) associated to the root α_{1}.

First, we identify the action of g on Y. Suppose π : X → ℙ^{2} is the blowup of 6 points a_{1}, …, a_{6}. Identifying each e_{i} with the class of each exceptional divisor E_{i} := π^{−1}(a_{i}) in the cubic surface X we may determine the action of g. Indeed, for i ≠ j ≠ k taken from {1, 2, 3}, the line E_{i} is taken to the strict transform of the line between a_{j} and a_{k}; while E_{4}, E_{5}, E_{6} are all left fixed. Recall that the standard quadratic transform s : ℙ^{2} −−→ ℙ^{2} at the points a_{1}, a_{2}, a_{3} is the map obtained by blowing up the points and then blowing down the strict transforms of the lines between them. In this language, g : Y → Y is given by

[a_{1}, …, a_{6}] ↦ [s(a′_{1}), s(a′_{2}), s(a′_{3}), s(a_{4}), s(a_{5}), s(a_{6})]

where a′_{1} is any point on the line between a_{2} and a_{3} (and similarly for a′_{2} and a′_{3}).

We now construct an action of g on C_{sm}^{6} following H. Pinkham [P80]. If C ⊂ ℙ^{2} is a cuspidal cubic, then, for any three points u_{1}, u_{2} and u_{3} in the smooth locus C_{sm} of C, C′ = s(C) is also a cuspidal cubic in ℙ^{2}. Since any two cuspidal cubics in ℙ^{2} are linear translates of each other, there exists an l ∈ PGL_{3} such that l(C′) = C. Composing s with l, one obtains a rational map l · s : C_{sm} −−→ C_{sm} which is regular on C_{sm} \ {u_{1}, u_{2}, u_{3}}. Let u′_{1} be the unique third intersection point of C with the line passing through u_{2} and u_{3} (and similarly for u′_{2} and u′_{3}). We define a map g : C_{sm}^{6} → C_{sm}^{6} via

(u_{1}, …, u_{6}) → (l · s(u′_{1}), l · s(u′_{2}), l · s(u′_{3}), l · s(u_{4}), l · s(u_{5}), l · s(u_{6})) .

By construction, we see that f is g-equivariant.

Note that the choice of l and thus of the map l · s : C_{sm} −−→ C_{sm} above is not unique. Pinkham’s observation [P80, pp. 196–197] is that there is a choice of l such that the resulting map g gives rise to a linear representation of W (E_{6}) = ⟨g, S_{6}⟩ on C_{sm}^{6} ≃ 𝔸^{6}. In fact, C_{sm}^{6} can be identified with a Cartan subalgebra of the Lie algebra of type E_{6} with the standard action of the Weyl group. This construction is valid over any field k of characteristic ≠ 2, 3. This completes the proof of Lemma 6.1 and thus of Theorem 1.3(a). □

7. Proof of Theorem 1.3(b) and (c)

As we have previously pointed out, ed(G) ⩽ dim(V); see, e.g., [R10, (2.3)]. In the case where G = G(m, m, n) and m ⩾ 2 and (m, n) are relatively prime, no element of G acts as a scalar on V. The natural G-equivariant dominant rational map V −−→ ℙ(V) tells us that ed(G) ⩽ dim(V) − 1.

It now suffices to show that for every irreducible G ⊂ GL(V) generated by pseudoreflections there exists a prime p such that

a(p) = \begin{cases}dim(V) − 1, & if G ≃ G(m, m, n) with m, n relatively prime, \\ dim(V), & otherwise. \end{cases}

Indeed, Theorem 1.1(b) will then tell us that ed(G) ⩾ ed(G; p) ⩾ a(p) ⩾ dim(V) − 1 in the first case and ed(G) ⩾ ed(G; p) ⩾ a(p) ⩾ dim(V) in the second. Since we have established the opposite inequalities, this will complete the proof in both cases.

By Springer’s theorem (1.4), a(p) is equal to the number of invariant degrees d_{i} which are divisible by p. In the case where G = G(m, m, n), m ⩾ 2 and (m, n) are relatively prime, the degrees d_{i} are m, 2m, …, (n − 1)m, and n. Taking p to be a prime divisor of m, we see that a(p) = n − 1 = dim(V) − 1, as desired.

For all other groups of the form G = G(m, l, n), with m ⩾ 2 the degrees d_{i} are m, 2m, …, (n − 1)m, and \frac{mn}{l}. All of them are divisible by every prime factor p of gcd(m, \frac{mn}{l}) > 1. Hence, in this case a(p) = n = dim(V), as desired.

Finally, in the case where m = 1, G(m, l, n) = G(1, 1, n) = S_{n} is excluded by our hypothesis.

This leaves us with the exceptional groups ST_{4} – ST_{37}. If G ≠ ST_{25}, ST_{35} then every degree d_{i} of G is divisible by 2. If G = ST_{25} then every degree d_{i} of G is divisible by 3. Finally, ST_{35} = W (E_{6}) was treated in part (a). □

Remark 7.1. Our proof shows that for every G in the statement of Theorem 1.3 there is a prime p such that ed(G) = a(p) = ed(G; p).

Remark 7.2. Pinkham’s construction applies in greater generality than the case of W (E_{6}) used in Lemma 6.1. In particular, one can use it to construct a dominant rational W (E_{7})-equivariant map 𝔸^{7} −−→ Z, where Z is a dense open subset of the 6-dimensional variety (ℙ^{2})_{ss}^{7}// PGL_{3}. Here the subscript _{ss} denotes the semistable locus. Since we know from part (c) that ed W (E_{7}) = 7, this gives an alternative (indirect) proof of the classical fact that the Coble representation of W (E_{7}) on (ℙ^{2})_{ss}^{7}// PGL_{3} is not faithful; see [Dol83, p. 293] or [DO88, p. 122].

8. Proof of Theorem 1.4: preliminaries

First we observe that part (b) is an immediate consequence of part (a). Indeed, combining the first inequality in (1.5) with part (a), we have

max_{p} ed(G; p) ⩽ pmed(G) ⩽ max_{p} a(p) ,

Theorem 1.1(b) now tells us that a(p) = ed(G; p) for each prime p, and part (b) follows.

We now proceed with the proof of Theorem 1.4(a). Let G be a finite group and G ↪ GL(V) be a linear representation defined over k. We will assume throughout that char(k) does not divide |G|. Consider the closed subscheme

B := ⋃_{g∈G, ζ≠1} V (g, ζ) or equivalently, B = ⋃_{\begin{matrix}g ∈ G, ζ^{p} = 1 \\ ζ ≠ 1, p prime \end{matrix}} V (g, ζ) ,

where ζ ranges over the roots of unity in k̄. Note that, although each V (g, ζ) is defined only over k(ζ), their union B is defined over k.

The following lemma may be viewed as a variant of [Spr74, Proposition 3.2].

Lemma 8.1. Let m ≥ |G| be an integer. Suppose v ∈ V has the property that f(v) = 0 for every G-invariant homogeneous polynomial f of degree m. Then v ∈ B.

Proof. We may assume v ≠ 0. Let \overline{v} ∈ ℙ(V) be the projective point associated to v. Denote the G-orbit of \overline{v} by \overline{v}_{1} = \overline{v}, \overline{v}_{2}, …, \overline{v}_{r} ∈ ℙ(V). Note that r ≤ |G| ≤ m.

We claim that there exists a homogeneous polynomial h ∈ k[V] of degree m such that h(\overline{v}_{1}) ≠ 0 but h(\overline{v}_{i}) = 0 for any i = 2, …, r. To construct h, for every i = 2, …, r choose a linear form l_{i} ∈ V^{∗} such that l_{i}(\overline{v}_{i}) = 0 but l_{i}(\overline{v}_{1}) ≠ 0. Now set h = l_{2}^{m+2−r}l_{3} … l_{r}. This proves the claim.

We now define a G-invariant homogeneous polynomial f of degree m by summing the translates of h over G:

(8.1)  f(v′) = ∑_{g∈G} h(g · v′) ∀v′ ∈ V .

By our assumption, f(v) = 0.

Let S ⊂ G be the stabilizer of \overline{v}, i.e., the subgroup of elements s ∈ G such that v is an eigenvector for s. Then s(v) = χ(s)v for some multiplicative character χ: S → k^{∗}. It now suffices to show that χ(s) ≠ 1 for some s ∈ S. Indeed, if we denote χ(s) by ζ, for this s, then v ∈ V (s, ζ) ⊂ B, as desired.

To show that χ(s) ≠ 1 for some s ∈ S, recall that by our choice of h, h(g · v) = 0 unless g ∈ S. Thus

0 = f(v) = ∑_{s∈S} h(s · v) = ∑_{s∈S} h(χ(s)v) = ∑_{s∈S} χ(s)^{m}h(v) .

If χ(s) = 1 for every s ∈ S, this yields 0 = |S| · h(v). This is a contradiction since h(v) ≠ 0, and we are assuming that char(k) does not divide |G|. Thus χ(s) ≠ 1 for some s ∈ S, as claimed. □

Denote the direct sum of V and the trivial 1-dimensional representation of G by W := V × k. We will identify V with V × {0} ⊂ W and ℙ(V) with the hyperplane ℙ(V × {0}) in ℙ(W). Set n := dim(V) = dim(ℙ(W)).

Lemma 8.2. Consider the rational map

ψ_{m}: ℙ(W) −−→ ℙ^{N}

given by the linear system k[W]_{m}^{G} of G-invariant homogeneous polynomials of degree m on W. Denote the closure of the image of ψ_{m} by Y ⊂ ℙ^{N}. Assume m ⩾ |G|. Then

(a) The map ψ_{m} is regular away from ℙ(B).

(b) ψ_{m}: ℙ(W) −−→ Y induces an isomorphism between k(Y) and the field of Ginvariant rational functions on ℙ(W).

(c) Let 1 ≤ i ≤ N − 1, Gr(i, N) be the Grassmannian of (projective) i-dimensional linear subspaces L ⊂ ℙ^{N}, and I = {(y, L) | y ∈ L} ⊂ Y × Gr(i, N) be the incidence correspondence. Then I is irreducible over k̄. Furthermore, if k is infinite then k-points are dense in I.

Proof. Let z be the coordinate along the second factor of k in W = V × k.

(a) We may assume without loss of generality that k is algebraically closed. Since z^{m} ∈ k[W]_{m}^{G}, we see that the indeterminacy locus of ψ_{m} consists of points (v : a) ∈ ℙ(W) with a = 0 and f(v) = 0 for every f ∈ k[V]_{m}^{G}, where k[V]_{m}^{G} denotes the k-vector space of G-invariant homogeneous polynomial on V of degree m. By Lemma 8.1, v ∈ B. Thus (v : a) ∈ ℙ(B) ⊂ ℙ(V × {0}) ⊂ ℙ(W), as claimed.

(b) We see that ψ_{m}^{∗}: k(Y) ↪ k(ℙ(W))^{G}. To show that this inclusion is an isomorphism, we restrict ψ_{m} to the dense open subset V ⊂ ℙ(W) given by z = 1. This restriction is the morphism

\begin{matrix}V → 𝔸^{N} \\ v ↦ (f_{1}(v), …, f_{N}(v)) , \end{matrix}

where f_{1}, …, f_{N} form a basis of the vector space of k[V]_{⩽m}^{G} of G-invariant polynomials of degree ⩽ m. Consequently, f_{1}, …, f_{N} ∈ ψ_{m}^{∗} k(Y). By the Noether bound, k[V]^{G} is generated by polynomials of degree ⩽ |G| as a k-algebra; see [DK01, Section 3.8]. Since |G| ⩽ m, we conclude that ψ_{m}^{∗} k(Y) contains k[V]^{G} and thus its fraction field k(V)^{G}. Since V is a G-invariant dense open subset of ℙ(W), we have k(V) = k(ℙ(W)). Therefore, ψ_{m}^{∗} k(Y) ⊃ k(V)^{G} = k(ℙ(W))^{G}, as desired.

(c) The fiber of any y ∈ Y (k̄) under the natural projection π_{1} : I → Y is k̄-isomorphic to a smaller Grassmannian Gr(i − 1, N − 1). Irreducibility of I now follows from the fact that Y is irreducible and the fibers of π_{1} are all of the same dimension.

To prove the second assertion, let U ⊂ I be a dense open subset defined over k. Our goal is to show that U contains a k-point. Let π_{1}|_{U} : U → Y be the restriction of the natural projection π_{1} : I → Y to the first factor. By Chevalley’s theorem, the image of π_{1}|_{U} contains a dense open subset U′ of Y. Note that k-points are dense in Y since they are dense in ℙ^{N} and ϕ_{m} : ℙ^{N} −−→ Y is dominant. Thus, there exists a k-point y_{0} in U′. The fiber I_{0} := π_{1}^{−1}(y_{0}) = {(y_{0}, L) | y_{0} ∈ L} intersects U non-trivially. Since I_{0} is isomorphic to the Grassmannian Gr(i − 1, N − 1) defined over k, k-points are dense in I_{0}. Hence, there is a k-point in I_{0} ∩ U. □

Remark 8.3. E. Noether showed that k[V]^{G} is generated by polynomials of degree ⩽ |G| as a k-algebra under the assumption that char(k) = 0. The more general variant of the Noether bound used in the proof of Lemma 8.2(b) (where char(k) > 0 is allowed, as long as char(k) does not divide |G|) is due to P. Fleischmann, J. Fogarty, and D. Benson. For details and further references, see [DK01, Section 3.8].

9. Conclusion of the proof of Theorem 1.4(a)

Set d := dim(B) = max_{p} a(p). Our goal is to construct a d-dimensional irreducible Gvariety X_{0} which is p-versal for every prime p. This would imply pmed(G) ⩽ dim(X_{0}) = d, as desired.

To construct X_{0}, let q be a prime integer such that q > |G| and q ≠ char(k), and let ψ_{q}: ℙ(W) −−→ Y ⊂ ℙ^{N} be the rational map given by the linear system of G-invariant homogeneous polynomials of degree q. By part (a) of Lemma 8.2, ψ_{q} is regular away from B, and by part (b), ψ_{q} is generically a G-torsor. Let Y_{0} be a dense open subset of Y over which ψ_{q} is a G-torsor (and in particular, etale).

Recall that n = dim(V) = dim(ℙ(W)). Let L be a linear subspace of ℙ^{N} of codimension n − d defined over k (we will impose further conditions on L soon). Let X be the preimage in ℙ(W \ B) of the set L ∩ Y ⊂ ℙ^{N}; similarly, let X_{0} be the preimage of L ∩ Y_{0}. In other words, X is the scheme-theoretic intersection of G-invariant hypersurfaces H_{1}, …, H_{n−d} of degree q in ℙ(W \ B),

(9.1)  X = (H_{1} ∩ ⋯ ∩ H_{n−d}) \ ℙ(B) .

Note that, by the fiber dimension theorem dim(X) = dim(X_{0}) = d. We think of L as a k-point in Gr(N − (n − d), N). We will choose it subject to the following conditions:

(i) X is irreducible,

(ii) X_{0} is smooth,

(iii) L passes through a k-point of Y_{0}. We claim that there exists a k-point L of Gr(N − n + d, N) satisfying (i), (ii) and (iii). Indeed, by variants of Bertini’s theorem [Zha95, Corollaire 1.7] and [Jou83, Corollaire 6.11], conditions (i) and (ii) are satisfied by all L contained in dense open subsets U_{1} ⊂ Gr(N − n + d, N) and U_{2} ⊂ Gr(N − n + d, N) respectively.

Let I = {(y, L) | y ∈ L} ⊂ Y × Gr(N − n + d, N) be the incidence correspondence. Both natural projections π_{1} : I → Y and π_{2} : I → Gr(N −n+d, N) are dominant. Hence, π_{2}^{−1}(U_{1}), π_{2}^{−1}(U_{2}) and π_{1}^{−1}(Y_{0}) are non-empty open dense subsets of I. By Lemma 8.2(c), there exists a k-point (y, L) in their intersection. Thus L ∈ Gr(N − n + d, N) satisfies conditions (i), (ii) and (iii). This proves the claim.

We now fix L and X = π^{−1}(Y ∩ L), X_{0} = π^{−1}(Y ∩ L) for the rest of the proof. Note that by our choice of Y_{0}, G acts faithfully on π^{−1}(y) ∈ X_{0}, and hence, on X_{0}. It remains to show that the G-action on X_{0} is p-versal for every prime p.

Case 1: p = q. Recall that the G-action on X_{0} is p-versal if and only the G_{p}-action on X_{0} is p-versal, where G_{p} is a Sylow p-subgroup of G; see [DR13, Corollary 8.6]. Since q > |G|, we have G_{q} = {1}. Thus to show that X_{0} is q-versal it suffices to show that X_{0} has a 0-cycle of degree prime to q; see [DR13, Lemma 8.2 and Theorem 8.3].

To construct this 0-cycle, recall that by (iii) there exists a k-point x ∈ L ∩ Y_{0}. By Lemma 8.2(b) the preimage ψ_{q}^{−1}(x) is a 0-cycle in X_{0} of degree |G|, which is prime to q, since q > |G|.

Case 2: p ≠ q. To show that the G-action on X_{0} is p-versal it suffices to prove that for every field extension K/k and every G-torsor T → Spec(K), the twisted K-variety T X_{0} contains a 0-cycle Z of degree q^{n−d} over K; see [DR13, Section 8].

Since the G-action on ℙ(W) lifts to a linear G-action on W, Hilbert’s Theorem 90 tells us that ^{T} ℙ(W) = ℙ(W_{K}) is a projective space over K; see, e.g., [DR13, Lemma 10.1]. Note that

dim ^{T} ℙ(B) = dim ℙ(B) = d − 1

and, since dim(X) = d and X is irreducible,

dim(^{T} X \^{T} X_{0}) = dim(X \ X_{0}) ⩽ d − 1 .

Thus, a linear subspace M of codimension d in ℙ(W_{K}) in general position misses both T ℙ(B) and ^{T} X \^{T} X_{0}. Let Z be the intersection cycle obtained by intersecting ^{T} X_{0} with the linear subspace M. By (9.1),

T X = (^{T} H_{1} ∩ ⋯ ∩ ^{T} H_{n−d}) \ ^{T} ℙ(B) .

Thus,

M ∩ ^{T} H_{1} ∩ ⋯ ∩ ^{T} H_{n−d} = M ∩ ^{T} X = M ∩ ^{T} X_{0} = Z .

By [DR13, Lemma 10.1(c)], each ^{T} H_{i} is a hypersurface of degree q in ℙ(W_{K}). We may think of M has an intersection of d hyperplanes M_{1}, …, M_{d}. Hence, by Bezout’s theorem [Ful85, Proposition 8.4],

deg_{K}(Z) = deg(^{T} H_{1}) ⋯ deg(^{T} H_{n−d}) · deg(M_{1}) ⋯ deg(M_{d})

= \underbrace{q ⋯ q}_{n − d times} · \underbrace{1 ⋯ 1}_{d times} = q^{n−d} ,

as claimed. □

10. Examples

In this section we illustrate Theorem 1.4(b) with two examples. Example 10.1 shows that the inequality of Theorem 1.4(a) is in fact an equality, for the natural n-dimensional representation V of the alternating group A_{n}. Note that Theorem 1.4(b) cannot be applied to A_{n} ⊂ GL(V), since A_{n} contains no pseudo-reflections. Nevertheless, the conclusion of Theorem 1.4(b) continues to hold in this case. On the other hand, Example 10.2 shows that for G = ℤ/5ℤ ⋊ ℤ/4ℤ the inequality of Theorem 1.4(a) is strict for every faithful representation G ↪ GL(V).

Example 10.1. pmed(A_{n}) = ed(A_{n}; 2) = 2 ⌊\frac{n}{4}⌋ for any n ⩾ 4.

Proof. Since A_{n} contains an elementary abelian subgroup of rank 2 ⌊\frac{n}{4}⌋ generated by

(12)(34), (13)(24), (56)(78), etc., we have pmed(A_{n}) ⩾ ed(A_{n}; 2) = 2 ⌊\frac{n}{4}⌋; see [BR97, Theorem 6.7(c)].

We will now deduce the opposite inequality,

(10.1)  pmed(A_{n}) ≤ 2 ⌊\frac{n}{4}⌋

from Theorem 1.4(a). Let V = k^{n} be the natural representation of S_{n}. One checks that for any g ∈ S_{n} and any prime p, the dimension of the eigenspace V (g, ζ_{p}) is the number of cycles of length divisible by p in the cycle decomposition of g. Thus

a(p) = max_{g∈A_{n}} dim V (g, ζ_{p}) = \begin{cases}⌊n/p⌋, if p is odd, and \\ 2 ⌊n/4⌋, if p = 2, \end{cases}

Since we are assuming that n ≥ 4, the maximal value of a(p) is attained at p = 2. The inequality (10.1) now follows from Theorem 1.4(a), as desired. □

Example 10.2. Let G = ℤ/5ℤ ⋊ ℤ/4ℤ, where ℤ/4ℤ acts faithfully on ℤ/5ℤ. Assume ζ_{20} ∈ k. Then

(a) pmed(G) = 1, but

(b) a_{ϕ}(2) ⩾ 2 for every faithful representation ϕ: G ↪ GL(V).

Proof. Since the Sylow subgroups of G are ℤ/5ℤ and ℤ/4ℤ, part (a) follows from Proposition 11.1.

(b) Each of the four characters ℤ/4ℤ → k^{∗} induces a 1-dimensional representation G → GL_{1}. We will denote these representations by ϕ_{0} = id, ϕ_{1}, ϕ_{2}, and ϕ_{3}. Let ϕ_{4} = Ind_{ℤ/5ℤ}^{G}(χ), where χ is a non-trivial multiplicative character ℤ/5ℤ → k^{∗}. We see that ϕ_{4} is a faithful irreducible 4-dimensional representation of G (irreducibility follows, e.g, from Mackey’s criterion) and a_{ϕ_{4}}(2) = 2. Since dim(ϕ_{0})^{2} + ⋯ + dim(ϕ_{4})^{2} = 4 · 1^{2} + 4^{2} = 20 = |G|, ϕ_{0}, …, ϕ_{4} are the only irreducible representations of G. Moreover, since ℤ/5ℤ lies in the kernel of ϕ_{0}, …, ϕ_{3}, every faithful representation ϕ: G ↪ GL(V) must contain a copy of ϕ_{4}. Thus a_{ϕ}(2) ⩾ a_{ϕ_{4}}(2) = 2. □

Remark 10.3. A. Ledet showed that ed(ℤ/5ℤ ⋊ ℤ/4ℤ) = 2; see [Le02, p. 426]. Note that in [Le02] this group is denoted by C_{5}.

11. A-groups

Let G be a finite group, p be a prime and G_{p} be a Sylow p-subgroup of G. Recall that G is called an A-group if G_{p} is abelian for every p; see, e.g., [Itô52, Wal69, Bro71]. For the rest of this section, with the exception of Conjecture 11.5 below, we will assume that the base field k is of characteristic zero and ζ_{e} ∈ k, where e is the exponent of G.

Proposition 11.1. Let G be an A-group. Then

pmed(G) = max_{p} ed(G; p) = max_{p} rank(G_{p})

where the maximum is taken over all primes p.

Here, as usual, by the rank of a finite abelian group H we mean the minimal number of generators of H.

Proof. The second equality is well known; see, e.g., [RY00, Corollary 7.3]. Note also that this is a very special case of (1.1). In view of (1.5), in order to prove the first equality, we only need to show that pmed(G) ⩽ max_{p} rank(G_{p}).

Let p_{1}, …, p_{r} be the prime divisors of |G| and d = max rank(G_{p_{i}}), as i ranges from 1 to r. By [RY01, Theorem 8.6] there exists a faithful primitive d-dimensional G-variety Y with smooth k-points y_{1}, …, y_{r} such that G_{p_{i}} ⊂ Stab_{G}(y_{i}) for i = 1, …, r.

Recall that “primitive” means that G transitively permutes the irreducible components of Yk̄. We claim that any such Y is, in fact, absolutely irreducible. Let us assume this claim for a moment. The G-orbit of y_{i} is a zero cycle of degree prime to p_{i}. Thus for any given prime p, the degree of one of these orbits is prime to p. By [DR13, Corollary 8.6(b)], this implies that Y is p-versal for every p. Hence, pmed(G) ≤ dim(Y) = d, and the proposition follows.

It remains to show that Y is absolutely irreducible. After replacing k by its algebraic closure k̄, we may assume that k is algebraically closed. Let Y_{0} be an irreducible component of Y and H be the stabilizer of Y_{0} in G. Our goal is to prove that H = G. Since G acts transitively on the irreducible components of Y, this will imply that Y = Y_{0}.

Since y_{i} is a smooth point of Y, it lies on exactly one irreducible component of Y, say on g_{i}(Y_{0}) for some g_{i} ∈ G. Since y_{i} is G_{p_{i}}-invariant, y_{i} also lies on gg_{i}(Y_{0}) for every g ∈ G_{p_{i}}. In other words, gg_{i}(Y_{0}) = g_{i}(Y_{0}) for every g ∈ G_{p_{i}} or equivalently, g_{i}^{−1}G_{p_{i}}g_{i} ⊂ H for every i = 1, …, s. This shows that H contains a Sylow p_{i}-subgroup of G for i = 1, …, r. Hence, |H| is divisible by |G_{p_{i}}| for every i = 1, …, r. We conclude that |H| is divisible by |G| = |G_{p_{1}}| · · · · · |G_{p_{s}}| and hence, H = G. □

Remark 11.2. The above argument relies, in a key way, on [RY01, Theorem 8.6]. This theorem is proved in [RY01] over an algebraically closed field of characteristic 0 but the proof goes through for any k as above. The condition that ζ_{e} ∈ k, is necessary; it is not mentioned in [RY01, Remark 9.9] due to an oversight.

Example 11.3. If G is a non-abelian group of order pq, where p and q are odd primes. Then Proposition 11.1 tells us that pmed(G) = 1. On the other hand, ed(G) ⩾ 2; see [BR97, Theorem 6.2]. This is, perhaps, the simplest example where pmed(G) < ed(G).

Remark 11.4. Non-abelian simple A-groups are classified in [Bro71, Theorem 3.2]: they are J_{1}, the first Janko group, and PSL_{2}(q) for q > 3 and q ≡ 0, 3, or 5 (mod 8). By Proposition 11.1,

pmed(G) = \begin{cases}3, if G ≃ J_{1}, \\ 2, if G ≃ PSL_{2}(q), with q as above. \end{cases}

On the other hand, by [Bea11], ed(G) ⩾ 4 for any of these groups, except for G ≃ PSL_{2}(5) and (possibly) PSL_{2}(11).

It is natural to conjecture the following generalization of [RY01, Theorem 8.6].

Conjecture 11.5. Let d be a positive integer. Suppose G is a finite group with subgroups H_{1}, …, H_{r} such that rdim_{k}(H_{i}) ≤ d for all i = 1, …, r. Then there exists a d-dimensional k-variety X with a faithful G-action and smooth k-points x_{1}, …, x_{r} ∈ X such that H_{i} fixes x_{i} for each i = 1, …, r.

Here we are not imposing any restrictions on the base field k. Note that each H_{i} must act faithfully on the tangent space of the corresponding x_{i} and so the condition that the representation dimension of each H_{i} should be ≤ d is necessary.

Of particular interest is the special case where p_{1}, …, p_{r} are the distinct primes dividing |G|, each H_{i} is a Sylow p_{i}-subgroup, and d is the maximum of ed_{k}(G; p) = rdim_{k}(H_{i}). If Conjecture 11.5 could be established in this special case, then the argument we used in the proof of Proposition 11.1 would show that the G-action on X is p-versal for every prime p and, consequently, that (1.6) holds for G. We have not been able to prove (1.6) by this method beyond the case of A-groups.

Acknowledgements

The authors are grateful to I. Dolgachev for suggesting the geometric construction used in the proof of Lemma 6.1. We would also like to thank M. García-Armas, G. I. Lehrer, R. Lötscher, M. MacDonald, and J.-P. Serre for helpful comments.

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Alexander Duncan Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Zinovy Reichstein Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada