COHOMOLOGY OF WEYL GROUPS WITH APPLICATIONS TO TOPOLOGY

KASPER K. S. ANDERSEN

Preface

This masters thesis is written for the Cand. Scient. degree in mathematics at The University of Copenhagen under the supervision of Jesper Michael Møller.

The aim is to study finite p-adic pseudoreflection groups, compute their low-dimensional cohomology groups and use these results in the theory of homotopy Lie groups.

The contents are as follows. Section 1 contains definitions and elementary results about pseudoreflection groups and states the basic theorem linking these to invariant theory.

The second section describes the classification of finite pseudoreflection groups over ℚ, ℝ and ℂ. We also give a complete list of isomorphisms among groups in the classification list, thus allowing us to state a strong rigidity theorem.

The short section 3 contains the classification of pseudoreflection groups over ℚ_{p}ˆ .

What we are really interested in though, is a classification of pseudoreflection groups over ℤ_{p}ˆ . Section 4 contains a discussion and some basic results to be used later on.

Section 5 is devoted to some specific examples of particular interest. We collect here a number of group theoretic facts about these cases.

Section 6 starts with a discussion of lattices for irreducible pseudoreflection groups and then gives Notbohm’s results for odd primes.

Section 7 contains the actual cohomology computations.

Section 8 is a brisk introduction to the theory of p-compact groups. It also contains an overview of the recent developments concerning the Steenrod problem.

In section 9 we apply the computations from section 7 to prove our main theorem. We also give a discussion of self-maps and determinism of p-compact groups. Finally we study some concrete examples, more specifically they correspond to the groups we studied in section 5.

We end by a short section discussing open questions.

During this project I have made use of the computer algebra system Magma [12]. I believe in computers as an experimental tool which might be very useful in testing and formulating conjectures. Of course the emphasis should not be removed from real mathematics and in the final work the computer calculations plays only a very minor role.

The sections 1–6 are self-contained. In section 7 knowledge of elementary methods in group cohomology is necessary. It can be found in any of the books [11, 33, 82] for example. The last sections however are really written for the working algebraic topologist and requires a thorough knowledge of p-compact groups. Some excellent survey articles are [59, 47].

Acknowledgments

First of all I would like to thank Jesper Møller for being a good advisor, who has provided me with constant optimism, interest and support all along. Also I would like to thank him for introducing me to algebraic topology and suggesting that I participated in the ’Emphasis year in Homotopy Theory’ at The Field Institute in Toronto.

Thanks also to Jesper Grodal for close collaboration and countless discussions ever since we met, for spending the year in Canada with me and for being such a close personal friend in general. He also showed continued interest in my work and has given me valuable suggestions for corrections and improvements.

So to the entire algebraic topology group in Copenhagen : Jesper and Jesper, thank you !

From the department I would also like to thank Christian U. Jensen and Jørn B. Olsson for useful discussions and help.

I am very grateful to The Fields Institute for giving me the opportunity to participate in the emphasis year. I would like to thank the organizers for good courses and for providing such a stimulating atmosphere. Most of all I would like to thank all the participants for making the stay so profitable both mathematically and not at least socially. Most notably I must mention the other long term participants W. Chachólski, J. Grodal, P. Lambrechts, J. Scherer and B. Schuster. Even though I’m not a rational homotopy theorist I must also mention M. Cuvilliez, L. Fernandez, S. Ghorbal, T. Kahl, F. Simoncini and L. Vandembroucq.

Thanks also to D. Milne and S. Witherspoon for starting the group cohomology seminar.

I would like also to thank the organizers and participants of the AMS Summer Research Institute in Seattle, of the Advanced Course at the CRM in Barcelona, and of the conferences in Barcelona and Kazimierz.

All of this would not have been possible without financial support, and I’m grateful to have this opportunity to thank Det Internationale Kontor, Det Naturvidenskabelige Fakultet, Knud Højgaards Fond, Julie Damms Fond and Matematisk Institut.

During the Kazimierz conference and after, I had useful discussions with A. Viruel and F. Neumann, whom I also would like to thank for telling me about their work. I would also like to thank D. Benson for useful help.

F. Neumann, J. Scherer and A. Viruel all read various versions of the manuscript, and I’m grateful to them for pointing out many misprints and suggesting other improvements.

Finally thanks to my family and friends for their interest and just for being there.

Copenhagen, August 1997

Kasper K. S. Andersen

Contents

Preface i Acknowledgments ii 1. Pseudoreflection groups and invariant theory 1 2. Finite pseudoreflection groups over ℚ, ℝ and ℂ 4 3. Pseudoreflection groups over ℚ_{p}ˆ 10 4. Integral representations 12 5. The Aguadé cases 15 6. Classification of lattices 19 7. Cohomology of Weyl groups 22 8. p-compact groups 26 9. Normalizers, Self-maps and Determinism 29 10. Further directions 32 References 33

1. Pseudoreflection groups and invariant theory

In this chapter we introduce our main objects of study, namely pseudoreflection groups, and discuss their elementary properties and the relationship with invariant theory which is the real reason these objects are of interest.

Let R denote an integral domain with quotient field K.

Definition 1.1. An element g ∈ GL_{n}(R) is called a pseudoreflection if it has finite order and the rank of g −1 is 1 (as an element of M_{n}(K)). A subgroup of GL_{n}(R) generated by pseudoreflections is called a pseudoreflection group. If ρ : G ↪ GL_{n}(R) is a faithful representation and ρ(G) is a pseudoreflection group, then we call (G, ρ) a pseudoreflection group. Often, working with a particular representation ρ, we will just abuse the notation and call G a pseudoreflection group. The pseudoreflection group is said to be irreducible if the representation ρ : G ↪ GL_{n}(K) is irreducible.

Proposition 1.2. Let g ∈ GL_{n}(R) be a pseudoreflection. The characteristic polynomial of g has 1 as a root of multiplicity at least n − 1 and one other root λ ∈ R which is an m-th root of unity, where m denotes the order of g. If λ ≠ 1 then g is diagonalizable over K and λ is a primitive m-th root of unity. If λ = 1 then K has characteristic p, the order of g is p and g is similar to a matrix of the form

[ \begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & 1 & ... & ⋮ \\ 0 & 0 & ⋯ & 1 & 1 \\ 0 & 0 & 0 & ⋯ & 1 \end{matrix} ]

Proof: The first two statements are obvious. Assume that λ = 1. For a vector x outside the (n − 1)-dimensional hyperplane V^{g} fixed by g, we must have g · x = x + v where v ∈ V^{g}, since if g · x = μx + w with μ ≠ 1 and w ∈ V^{g} then (μ − 1)x + w is an eigenvector for g with eigenvalue μ ≠ 1 which is a contradiction. So g · x = x + v for some v ∈ V^{g}. Obviously v ≠ 0 since x ∉ V^{g}. Therefore we can choose a basis for V^{g} containing v and supplement it by x to obtain a basis of V. Thus in this basis g has the form stated. By induction we get g^{n} · x = x + nv, which shows the remaining part of the proposition. □

Proposition 1.3. Let ρ : G ↪ GL_{n}(K) = GL(V) be a finite pseudoreflection group. Assume that |G| is invertible in K. Then we have splittings G = G_{1} × … × G_{r}, V = V_{1} ⊕ … ⊕ V_{r} such that G_{i} acts trivially on V_{j} for j ≠ i and G_{i} acts faithfully as a finite irreducible pseudoreflection group on V_{i}. These splittings are uniquely determined (up to reordering of the factors) by the representation ρ.

Proof: By the assumption that |G| is invertible in K the representation ρ is completely reducible by Maschke’s theorem. We may thus write ρ = ρ_{1} ⊕ … ⊕ ρ_{r} and V = V_{1} ⊕ … ⊕ V_{r} where V_{i} is the G-invariant subspace corresponding to the irreducible representation ρ_{i}. Now let G_{i} be the subset of G that acts trivially on all V_{j}, j ≠ i. It is obvious that G_{i} is a normal subgroup and since ρ is faithful, G_{i} and G_{j} intersect trivially for i ≠ j. Obviously elements from G_{i} commute with elements from G_{j} for i ≠ j. If g ∈ G is a pseudoreflection then there exists an i such that g acts trivially on V_{j} for all j ≠ i and acts as a pseudoreflection on V_{i}. Thus g ∈ G_{i} for a unique i. Since the elements of G_{j}, j ≠ i act trivially on V_{i} we see that G_{i} is generated by its pseudoreflections and that G = G_{1} × … × G_{r}. Finally note that by definition of G_{i} we see that the restriction of ρ to G_{i} is a faithful irreducible representation G_{i} ↪ GL(V_{i}). The uniqueness of the splittings may be seen as follows. The V_{i}’s are uniquely determined (up to reordering) by the irreducible components of ρ. If G = G̃_{1} × … G̃_{r} is another splitting satisfying the condition that G̃_{i} acts trivially on V_{j} for j ≠ i then G̃_{i} ⊆ G_{i} by definition. But then since G = G̃_{1} × … G̃_{r} we conclude G̃_{i} = G_{i}. □

In the case where the G-action has fixed points, the representation V^{G} splits as a direct sum of one-dimensional trivial representations, and the corresponding groups in the product decomposition are trivial. An alternative way of stating the result is to say that we have splittings G = G_{1} × … × G_{s}, V = V^{G} ⊕ V_{1} ⊕ … ⊕ V_{s} with the same properties as before, but this time the G_{i}’s are nontrivial finite irreducible pseudoreflection groups.

We briefly review some of the main results from the invariant theory of finite groups. We only recall the facts we will use in the following, the reader should consult for example the books [5, 74] for more details and further references.

Let K be a field, and let V be an n-dimensional vector space over K. Let V^{∗} be the dual vector space of V and denote by K[V] the symmetric algebra over V^{∗}.

So, if e_{1}, …, e_{n} is a basis of V and x_{1}, …, x_{n} is its dual basis, then K[V] is the polynomial algebra K[x_{1}, …, x_{n}]. We can grade this by putting the degree of x_{i} equal to 1 for all i. It should be noted that this is in contrast to normal conventions of algebraic topology where the degrees are doubled for odd primes. Since we only do one concrete calculation (Example 1.8 below) this will not cause any confusion.

We have an action of GL_{n}(K) on V^{∗} given by (g · φ)(v) = φ(g^{−1} · v) for g ∈ GL_{n}(K), φ ∈ V^{∗} and v ∈ V. This obviously extends to an action of GL_{n}(K) on K[V] = K[x_{1}, …, x_{n}].

Note that if R is a subring of K then this action also gives rise to an action of GL_{n}(R) on R[x_{1}, …, x_{n}]. If L is a free R-module of rank n then we will also denote R[x_{1}, …, x_{n}] by R[L]. If G is a subgroup of GL_{n}(R) we denote by R[x_{1}, …, x_{n}]^{G} (or R[L]^{G}) the R-algebra of G-invariant polynomials.

The connection between pseudoreflection groups and invariant theory is due to the following:

Theorem 1.4. Let V be a finite dimensional faithful representation of a finite group G over a field K. If K[V]^{G} is a polynomial ring then G is a pseudoreflection group. In this case, if K[V]^{G} = K[f_{1}, f_{2}, …, f_{n}], then ∏_{i=1}^{n} deg(f_{i}) = |G|.

Conversely, if G is a pseudoreflection group and |G| is invertible in K, then K[V]^{G} is a polynomial ring. In this case, if K[V]^{G} = K[f_{1}, f_{2}, …, f_{n}], we also have that ∑_{i=1}^{n}(deg(f_{i}) − 1) is the number of pseudoreflections in G.

Proof: See [5, 7.2.1]. □

This theorem has a curious history. It was first observed by classification and case-by-case checking for complex pseudoreflection groups by Shephard and Todd [72]. Later Chevalley [13] gave a proof in the real case without using the classification and Serre pointed out how to modify the argument so that it also worked in the complex case. The general version given above is due to Bourbaki [7, Chapitre 5 §5 Exercise 7,8] and Smith [73].

Remark 1.5. In the case where the invariant ring is polynomial, K[V]^{G} = K[f_{1}, …, f_{n}], the degrees of the polynomials f_{i} are uniquely determined. In fact the Poincaré series of K[V]^{G} is equal to ∏_{i=1}^{n}(1 − t^{deg(f_{i})})^{−1} from which we can easily read off the degrees. Thus we may speak of the degrees of the pseudoreflection group G if it has polynomial invariants.

Example 1.6. In the case where |G| is not invertible in K the invariant ring K[V]^{G} need no longer be polynomial. In fact the Weyl group of F_{4} at the prime p = 3 does not have polynomial invariants [80], [74, p. 213-218, 301-302], nor has the Weyl group of P U(p) at the prime p if p ≥ 5 [25, 7.1]. A simpler (purely algebraic) example is given in [54, 3.8] (a proof is given in [43, Example 19]).

Even if K[V]^{G} = K[f_{1}, f_{2}, …, f_{n}] is a polynomial ring, the number of pseudoreflections is not necessarily given by the formula ∑_{i=1}^{n}(deg(f_{i}) − 1). An example may be found in [5, 3.13.1].

Computing invariants over a finite field whose characteristic is not prime to the order of the group is thus not an easy matter. Consider a subgroup G ↪ GL_{n}(𝔽_{q}). The invariants 𝔽_{q}[V]^{G} obviously contains 𝔽_{q}[V]^{GL_{n}(𝔽_{q})}, which may thus be considered as a universal set of modular invariants. This is called the Dickson algebra after L. E. Dickson [23] who was the first to compute it. More recent references are [74, section 8.1] and [84].

Theorem 1.7. Let q be a prime power and let n ≥ 1 be arbitrary. Define f(X) ∈ 𝔽_{q}[V][X] by f(X) = ∏_{v∈V} (X − v^{∗}). Then this polynomial has the form

f(X) = X^{q^{n}} + ∑_{i=0}^{n−1}(−1)^{n−i}c_{n,i}X^{q^{i}},

which define polynomials c_{n,i} ∈ 𝔽_{q}[V], 0 ≤ i ≤ n − 1, of degree q^{n} − q^{i}. These are called the Dickson invariants and the invariant ring of GL_{n}(𝔽_{q}) is polynomial on these, 𝔽_{q}[V]^{GL_{n}(𝔽_{q})} = 𝔽_{q}[c_{n,n−1}, …, c_{n,0}]. □

More practical formulas for computing the invariants may be found in the references above, where also the Steenrod algebra action (in the case q is a prime) and other applications to topology may be found. We end by a particular example.

Example 1.8. We compute the Dickson invariants for n = 2, q = 3 and determine the Steenrod algebra structure. We switch to topological conventions in our grading, thus doubling the degrees. We consider the action of GL_{2}(𝔽_{3}) on the polynomial algebra 𝔽_{3}[t_{1}, t_{2}], where |t_{1}| = |t_{2}| = 2. We find by using the formulas the two Dickson invariants

x_{12} = c_{2,1} = t_{1}^{6} + t_{1}^{4}t_{2}^{2} + t_{1}^{2}t_{2}^{4} + t_{2}^{6}
x_{16} = c_{2,0} = t_{1}^{6}t_{2}^{2} + t_{1}^{4}t_{2}^{4} + t_{1}^{2}t_{2}^{6}.

The Steenrod algebra is generated as an algebra by the Bockstein β and the powers P^{3^{i}}, i ≥ 0. Since all elements in our case have even degrees and since x_{16} sits in degree 16, we only need to consider P^{1} and P^{3}. We get

P^{1}x_{12} = x_{16}, P^{1}x_{16} = 0, P^{3}x_{12} = −x_{12}^{2}, P^{3}x_{16} = −x_{12}x_{16},

and this determines the action.

2. Finite pseudoreflection groups over ℚ, ℝ and ℂ

In this section we discuss the classification of finite pseudoreflection groups over the fields ℚ, ℝ and ℂ. The classification of finite real pseudoreflection groups was first obtained by Coxeter [17]. In this case there is a bijective correspondence between conjugacy classes of these and the so called Coxeter-Dynkin diagrams. We refer the reader to the treatments in [7, 37]. In particular Coxeter also classified the finite pseudoreflection groups over ℚ, which are usually called finite crystallographic reflection groups. These occur as Weyl groups of the classical compact Lie groups, see for example [7, 8].

The classification of finite complex pseudoreflection groups was first obtained by Shephard and Todd [72]. A more modern treatment is given by Cohen [15] who extended the definition of Coxeter-Dynkin diagrams to obtain the classification. Recall that by Proposition 1.3 it suffices to classify the finite irreducible complex pseudoreflection groups.

Theorem 2.1. The finite irreducible complex pseudoreflection groups fall up to conjugacy into three infinite families denoted by 1 − 3 and 34 sporadic cases denoted by 4 − 37. □

Number Rank Order Order of center

Degrees

1  n  (n + 1)!  1 for n ≥ 2, 2 for n = 1  2, 3, …, n + 1
2  n  n!m^{n}/r  m(n, r)/r  m, 2m, …, (n − 1)m, mn/r
3  1  n  n  n
4  2  24 = 2^{3} · 3  2  4, 6
5  2  72 = 2^{3} · 3^{2}  6  6, 12
6  2  48 = 2^{4} · 3  4  4, 12
7  2  144 = 2^{4} · 3^{2}  12  12, 12
8  2  96 = 2^{5} · 3  4  8, 12
9  2  192 = 2^{6} · 3  8  8, 24
10  2  288 = 2^{5} · 3^{2}  12  12, 24
11  2  576 = 2^{6} · 3^{2}  24  24, 24
12  2  48 = 2^{4} · 3  2  6, 8
13  2  96 = 2^{5} · 3  4  8, 12
14  2  144 = 2^{4} · 3^{2}  6  6, 24
15  2  288 = 2^{5} · 3^{2}  12  12, 24
16  2  600 = 2^{3} · 3 · 5^{2}  10  20, 30
17  2  1200 = 2^{4} · 3 · 5^{2}  20  20, 60
18  2  1800 = 2^{3} · 3^{2} · 5^{2}  30  30, 60
19  2  3600 = 2^{4} · 3^{2} · 5^{2}  60  60, 60
20  2  360 = 2^{3} · 3^{2} · 5  6  12, 30
21  2  720 = 2^{4} · 3^{2} · 5  12  12, 60
22  2  240 = 2^{4} · 3 · 5  4  12, 20
23  3  120 = 2^{3} · 3 · 5  2  2, 6, 10
24  3  336 = 2^{4} · 3 · 7  2  4, 6, 14
25  3  648 = 2^{3} · 3^{4}  3  6, 9, 12
26  3  1296 = 2^{4} · 3^{4}  6  6, 12, 18
27  3  2160 = 2^{4} · 3^{3} · 5  6  6, 12, 30
28  4  1152 = 2^{7} · 3^{2}  2  2, 6, 8, 12
29  4  7680 = 2^{9} · 3 · 5  4  4, 8, 12, 20
30  4  14400 = 2^{6} · 3^{2} · 5^{2}  2  2, 12, 20, 30
31  4  64 · 6! = 2^{10} · 3^{2} · 5  4  8, 12, 20, 24
32  4  216 · 6! = 2^{7} · 3^{5} · 5  6  12, 18, 24, 30
33  5  72 · 6! = 2^{7} · 3^{4} · 5  2  4, 6, 10, 12, 18
34  6  108 · 9! = 2^{9} · 3^{7} · 5 · 7  6  6, 12, 18, 24, 30, 42
35  6  72 · 6! = 2^{7} · 3^{4} · 5  1  2, 5, 6, 8, 9, 12
36  7  8 · 9! = 2^{10} · 3^{4} · 5 · 7  2  2, 6, 8, 10, 12, 14, 18
37  8  192 · 10! = 2^{14} · 3^{5} · 5^{2} · 7  2  2, 8, 12, 14, 18, 20, 24, 30

Table 1. The finite irreducible complex pseudoreflection groups

The first family consists of the symmetric groups Σ_{n+1}, n ≥ 1. The (n + 1)-dimensional representation obtained by permuting coordinates, splits as a direct sum of a one-dimensional representation given by the fixed line spanned by the vector e_{1} + … + e_{n+1} and its orthogonal complement consisting of all vectors with sum of coordinates equal to 0. This n-dimensional representation is irreducible, and since the transpositions act as reflections and they generate Σ_{n+1} it is actually a pseudoreflection representation. The representation is defined over ℚ and is thus among the groups classified by Coxeter. The associated Coxeter-Dynkin diagram has the form A_{n}. It is the Weyl group of the compact Lie groups U(n + 1) and SU(n + 1) and their central quotients.

The second family consists of the groups G(m, r, n) where r|m. They are defined as follows. Let A(m, r, n) consist of all diagonal matrices of the form

[ \begin{matrix}θ_{1} & 0 & … & 0 \\ 0 & θ_{2} & ⋮ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & θ_{n} \end{matrix} ]

with θ_{i}^{m} = 1, 1 ≤ i ≤ n, and (θ_{1} … θ_{n})^{m/r} = 1. This is an abelian group of order m^{n}/r which has an action of Σ_{n} given by permutation of the entries in the diagonal. Denoting the permutation representation Σ_{n} ↪ GL_{n}(ℂ) by ρ, the group G(m, r, n) is the subgroup of GL_{n}(ℂ) generated by A(m, r, n) and ρ(Σ_{n}). In other words, G(m, r, n) is the semidirect product A(m, r, n) ⋊ Σ_{n}, and its order is n!m^{n}/r. Note that we have a canonical inclusion G(m, r, n) ⊆ G(m′, r′, n′) if m|m′, \frac{m}{r}|\frac{m′}{r′} and n ≤ n′.

In case n = 1 our group is nothing but a cyclic group of order m/r. These are separated as a special case and they constitute the third infinite family (see below). For n ≥ 2 the group G(m, r, n) is irreducible iff m ≥ 2 and (m, r, n) ≠ (2, 2, 2) (see [15, 2.4]). The group G(2, 2, 2) is isomorphic to C_{2} × C_{2}, which is abelian and thus only has irreducible representations of dimension one. To summarize, the allowed parameters are m ≥ 2, r|m, n ≥ 2 and (m, r, n) ≠ (2, 2, 2), and we will always assume these conditions to be satisfied when speaking of groups from this family.

The groups G(m, r, n) include most of the Coxeter groups. First of all G(m, m, 2) is conjugate to the dihedral group D_{2m}, which has Coxeter-Dynkin diagram I_{2}(m). For m = 3, D_{6} is conjugate to Σ_{3} and is thus the Weyl group of U(3) and SU(3) (and their common quotient P U(3)). For m = 4, D_{8} is the Weyl group of SO(5) and Sp(2). For m = 6, D_{12} is the Weyl group of G_{2}.

The only other cases where G(m, r, n) is a real reflection group are the cases G(2, 1, n), n ≥ 2 and G(2, 2, n), n ≥ 3. The first groups G(2, 1, n) have Coxeter-Dynkin diagrams B_{n} = C_{n} and are the Weyl groups of the compact Lie groups SO(2n + 1) and Sp(n). The groups G(2, 2, n) have Coxeter-Dynkin diagrams D_{n} and are the Weyl groups of the compact Lie groups SO(2n).

The third family consists simply of a cyclic group C_{n} in the faithful one-dimensional representation shich maps a generator of C_{n} to a primitive n’th root of unity ζ_{n}.

In addition to the above mentioned examples the classification of Coxeter groups consists only of the following examples: The non-crystallographic groups number 23 and 30 which have the Coxeter-Dynkin diagrams H_{3} and H_{4} respectively. They are more closely discussed in for example [37, Section 2.13], [74, p. 197-198] and [5, p. 80]. Furthermore we have the crystallographic groups with numbers 28, 35, 36 and 37 which are respectively the Weyl groups of the compact Lie groups F_{4}, E_{6}, E_{7} and E_{8}.

Remark 2.2. It should be emphasized that the classification is up to conjugacy of subgroups in GL_{n}(ℂ), and not up to equivalence of representations. The result may also be reformulated as follows.

Let R be any ring. We say that two representations ρ_{1}, ρ_{2} : G → GL_{n}(R) are weakly equivalent if there is an automorphism φ of G such that ρ_{2} is equivalent to ρ_{1}φ. It is easy to see that for faithful representations this is the same as saying that the groups ρ_{1}(G) and ρ_{2}(G) are conjugate within GL_{n}(R).

Thus the classification theorem lists exactly the weak equivalence classes of faithful irreducible complex pseudoreflection representations of finite groups.

As an illustration consider the dihedral groups D_{m} of order m. It is not hard to compute all the irreducible representations, see for example [20, 10.11]. The result is that for m ≡ 0 (mod 4) there are 4 one-dimensional and \frac{m}{4} − 1 two-dimensional representations, while for m ≡ 2 (mod 4) there are 2 one-dimensional and \frac{m−2}{4} two-dimensional representations. All the two-dimensional representations are pseudoreflection representations. Thus for all m ≥ 10 we have more than one pseudoreflection representation. However they are all weakly equivalent, since it is easily checked that the images of the representations are all conjugate. Therefore these groups appear only once in the table.

Concerning the centers of the groups we have the following very useful observation:

Proposition 2.3. The centers of the finite irreducible complex pseudoreflection groups are all cyclic groups consisting of scalar matrices, and their orders are the ones listed in the table.

Proof: To each finite complex irreducible pseudoreflection group G ⊆ GL_{n}(ℂ) we can associate the image of G in PGL_{n}(ℂ). This turns out to be a finite collineation group generated by homologies. All such have been classified, and Shephard and Todd’s classification of finite irreducible complex pseudoreflection groups is in fact based on this classification. By case-by-case checking we see that all the collineation groups have trivial center. The observation now follows from [72, §1 and Table VII]. □

Recall that ℂ[V]^{G} is polynomial by Theorem 1.4 and we may thus speak of the degrees of G, cf. Remark 1.5. This is the information listed in the last column of the table. It should again be noted that this is algebraic grading, the degrees should be doubled if we are working topologically at an odd prime.

An abstract group may appear more than once in the table above. However not too many repetitions occur as is shown in Theorem 2.6. A corollary is a strong rigidity theorem stating that an abstract group admits at most one representation as a finite irreducible complex pseudoreflection group (up to weak equivalence).

First we need a technical lemma concerning the commutator subgroup of certain semidirect products. We use the following notation. For two elements x, y in a group G, the conjugate of y by x, x^{−1}yx, is denoted by y^{x}. The commutator subgroup of a group G is denoted by G′ = G^{(1)} and we define the iterated commutator subgroups by G^{(i+1)} = (G^{(i)})′. The derived length of a group G is the smallest i such that G^{(i)} = 1. It is well known that G is solvable iff the derived length is finite.

Lemma 2.4. Let A be an abelian group (written multiplicatively) and let H be a group acting on A. Then we have (A ⋊ H)′ = Ã ⋊ H′ where Ã is the subgroup of A generated by all elements of the form a^{h}a^{−1}, a ∈ A, h ∈ H.

Proof: This follows directly from [34, 1.1.3]. □

We use this to compute the derived length of the groups G(m, r, n) for small n.

Proposition 2.5. The group G(m, r, n) is solvable iff 2 ≤ n ≤ 4, and in this case the derived length equals n. In particular none of the groups G(m, r, n) are abelian. Furthermore we have G(m, r, n)′ = A(m, m, n) ⋊ A_{n} for n ≥ 3.

Proof: Since A(m, r, n) is always abelian and Σ_{n} is solvable exactly when n ≤ 4 it is clear that G(m, r, n) is solvable iff 2 ≤ n ≤ 4. By Lemma 2.4 we obviously have G(m, r, n)^{(i)} ⊆ A(m, r, n) ⋊ Σ_{n}^{(i)} for all i ≥ 0. Thus if l is the derived length of Σ_{n} we have

G(m, r, n)^{(l+1)} = (G(m, r, n)^{(l)})′ ⊆ A(m, r, n)′ = 1.

Thus we see that the derived length of G(m, r, n) is at most equal to l + 1. Since Σ_{2}, Σ_{3} and Σ_{4} have derived length respectively 1, 2 and 3 we get the upper bound.

Let ζ_{m} denote a primitive m’th root of unity. Since G(m, r, 2) contains the matrices

[ \begin{matrix}ζ_{m} & 0 \\ 0 & ζ_{m}^{−1} \end{matrix} ] , [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ]

which only commute for m = 2, we see that G(m, r, 2) is non-abelian except possibly for the two cases G(2, 1, 2) and G(2, 2, 2). The group G(2, 2, 2) is excluded since it is not irreducible and G(2, 1, 2) = (ℤ_{2}) ≀ (ℤ_{2}) ≅ D_{8} is non-abelian. So all groups G(m, r, 2) are non-abelian, which shows that the derived length is exactly 2.

Now assume n ≥ 3. Denote a diagonal matrix with entries θ_{1}, …, θ_{n} by diag(θ_{1}, …, θ_{n}), for short. Let i, j ∈ {1, …, n} be distinct and choose k ∈ {1, …, n} distinct from i and j. Letting a be the diagonal matrix with all 1’s except for ζ_{m} in the j’th row and ζ_{m}^{−1} in the k’th row we see that a ∈ A(m, r, n). Let σ denote the transposition (i, j). We easily see a^{σ}a^{−1} equals the diagonal matrix with all 1’s except for ζ_{m} in the i’th row and ζ_{m}^{−1} in the j’th row. It is easily seen that the subgroup generated by these matrices is A(m, m, n). On the other hand all matrices of the form a^{σ}a^{−1} have determinant 1 so by Lemma 2.4 we conclude that G(m, r, n)′ = A(m, m, n)⋊Σ′_{n} = A(m, m, n)⋊A_{n}.

For n = 3, by what already has been proved, we see that G(m, r, 3)′ = A(m, m, 3) ⋊ A_{3}. This is not abelian since it contains the diagonal matrix diag(ζ_{m}, ζ_{m}^{−1}, 1) and the cycle (1, 2, 3). Thus G(m, r, 3)′′ ≠ 1 and the derived length is precisely 3.

For n = 4 first note that G(m, r, 4)′ = A(m, m, 4) ⋊ A_{4}. Since A(m, m, 4) contains the diagonal matrix a = diag(ζ_{m}, ζ_{m}^{−1}, 1, 1) and A_{4} contains the permutation σ = (1, 2, 3) we see that G(m, r, 4)′′ contains the diagonal matrix b = a^{σ}a^{−1} = diag(ζ_{m}^{−2}, ζ_{m}, ζ_{m}, 1). Since G(m, r, 4)′′ also contains Σ′′_{4} = V_{4} (Klein’s four group), we see that G(m, r, 4)′′ is not abelian since b does not commute with the element (1, 2)(3, 4) in V_{4}. Thus G(m, r, 3)^{(3)} ≠ 1 and the derived length is precisely 4. □

Theorem 2.6. The only isomorphisms occurring in the list of finite irreducible complex pseudoreflection groups are the following

Σ_{3} ≅ G(3, 3, 2), Σ_{4} ≅ G(2, 2, 3), Σ_{2} ≅ C_{2}, G(2, 1, 2) ≅ G(4, 4, 2).

Proof: We divide the proof into several cases corresponding to the classification.

\underline{Case 1:} We first look for isomorphisms between pairs of groups in the list of sporadic cases number 4 − 37. By checking the order of the group and the center in the table above we easily see that the only possibilities of isomorphisms are given by the pairs (G_{8}, G_{13}) and (G_{10}, G_{15}). However these are not isomorphic as shown in [72, p. 281].

\underline{Case 2:} Next we look for possible isomorphisms between one of the sporadic groups and one of the groups coming from one of the three infinite families 1, 2 and 3. Again by comparing the order of the group and its center we see that there can be no such isomorphism between a sporadic group and a group belonging to one of the families 1 or 3. Also we see that there are only 3 possible isomorphisms between a sporadic group and a group G(m, r, n) from the family number 2 in the case n ≥ 3. They are given by the following list

(G_{12}, G(2, 1, 3)), (G_{25}, G(6, 2, 3)), (G_{26}, G(6, 1, 3))

The groups G_{12}, G_{25} and G_{26} have derived lengths respectively 4, 5 and 5 and thus the groups may be distinguished by Proposition 2.5. Assume now that n = 2. By Proposition 2.5 the group G(m, r, 2) has derived length 2. Checking the cases 4 − 37 we see that none of these groups has G′ abelian and thus have derived length at least 3. Thus for n = 2 there are no isomorphisms.

\underline{Case 3:} Since the families 1 and 3 represents respectively the symmetric and the cyclic groups, we see that there can be no isomorphisms within these families and the only isomorphism between groups coming from the two families is Σ_{2} ≅ C_{2}. By the preceding proposition the groups G(m, r, n) are always non-abelian, so there cannot be any isomorphisms between a pair of groups from the families 2 and 3.

\underline{Case 4:} We now investigate which isomorphisms can occur between a group belonging to the family 1 and a group from the family 2. Assume that Σ_{k} ≅ G(m, r, n) where k ≥ 2, m ≥ 2, r|m and n ≥ 2. Since G(m, r, n) contains the abelian normal subgroup A(m, r, n) of index n! whereas Σ_{k} for k ≥ 5 only contains one abelian normal subgroup namely the trivial group, we conclude that A(m, r, n) is trivial if k ≥ 5. Since on the other hand |A(m, r, n)| = m^{n−1} · m/r ≠ 1 we see that we must have k ≤ 4. This leaves only finitely many cases and comparing again the order of the groups and their centers we see that only two pairs are left namely (Σ_{3}, G(3, 3, 2)) and (Σ_{4}, G(2, 2, 3)). In both cases we actually have that the groups are conjugate within respectively GL_{2}(ℂ) and GL_{3}(ℂ) since they all are Coxeter groups with associated Coxeter-Dynkin diagrams of the type A_{2} = I_{2}(3) and A_{3} = D_{3} respectively (see for example [7] or [37]).

\underline{Case 5:} Finally we have to look at isomorphism between two groups both belonging to family number 2. Assume that we have an isomorphism G(m, r, n) ≅ G(m′, n′, r′).

The composition factors of G(m, r, n) are given by the composition factors of A(m, r, n) and Σ_{n}. Thus these are all cyclic groups of prime order for n ≤ 4 and for n ≥ 5 they are all cyclic of prime order except for one which is the alternating group A_{n} of degree n. Thus we have either n = n′ ≥ 5 or 2 ≤ n, n′ ≤ 4.

Assume first that n = n′ ≥ 5. Note that A(m, r, n) is an abelian normal subgroup of G(m, r, n). In particular it is solvable. By Noether’s isomorphism theorem it is clear that it is the largest normal solvable subgroup of G(m, r, n), i.e. it is the Fitting subgroup. Thus we have an isomorphism A(m, r, n) ≅ A(m′, r′, n). It is not hard to see that A(m, r, n) ≅ (ℤ/m)^{n−1} ×ℤ/(m/r) which shows m = m′ and m/r = m′/r′ by the structure theorem of finitely generated abelian groups. Thus we have no repetitions in this case.

In the rest of the proof assume that 2 ≤ n, n′ ≤ 4. By Proposition 2.5 the derived length of G(m, r, n) is n so n = n′. For n ≥ 3 we also have G(m, r, n)′ = A(m, m, n) ⋊ A_{n}. By comparing orders we thus immediately get m^{n−1}n!/2 = m′^{n−1}n!/2 which shows m = m′. By considering the order of G(m, r, n) we also get r = r′.

We are now reduced to the case n = n′ = 2. The order of the center of G(m, r, 2) equals \frac{m}{r}(2, r) which leads to the equation \frac{m}{r}(2, r) = \frac{m′}{r′} (2, r′). We may assume thus that (2, r) divides (2, r′). Denoting the quotient by k we have k ∈ \{1, 2\}. We thus have \frac{m}{r} = \frac{m′}{r′} k. Combining this with the fact that the orders of G(m, r, 2) and G(m′, r′, 2) are equal we see that mk = m′. If k = 1 we see that m = m′ and r = r′. Thus in this case there are no repetitions.

Assume now that k = 2, i.e. that m′ = 2m. This immediately gives r′ = 4r. Thus, since we are assuming (2, r′) = 2(2, r), we must have r odd. Also since we must have r′|m′ we see that m is even. To end the proof we count the number of elements of order 2. Since G(m, r, 2) consists of all matrices of the forms

[ \begin{matrix}θ_{1} & 0 \\ 0 & θ_{2} \end{matrix} ] , [ \begin{matrix}0 & θ_{1} \\ θ_{2} & 0 \end{matrix} ]

satisfying θ_{1}^{m} = θ_{2}^{m} = (θ_{1}θ_{2})^{m/r} = 1 it is easily shown that the number of elements of order 2 is given by

m  for m odd
m + 1 for m even and m/r odd
m + 3 for m even and m/r even

As a consequence we conclude that |m − m′| ≤ 3. Since m′ = 2m we get m = 2 or m = 3. Since m is even and r is odd we see that m = 2 and r = 1. In this case the groups in question are G(2, 1, 2) and G(4, 4, 2). In fact these are conjugate within GL_{2}(ℂ), since they both are Coxeter groups with associated Coxeter-Dynkin diagram of the form B_{2} = I_{2}(4) (see [37] or [7]). □

Corollary 2.7 (Strong rigidity). Assume that the group G has two representations ρ_{i} : G ↪ GL(n_{i}, ℂ), i = 1, 2, as a finite irreducible complex pseudoreflection group. Then n := n_{1} = n_{2} and ρ_{1}(G) and ρ_{2}(G) are conjugate within GL_{n}(ℂ) (i.e. the representations are weakly equivalent).

Proof: By Theorem 2.6 we just have to check this for each of the cases listed there. The case of Σ_{2} ≅ C_{2} in its one-dimensional representation, is obvious. Checking the proof of Theorem 2.6 we see that in all other cases we have already shown that the groups occurring are conjugate by noting that they are conjugate to real reflection groups with the same Coxeter-Dynkin diagrams. □

Remark 2.8. Note that it is necessary that both representations are irreducible. First of all we could otherwise add copies of the trivial representation, but even if we assume that both representations are fixed point free, the result does not hold. An example is given by the group C_{6} ≅ C_{2} × C_{3} which has an irreducible pseudoreflection representation of degree one and a fixed point free pseudoreflection representation of degree two.

As a corollary we obtain the rigidity theorem of Dwyer, Miller and Wilkerson [28, 5.5].

Corollary 2.9. If W ↪ GL_{n}(ℂ) is a pseudoreflection group and σ is a field automorphism of ℂ then the group σ(W) is conjugate to W in GL_{n}(ℂ).

Proof: By Proposition 1.3 it suffices to show this in the irreducible case, in which it follows directly from Theorem 2.7. □

It should be remarked that the proof in [28] also uses the classification but is substantially shorter since we directly know that W and σ(W) have the same rank and degrees etc.

3. Pseudoreflection groups over ℚ_{p} ˆ

In this section we classify the finite pseudoreflection groups over ℚ_{p}ˆ following Clark and Ewing [14]. The following precise statement is due to Dwyer, Miller and Wilkerson [28, 1.5].

Theorem 3.1. For any numbers r and n there is a bijection between the sets

(1) Conjugacy classes of finite p-adic pseudoreflection subgroups W ⊆ GL_{r}(ℚ_{p}ˆ ) of order |W | = n.

(2) Conjugacy classes of finite complex pseudoreflection subgroups W ⊆ GL_{r}(ℂ) of order |W | = n, such that character field of W can be embedded in ℚ_{p}ˆ .

The character field is simply the field obtained from ℚ by adjoining all values of the associated character. Proof: (Sketch) The proof follows easily from Clark and Ewing [14] using Corollary 2.9. See [28] for more details. Note in particular that the assumption in [28] that p ∤ n is not used in this part of the proof. □

Thus the problem is reduced to finding the character fields of the finite irreducible complex pseudoreflection groups and determining the primes p for which these are contained in ℚ_{p}ˆ . In most cases the determination of the character field is straightforward and the result is given in the table below. Here ζ_{m} denotes a primitive m’th root of unity. Only in the case of the groups G(m, r, n) (number 2 in the list of the previous section) there are some complications. Indeed we have the following result from [14]:

Proposition 3.2. Let m ≥ 2, n ≥ 2 and r|m with (m, r, n) ≠ (2, 2, 2) (i.e. we make sure G(m, r, n) is irreducible). If n ≥ 3 or if n = 2 and r < m the character field equals ℚ(ζ_{m}). If n = 2 and r = m the group G(m, r, n) = G(m, m, 2) is the dihedral group D_{2m} of order 2m, and the character field is given by ℚ(ζ_{m} + ζ_{m}^{−1}).

In the following we will denote the first case by number 2a and the second case by number 2b. Note that in case 2b we automatically have m ≥ 3. Using the following proposition we can now determine the list of primes in each case.

Proposition 3.3. Let p be a prime and let m ≥ 2.

(1) Let a be a squarefree integer such that p ∤ a. Then

\sqrt{a} ∈ ℚ_{p}ˆ ⟺ \{ \begin{matrix}a ≡ 1 (mod 8) & for p = 2 \\ (a/p) = +1 & for p odd, \end{matrix}

where (a/p) is the Legendre symbol.

(2) In particular we have \sqrt{−2} ∈ ℚ_{p}ˆ iff p ≡ 1, 3 (mod 8), \sqrt{5} ∈ ℚ_{p}ˆ iff p ≡ 1, 4 (mod 5) and \sqrt{−7} ∈ ℚ_{p}ˆ iff p ≡ 1, 2, 4 (mod 7).

(3) The torsion part of the unit group of ℚ_{p}ˆ is ℤ/2 for p = 2 and ℤ/(p − 1) for p odd. Thus for m ≥ 3 we have ζ_{m} ∈ ℚ_{p}ˆ iff p ≡ 1 (mod m).

(4) We have ζ_{m} + ζ_{m}^{−1} ∈ ℚ_{p}ˆ iff p ≡ ±1 (mod m) or p = 2 and m ∈ {2, 4, 6} or p = 3 and m ∈ {3, 6}.

Proof: (1) See [6, 1.6.1 and 1.6.2].

(2) This follows immediately from (1) using the quadratic reciprocity theorem.

(3) See [70, II.§3.2].

(4) This requires some facts from algebraic number theory. We may assume m ≥ 3.

We first prove that if p ∤ m then ζ_{m} + ζ_{m}^{−1} ∈ ℚ_{p}ˆ iff p ≡ ±1 (mod m).

In this case the extension ℚ_{p}ˆ (ζ_{m})/ℚ_{p}ˆ is unramified and Galois with cyclic Galois group generated by the Frobenius automorphism σ given by σ(ζ_{m}) = ζ_{m}^{p}, [69, IV.§4.16].

If ζ_{m} + ζ_{m}^{−1} ∈ ℚ_{p}ˆ then ℚ_{p}ˆ = ℚ_{p}ˆ (ζ_{m} + ζ_{m}^{−1}) and the extension ℚ_{p}ˆ (ζ_{m})/ℚ_{p}ˆ is of degree at most 2. If the degree is 1 then by (3) we have p ≡ 1 (mod m). If the degree is 2 then σ(ζ_{m}) = ζ_{m}^{p} ≠ ζ_{m} since σ has order 2. Considering the polynomial (x − ζ_{m})(x − ζ_{m}^{−1}) = x^{2} − (ζ_{m} + ζ_{m}^{−1})x + 1 we see that σ(ζ_{m}) = ζ_{m}^{−1}. Thus p ≡ −1 (mod m).

Conversely if p ≡ ±1 (mod m) then we have either σ(ζ_{m}) = ζ_{m} or σ(ζ_{m}) = ζ_{m}^{−1}, and thus the element ζ_{m} + ζ_{m}^{−1} is fixed by σ. Therefore we have ζ_{m} + ζ_{m}^{−1} ∈ ℚ_{p}ˆ .

Consider now the general case. We can write m = p^{α}m′ with p ∤ m′, where we may now assume p^{α} ≥ 2.

Assume first that ζ_{m}+ζ_{m}^{−1} ∈ ℚ_{p}ˆ . The n’th Chebyshev polynomial of the first kind, T_{n}(x) ∈ ℤ[x], satisfies the relation T_{n}(cos(θ)) = cos(nθ) [1]. This gives T_{n}((ζ_{nn′} + ζ_{nn′}^{−1})/2) = (ζ_{n′} + ζ_{n′}^{−1})/2 for all n, n′. So if ζ_{m} + ζ_{m}^{−1} ∈ ℚ_{p}ˆ we get ζ_{m′} + ζ_{m′}^{−1} ∈ ℚ_{p}ˆ . If m′ ≥ 3 we thus have p ≡ ±1 (mod m′) by what has already been proved, while for m′ ≤ 2 we only get the restriction p ≠ 2 if m′ = 2.

Consider now the extension ℚ_{p}ˆ (ζ_{m})/ℚ_{p}ˆ . It is the composite of the unramified extension ℚ_{p}ˆ (ζ_{m′})/ℚ_{p}ˆ and the extension ℚ_{p}ˆ (ζ_{p^{α}})/ℚ_{p}ˆ . According to [69, IV.§4.17] the last extension is totally ramified and has degree φ(p^{α}). Thus the intersection of ℚ_{p}ˆ (ζ_{m′}) and ℚ_{p}ˆ (ζ_{p^{α}}) equals ℚ_{p}ˆ and the degree of ℚ_{p}ˆ (ζ_{m})/ℚ_{p}ˆ is equal to the product of [ℚ_{p}ˆ (ζ_{m′}) : ℚ_{p}ˆ ] and [ℚ_{p}ˆ (ζ_{p^{α}}) : ℚ_{p}ˆ ].

As above we get [ℚ_{p}ˆ (ζ_{m}) : ℚ_{p}ˆ ] ≤ 2. This shows φ(p^{α}) ≤ 2, and thus p^{α} = 2, 3 or 4. For p^{α} = 2 we get the possibilities m′ = 1, 3. For p^{α} = 3, 4 we have φ(p^{α}) = 2 so the extension ℚ_{p}ˆ (ζ_{m′})/ℚ_{p} ˆ must have degree 1. So p ≡ 1 (mod m′) if m′ ≥ 3. This leaves only the possibilities m′ = 1, 2 for p^{α} = 3 and m′ = 1 for p^{α} = 4. Summing up, the possibilities for (p, m) are m = 2, 4, 6 for p = 2 and m = 3, 6 for p = 3.

Conversely, since ζ_{m} + ζ_{m}^{−1} ∈ ℚ for m = 2, 3, 4 and 6, we are done. □

We distinguish the possible primes into two classes : If p ∤ |W | we say that p is a nonmodular prime for the p-adic pseudoreflection group W, and if p | |W | we say that p is a modular prime.

Number Order

Character field  Nonmodular primes  Modular primes
1  (n + 1)!  ℚ  p > n + 1  2 ≤ p ≤ n + 1
2a  n!mn/r  ℚ(ζm)  p ≡ 1 (mod m), p > n for m ≥ 3  p ≡ 1 (mod m), p ≤ n for m ≥ 3
p > n for m = 2  p ≤ n for m = 2
2b 2m ℚ(ζm + ζ{−1 \atop m}) p ≡ ±1 (mod m), (p, m) ≠ (2, 3) m = 3, 4, 6 for p = 2
m = 3, 6 for p = 3
3  n  ℚ(ζn)  p ≡ 1 (mod n) for p > 2 or n = 1, p = 2  n = 2, p = 2
4  24 = 23 · 3  ℚ(ζ3)  p ≡ 1 (mod 3)
5  72 = 23 · 32  ℚ(ζ3)  p ≡ 1 (mod 3)
6  48 = 24 · 3  ℚ(ζ12)  p ≡ 1 (mod 12)
7  144 = 24 · 32  ℚ(ζ12)  p ≡ 1 (mod 12)
8  96 = 25 · 3  ℚ(ζ4)  p ≡ 1 (mod 4)
9  192 = 26 · 3  ℚ(ζ8)  p ≡ 1 (mod 8)
10  288 = 25 · 32  ℚ(ζ12)  p ≡ 1 (mod 12)
11  576 = 26 · 32  ℚ(ζ24)  p ≡ 1 (mod 24)
12  48 = 24 · 3  ℚ(\sqrt{−2})  p ≡ 1, 3 (mod 8), p ≠ 3  p = 3
13  96 = 25 · 3  ℚ(ζ8)  p ≡ 1 (mod 8)
14  144 = 24 · 32  ℚ(ζ3, \sqrt{−2})  p ≡ 1, 19 (mod 24)
15  288 = 25 · 32  ℚ(ζ24)  p ≡ 1 (mod 24)
16  600 = 23 · 3 · 52  ℚ(ζ5)  p ≡ 1 (mod 5)
17  1200 = 24 · 3 · 52  ℚ(ζ20)  p ≡ 1 (mod 20)
18  1800 = 23 · 32 · 52  ℚ(ζ15)  p ≡ 1 (mod 15)
19  3600 = 24 · 32 · 52  ℚ(ζ60)  p ≡ 1 (mod 60)
20  360 = 23 · 32 · 5  ℚ(ζ3, \sqrt{5})  p ≡ 1, 4 (mod 15)
21  720 = 24 · 32 · 5  ℚ(ζ12, \sqrt{5})  p ≡ 1, 49 (mod 60)
22  240 = 24 · 3 · 5  ℚ(ζ4, \sqrt{5})  p ≡ 1, 9 (mod 20)
23  120 = 23 · 3 · 5  ℚ(\sqrt{5})  p ≡ 1, 4 (mod 5)
24  336 = 24 · 3 · 7  ℚ(\sqrt{−7})  p ≡ 1, 2, 4 (mod 7), p ≠ 2  p = 2
25  648 = 23 · 34  ℚ(ζ3)  p ≡ 1 (mod 3)
26  1296 = 24 · 34  ℚ(ζ3)  p ≡ 1 (mod 3)
27  2160 = 24 · 33 · 5  ℚ(ζ3, \sqrt{5})  p ≡ 1, 4 (mod 15)
28  1152 = 27 · 32  ℚ  p ≠ 2, 3  p = 2, 3
29  7680 = 29 · 3 · 5  ℚ(ζ4)  p ≡ 1 (mod 4), p ≠ 5  p = 5
30  14400 = 26 · 32 · 52  ℚ(\sqrt{5})  p ≡ 1, 4 (mod 5)
31  64 · 6! = 210 · 32 · 5  ℚ(ζ4)  p ≡ 1 (mod 4), p ≠ 5  p = 5
32  216 · 6! = 27 · 35 · 5  ℚ(ζ3)  p ≡ 1 (mod 3)
33  72 · 6! = 27 · 34 · 5  ℚ(ζ3)  p ≡ 1 (mod 3)
34  108 · 9! = 29 · 37 · 5 · 7  ℚ(ζ3)  p ≡ 1 (mod 3), p ≠ 7  p = 7
35  72 · 6! = 27 · 34 · 5  ℚ  p ≠ 2, 3, 5  p = 2, 3, 5
36  8 · 9! = 210 · 34 · 5 · 7  ℚ  p ≠ 2, 3, 5, 7  p = 2, 3, 5, 7
37  192 · 10! = 214 · 35 · 52 · 7  ℚ  p ≠ 2, 3, 5, 7  p = 2, 3, 5, 7

Table 2. The finite irreducible p-adic pseudoreflection groups

This distiction occurs several times, both in the theorems and their proofs. The nonmodular case is generally well understood both algebraically and topologically. This is basically due to the fact that various obstruction groups automatically vanish if the order of W is coprime to p. This may sound somewhat vague, but will become clearer later. Also, in the nonmodular case there are no problems with representation theory, as is shown by the theorems in the following section.

4. Integral representations

In the previous section we classified all finite pseudoreflection groups W ↪ GL_{n}(ℚ_{p}ˆ ). In this section we study integral and modular representations of finite pseudoreflection groups, i.e. representations W ↪ GL_{n}(ℤ_{p}ˆ ) and W ↪ GL_{n}(𝔽_{p}) and the connections between these. The general theory is quite well developed, see for example the monumental work [20, 21], but except for very small groups concrete calculations are amazingly complicated. However we will use only some very basic theorems which are simple to prove. We return to the problem again in section 6.

Our interest comes from the theory of homotopy Lie groups where one of the crucial facts is that a connected p-compact group X of rank n determines a finite pseudoreflection group W ↪ GL_{n}(ℤ_{p}ˆ ). Changing X up to homotopy equivalence only changes W up to conjugacy within GL_{n}(ℤ_{p}ˆ ). For definitions and references concerning p-compact groups we refer the reader to section 8.

It is not hard to show that any finite group G ↪ GL_{n}(ℚ_{p}ˆ ) is conjugate (within GL_{n}(ℚ_{p}ˆ )) to a subgroup of GL_{n}(ℤ_{p}ˆ ) (see Proposition 4.1 below). However this subgroup of GL_{n}(ℤ_{p}ˆ ) need not be uniquely determined and thus determining the total number of these different realizations (up to conjugacy within GL_{n}(ℤ_{p}ˆ )) becomes crucial.

In the following, if G → GL_{n}(ℤ_{p}ˆ ) is some representation, we denote the associated ℤ_{p}ˆ [G]-lattice (ℤ_{p}ˆ )^{n} simply by L. Composing with reduction modulo p, we get a representation G → GL_{n}(𝔽_{p}) and the associated 𝔽_{p}[G]-module L ⊗ 𝔽_{p} = (𝔽_{p})^{n} will just be denoted by L/p. Finally, if we start with a subgroup G ↪ GL_{n}(𝔽_{p}), we usually denote the associated 𝔽_{p}[G]-module (𝔽_{p})^{n} simply by V.

Proposition 4.1. Any representation of a finite group over ℚ_{p}ˆ is conjugate to a representation over ℤ_{p}ˆ .

Proof: See [20, 23.16]. □

Definition 4.2. Let ρ : G ↪ GL_{n}(ℚ_{p}ˆ ). By Proposition 4.1, G is conjugate to a subgroup of GL_{n}(ℤ_{p}ˆ ) within GL_{n}(ℚ_{p}ˆ ). We say that G has a unique lattice if this subgroup is unique up to conjugacy within GL_{n}(ℚ_{p}ˆ ), i.e. if given two subgroups G_{1}, G_{2} of GL_{n}(ℤ_{p}ˆ ) both conjugate to G within GL_{n}(ℚ_{p}ˆ ), then G_{1} and G_{2} are actually conjugate within GL_{n}(ℤ_{p}ˆ ). This may be reformulated as saying that if σ, σ′ : G ↪ GL_{n}(ℤ_{p}ˆ ) are both weakly equivalent to ρ as ℚ_{p}ˆ -representations, then they are weakly equivalent as ℤ_{p}ˆ -representations.

The following proposition will be used many times. The proof below was communicated to us by J. Møller. Another proof may be found in [81, p. 107] or [57, p. 346].

Proposition 4.3. Let ρ : G ↪ GL_{n}(ℤ_{p}ˆ ) be a finite group with associated lattice L. If L/p is an irreducible G-module, then the lattice L is unique.

Proof: Assume we have another representation ρ′ with lattice N, which is weakly equivalent to ρ as ℚ_{p}ˆ -representations. Then after composing with an automorphism of G, we may actually suppose N ⊗ ℚ ≅ L ⊗ ℚ as G-modules. Then for large enough r, p^{r}N is a submodule of L. Choose r minimal with this property, i.e. such that p^{r}N is a submodule of L, but not of pL. This shows that the map (p^{r}N)/p → L/p is nontrivial and therefore surjective, since we are assuming that L/p is irreducible. This shows that L = p^{r}N + pL, which by Nakayama’s lemma forces L = p^{r}N. Hence N ≅ L and we are done. □

Proposition 4.4. Assume G ↪ GL_{n}(ℚ_{p}ˆ ) is irreducible and p ∤ |G|. Then G has a unique lattice L, and L/p is irreducible.

Proof: This follows from [19, 75.6 and 76.15] and the preceding proposition. □

Concerning the reduction map from GL_{n}(ℤ_{p}ˆ ) to GL_{n}(𝔽_{p}) we have:

Proposition 4.5. Assume p > 2. If G ⊆ GL_{n}(ℤ_{p}ˆ ) is a finite subgroup, then the composition G ↪ GL_{n}(ℤ_{p}ˆ ) → GL_{n}(𝔽_{p}) is injective.

Proof: See [74, 10.7.1]. □

The situation in the non-modular case is summed up in the following addition to Theorem 3.1:

Theorem 4.6. [28, 1.5] For any numbers r, n and any odd prime p with p ∤ n there is a bijection between the sets

(1) Conjugacy classes of finite pseudoreflection subgroups W ⊆ GL_{r}(𝔽_{p}) of order |W | = n.

(2) Conjugacy classes of finite p-adic pseudoreflection subgroups W ⊆ GL_{r}(ℤ_{p}ˆ ) of order |W | = n.

(3) Conjugacy classes of finite p-adic pseudoreflection subgroups W ⊆ GL_{r}(ℚ_{p}ˆ ) of order |W | = n.

Proof: (Sketch) We have already seen the bijection between (2) and (3) by Proposition 4.1 and Proposition 4.4. To see the bijection between (1) and (2) use Proposition 4.5 and the fact that all pseudoreflections in W ⊆ GL_{r}(𝔽_{p}) have order coprime to p and are thus diagonalizable by Proposition 1.2. So we can lift pseudoreflections to pseudoreflections of the same order. The result follows. □

In the modular case the situation is not as straightforward as is shown by the following two examples.

Example 4.7. If the order of G ⊆ GL_{n}(𝔽_{p}) is divisible by p it is not always possible to lift G to a subgroup of GL_{n}(ℚ_{p}ˆ ). Let n ≥ 2. Then GL_{n}(𝔽_{p}) contains elements of order p since its order is divisible by p. On the other hand assume that x ∈ GL_{n}(ℚ_{p}ˆ ) has order p. Then x is a root of the polynomial λ^{p} − 1 = (λ − 1)(λ^{p−1} + … + λ + 1). The second factor is irreducible over ℚ_{p}ˆ since by [69, IV.§4.17] we have [ℚ_{p}ˆ (ζ_{p}) : ℚ_{p}ˆ ] = p − 1. Thus the minimal polynomial of x has degree p − 1 which shows that n ≥ p − 1 since x is also a root of its characteristic polynomial. So for 2 ≤ n < p − 1 it is not possible to lift GL_{n}(𝔽_{p}) to a subgroup of GL_{n}(ℚ_{p}ˆ ).

Example 4.8. [28] In fact the situation can be even worse. Consider the element

x = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ]

in GL_{2}(𝔽_{3}). It is a pseudoreflection of order 3 and it is easy to check that the invariant ring 𝔽_{3}[t_{1}, t_{2}]^{<x>} is polynomial on the generators t_{2} and t_{1}^{3} − t_{1}t_{2}^{2}. Moreover the subgroup < x > generated by x lifts to a subgroup of GL_{2}(ℤ_{3}ˆ ) but it does not have a lift which is a pseudoreflection subgroup. First of all it is easy to see that

\frac{1}{2} [ \begin{matrix}−1 & −1 \\ 3 & −1 \end{matrix} ]

is a lift of x to GL_{2}(ℤ_{3}ˆ ) which also has order 3. But x does not lift to a pseudoreflection in GL_{2}(ℚ_{3}ˆ ) since by Proposition 1.2 and Proposition 3.3 all pseudoreflections have order 2. This proves the claims.

In the general case we do not have a classification of pseudoreflection groups over 𝔽_{p}. As the examples above show there are many more such groups than pseudoreflection groups over ℚ_{p}ˆ since they do not need to lift and even if they do lift they need not lift to pseudoreflection groups. However some classification results have been obtained, see the references in [54].

About liftings in general we can say the following. Suppose G ⊆ GL_{n}(𝔽_{p}) is some subgroup. We are interested in lifting it to a subgroup of GL_{n}(ℤ_{p}ˆ ). The group GL_{n}(ℤ_{p}ˆ ) is the inverse limit of the system of epimorphisms

… → GL_{n}(ℤ/p^{k+1}) → GL_{n}(ℤ/p^{k}) → … → GL_{n}(ℤ/p)

Assume we have lifted G to a subgroup of GL_{n}(ℤ/p^{k}) for some k ≥ 1 and that the lift is unique. Consider the reduction map GL_{n}(ℤ/p^{k+1}) → GL_{n}(ℤ/p^{k}). The kernel is given by the set of matrices {1 + p^{k}A|A ∈ M_{n}(ℤ/p^{k+1})}. Since multiplication is given by the formula

(1 + p^{k}A)(1 + p^{k}B) = 1 + p^{k}(A + B) + p^{2k}AB = 1 + p^{k}(A + B)

we see that the kernel is isomorphic to M_{n}(ℤ/p) ≅ V ⊗ V^{∗}, where V = (𝔽_{p})^{n}. For any A ∈ M_{n}(ℤ/p^{k+1}) and X ∈ GL_{n}(ℤ/p^{k+1}) the computation X^{−1}(1 + p^{k}A)X = 1 + p^{k}X^{−1}AX shows that this isomorphism preserves the G-action. Thus the lifting obstructions are the same at each stage. We have proved:

Proposition 4.9. Assume G ⊆ GL_{n}(𝔽_{p}) and let V = (𝔽_{p})^{n} be the natural G-module. Then there is a lift of G to a subgroup of GL_{n}(ℤ_{p}ˆ ) if H^{2}(G, V ⊗V^{∗}) = 0 and this lift is unique up to conjugacy within GL_{n}(ℤ_{p}ˆ ) if H^{1}(G, V ⊗ V^{∗}) = 0. □

Example 4.10. Consider the group W which has number 12 in the classification tables. By Table 2 it is a 3-adic pseudoreflection group. It has order 48 = 2^{4} · 3 and rank 2 and is isomorphic to GL_{2}(𝔽_{3}). We will consider it again later in section 5. Here we show that all its pseudoreflection subgroups H have unique liftings by showing that H^{1}(H, V ⊗ V^{∗}) = 0. Note that we already have lifts in this case so we need not compute H^{2}(H, V ⊗ V^{∗}). By comparing with Table 2 we see that the 3-adic pseudoreflection subgroups of W are 1, ℤ/2, Σ_{3}, ℤ/2 × ℤ/2, D_{8}, D_{12} and W itself. In all cases there is only one conjugacy class isomorphic to the given subgroup, except in the case of Σ_{3} where we have two conjugacy classes. Since we work at p = 3 it is enough to show the result for Σ_{3}, D_{12} and W. Since both D_{12} and W contain Σ_{3} as a subgroup of index prime to 3 it suffices to consider the subgroups isomorphic to Σ_{3}. It is readily seen that the two conjugacy classes of groups isomorphic to Σ_{3} are SΣ_{3} generated by the matrices

σ = [ \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ] , τ = [ \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} ] ,

and P Σ_{3} generated by σ and −τ. Since both τ and −τ act on V ⊗ V^{∗} in the same way (−1 acts trivially), we see that the Σ_{3} action on V ⊗ V^{∗} is the same in both cases. It is now not hard to check directly that H^{1}(Σ_{3}, V ⊗ V^{∗}) = 0.

5. The Aguadé cases

In this section we look at a number of the irreducible modular cases, which were first considered in detail by Aguadé [3]. All of them are among the sporadic cases 4 − 37 and the order of the Weyl group W is divisible by p, but not by p^{2}. In fact the cases we will consider are characterized by this, apart from one single exception, namely W (E_{6}) (no. 35) at p = 5, which we will not consider. More precisely, comparing with Table 2 the cases we look at are (G_{12}, p = 3), (G_{29}, p = 5), (G_{31}, p = 5), (G_{34}, p = 7), (G_{36}, p = 5), (G_{36}, p = 7) and (G_{37}, p = 7). Another common feature of these groups is that all pseudoreflections have order 2.

We return to these groups in section 9, where we will show that their corresponding ring of mod-p invariants are the mod-p cohomology rings of certain uniquely determined spaces.

Aguadé proved that it is possible to embed a symmetric group Σ_{n+1} (n is the rank of W) in W in all cases and furthermore he obtained information on the normalizers of the Sylow p-subgroups.

Here we prove that each of the groups admits a unique lattice. We also examine Aguadé’s embeddings somewhat closer. This enables us to compute the normalizer N_{GL_{n}(ℚ_{p}ˆ )}(W) of the groups in question, which in turn will allow us to compute the self-maps of the spaces mentioned above.

Number Rank n p  Order  Center Z Σ
12  2  3  2^{4} · 3  2  Σ_{3}
29  4  5  2^{9} · 3 · 5  4  Σ_{5}
31  4  5 2^{10} · 3^{2} · 5  4  Σ_{5}
34  6  7 2^{9} · 3^{7} · 5 · 7  6  Σ_{7}
36  7  5 2^{10} · 3^{4} · 5 · 7  2  Σ_{8}
36  7  7 2^{10} · 3^{4} · 5 · 7  2  Σ_{8}
37  8  7 2^{14} · 3^{5} · 5^{2} · 7  2  Σ_{9}

Table 3. The Aguadé groups

We first give a description of each group to fix notation, and prove that all of them admit a unique lattice by using the criterion in Proposition 4.3. We will be rather brief, and the reader should consult [3] and the references therein for more information.

Example 5.1. \underline{The group number 12}. This group W is abstractly isomorphic to GL_{2}(𝔽_{3}) and has rank 2. Thus its reduction modulo 3 consists of the full general linear group by Proposition 4.5. Since this acts irreducibly on (𝔽_{3})^{2} this shows that W has a unique lattice.

Shephard and Todd [72] gave the following generators for the representation:

Z = [ \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ] , S = \frac{1}{\sqrt{2}} [ \begin{matrix}−1 & i \\ −i & 1 \end{matrix} ] , T = \frac{1}{\sqrt{2}} [ \begin{matrix}ε & ε \\ ε^{3} & ε^{7} \end{matrix} ] ,

Note that the matrix entries lie outside the character field, which is ℚ(\sqrt{−2}) in this case. By Theorem 3.1 the representation is equivalent to a representation taking values in the character field. A concrete representation with this property may be found in [74, p. 201-203].

Example 5.2. \underline{The groups number 29 and 31}. These groups are closely related to each other, in fact G_{29} is a subgroup of index 6 in G_{31}. They are generated by the following reflections (see [3] and [72, p.298]):

R_{1} = \frac{1}{2} [ \begin{matrix}1 & −1 & −1 & −1 \\ −1 & 1 & −1 & −1 \\ −1 & −1 & 1 & −1 \\ −1 & −1 & −1 & 1 \end{matrix} ] , R_{2} = [ \begin{matrix}0 & −i & 0 & 0 \\ i & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

R_{3} = [ \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , R_{4} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] , R_{5} = [ \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ] .

The group G_{29} is generated by R_{j}, 1 ≤ j ≤ 4 and G_{31} is generated by R_{j}, 1 ≤ j ≤ 5. By mapping i to a primitive fourth root of unity in ℤ_{5}ˆ we see that these representations are defined over GL_{4}(ℤ_{5}ˆ ).

It may be checked that G_{29} contains all permutation matrices in Σ_{4}. The permutations (1, 2) and (2, 3) equals respectively R_{3} and R_{4}, and a direct computation shows that also (3, 4) is contained in G_{29} (it equals the conjugate of R_{1} by R_{2}R_{3}R_{4}R_{2}R_{1}R_{3}R_{2}R_{4}). This, combined with the fact that G_{29} also contains the diagonal matrix R_{5} shows easily that the reduction modulo 5 of G_{29} acts irreducibly on (𝔽_{5})^{4} (compare the proof of Theorem 6.2 or the case of group number 34 below). Thus both G_{29} and G_{31} have unique lattices by Proposition 4.3.

Example 5.3. \underline{The group number 34}. This group has rank 6 and is according to Shephard [71, p. 373] generated by the 6 reflections in the hyperplanes x_{i} − x_{i−1} = 0, 2 ≤ i ≤ 5, x_{1} − ωx_{2} = 0 and x_{1} + … + x_{6} = 0, where ω is a primitive third root of unity. Note that this directly gives a representation of W in GL_{6}(ℤ_{7}ˆ ). It may also be checked that W contains the reflection in the hyperplane x_{5} − x_{6} = 0, and thus W contains all permutation matrices in Σ_{6}. We now show that W acts irreducibly on its reduction modulo 7 and thus has a unique lattice. Assume that 0 ≠ M is some W -invariant submodule of V = (𝔽_{7})^{6}. Choose a nonzero vector v in M. After acting by a suitable permutation we may assume that v_{1} ≠ 0 or v_{2} ≠ 0. By acting if necessary with the reflection R in the hyperplane x_{1} − ωx_{2} = 0, we may assume that v_{1} ≠ v_{2}. Setting λ = v_{1} − v_{2} ≠ 0 we see that M contains the vector w = v − (1, 2)v = (λ, −λ, 0, 0, 0, 0)^{t}. Thus M also contains the vector ωw + R · w = ((ω − ω^{2})λ, 0, 0, 0, 0, 0). It easily follows that M = V and thus V is irreducible as claimed.

Example 5.4. \underline{The groups number 36 and 37}. These are respectively the Weyl groups of E_{7} and E_{8}. Here we only consider them at the primes respectively p = 5, 7 for E_{7} and p = 7 for E_{8}. Explicit root systems are given in for example [37, p. 42–43] and we find by using similar arguments as for the group number 34, that in these cases L/p is irreducible. In fact the argument works for all primes dividing the order of the Weyl group except the case of L_{E_{7}}/2 which is not irreducible, cf. Theorem 6.2.

The following theorem collects the group theoretic results in [3] common to all the groups:

Theorem 5.5. Let W ↪ GL_{n}(ℚ_{p}ˆ ) be one of the cases in question. There is an embedding Σ_{n+1} ↪ W, and for all such, the representation Σ_{n+1} ↪ GL_{n}(ℚ_{p}ˆ ) is absolutely irreducible. Furthermore n + 1 ≥ p so the index [W : Σ_{n+1}] is coprime to p.

Denote by Σ any subgroup of W isomorphic to Σ_{n+1}. Let S denote a Sylow p-subgroup of Σ_{n+1} and let Z be the center of W.

Then we have Z ∩ Σ = 1, N_{W} (S) = Z × N_{Σ}(S), and N_{GL_{n}(ℚ_{p}ˆ )}(Σ) = (ℚ_{p}ˆ )^{∗} × Σ. Thus in particular we have N_{W} (Σ) = Z × Σ and N_{N_{W} (Σ)}(S) = N_{W} (S).

Proof: The existence of the embedding is shown by Aguadé [3] as already mentioned above.

For the cases we consider it is easy to see that Σ_{n+1} has no faithful representations in dimension less than n and thus the representation is irreducible. Recall that any irreducible representation of Σ_{n+1} is conjugate to a representation taking values in ℚ. So all irreducible representations are absolutely irreducible.

It is apparent that n + 1 ≥ p in all cases. Since also n + 1 ≥ 3 we see that the symmetric group Σ = Σ_{n+1} has no center. The fact that N_{W} (S) = Z × N_{Σ}(S) is proved by Aguadé [3, p. 36].

The final results may be shown as follows. Consider x ∈ N_{GL_{n}(ℚ_{p}ˆ )}(Σ). Then conjugation by x is an automorphism and since all automorphisms of Σ are inner in the cases we consider (we have n + 1 ≠ 2, 6, cf. [78, 2.18]) it follows that we can find y ∈ Σ such that xy^{−1} commutes with all elements in Σ. Then since the representation is absolutely irreducible we see by Schur’s lemma that xy^{−1} is equal to a scalar matrix. Since Σ has no center we get N_{GL_{n}(ℚ_{p}ˆ )}(Σ) = (ℚ_{p}ˆ )^{∗} × Σ and therefore also N_{W} (Σ) = Z × Σ recalling that Z consists of scalar matrices by Proposition 2.3. Using these results we get directly

N_{N_{W} (Σ)}(S) = N_{Z×Σ}(S) = Z × N_{Σ}(S) = N_{W} (S).
□

Remark 5.6. It is possible to choose for each of the groups in question, a representation W ↪ GL_{n}(ℤ_{p}ˆ ), such that the subgroup Σ above equals the stabilizer in W of the diagonal △ ⊆ L/p.

Moreover the representation W ↪ GL_{n}(ℤ_{p}ˆ ) equals the representation of Σ_{n+1} on the lattice associated to SU(n + 1). All of this actually may be verified directly by looking at the representations given by Aguadé [3]. This observation is due to Notbohm [61, p. 10].

We now proceed to the calculation of the normalizers.

Theorem 5.7. Let W ↪ GL_{n}(ℚ_{p}ˆ ) be one of the Aguadé groups. Then N_{GL_{n}(ℚ_{p}ˆ )}(W) is generated by W and the scalar matrices (ℚ_{p}ˆ )^{∗}.

Proof: (1) Let W = G_{12} ≅ GL_{2}(𝔽_{3}). Since all pseudoreflections have order 2, they have determinant −1. So all elements in W have determinant ±1. Since W contains the scalar matrix −1 we have an homomorphism χ : W → W given by x ↦ det(x) · x. Assume χ(x) = 1. Then x is a scalar matrix and thus in the center of W. But since the rank is even all elements in the center have determinant 1 and thus χ(x) = x. This shows x = 1 and thus χ is an automorphism. By a theorem of Hua [36, p. 101] the automorphism group of W is generated by χ and inner automorphisms.

Now let A ∈ N_{GL_{2}(ℚ_{3}ˆ )}(W). Then after modifying A by an element of W we may assume that conjugation by A either is the identity or that it equals χ. However χ does not preserve traces since the element ST has determinant −1 and trace −\sqrt{−2}. Therefore conjugation by A has to agree with the identity and we are done by Theorem 5.5.

(2) Let W denote any of the two groups G_{29} and G_{31}. We first describe the conjugacy classes inside W of subgroups isomorphic to the symmetric group Σ_{5}.

Note that, in the representation given, W is invariant under complex conjugation, which thus gives an automorphism φ of W (it suffices to note that φ(R_{2}) = R_{2}^{R_{3}} ∈ W).

Exactly the same argument as in the case of G_{12} shows that we have an automorphism χ given by x ↦ det(x) · x.

Denote by R the conjugate of R_{2} by R_{1}R_{4}R_{3}R_{2} (cf. [3, p. 32]) and let H_{1} be the subgroup generated by R_{1}, R_{3}, R_{4} and R. Let R′ = φ(R). Let H_{2} be the subgroup generated by R_{1}, R_{3}, R_{4} and R′, H_{3} the subgroup generated by −R_{1}, −R_{3}, −R_{4} and −R and finally let H_{4} be the subgroup generated by −R_{1}, −R_{3}, −R_{4} and −R′. These are all isomorphic to Σ_{5}. By using a computer it may be checked that any subgroups of W isomorphic to Σ_{5} is conjugate to either H_{1}, H_{2}, H_{3} or H_{4}. Note that we have H_{2} = φ(H_{1}), H_{3} = χ(H_{1}) and H_{4} = χφ(H_{1}).

Now assume A ∈ N_{GL_{4}(ℚ_{5}ˆ )}(W). Then, since H_{1}, H_{2}, H_{3} and H_{4} is a complete set of representatives for the conjugacy classes in W of subgroups isomorphic to Σ_{5}, after modifying A by an element of W we may assume that conjugation by A send H_{1} to one of the groups H_{i}, 1 ≤ i ≤ 4. Since all automorphisms of Σ_{5} are inner [78, 2.18] we may, after modifying A by a new element of W, assume that conjugation by A agrees on H_{1} with either the identity on W or with one of the automorphisms φ, χ or χφ. Since conjugation by A sends pseudoreflections to pseudoreflections we see that conjugation by A cannot agree with either χ or χφ.

Assume that conjugation by A agrees on H_{1} with φ. Then the automorphism given by composing conjugation by A with φ fixes H_{1} elementwise. By Theorem 5.5 this composition is therefore given by a conjugation. Thus we conclude that φ agrees on all of W with a conjugation by an element from GL_{4}(ℚ_{5}ˆ ). This is a contradiction since conjugation preserves traces, while the character field of W is given by ℚ(i) which is not invariant under complex conjugation.

Thus conjugation by A has to agree with the identity on H_{1} and we are done by Theorem 5.5.

(3) Consider now the group W = G_{34}. According to [15, p. 412] we have W/C_{3} ≅ SO^{−}(6, 𝔽_{3}), where C_{3} is the subgroup of order 3 in the center Z of W (recall that by Proposition 2.3 the center Z has order 6 and consists of scalar matrices). The group SO^{−}(6, 𝔽_{3}) is the group of matrices in SL(6, 𝔽_{3}) leaving the quadratic form f = x_{1}x_{2} + x_{3}x_{4} + x_{5}^{2} + x_{6}^{2} of Witt index 2 invariant. A direct computation shows that no matrices in GL(6, 𝔽_{3}) transforms f into −f. By using results of Dieudonné, [24, Theorems 15,16], it follows that any automorphism of SO^{−}(6, 𝔽_{3}) is the restriction of an automorphism of GO^{−}(6, 𝔽_{3}) and that all such are given by x ↦ α(x)gxg^{−1}, where α : GO^{−}(6, 𝔽_{3}) → {±1} is a homomorphism and g ∈ GO^{−}(6, 𝔽_{3}). From this it follows that any automorphism of W has the form x ↦ α(x)gxg^{−1}, where α : W → Z is a homomorphism and g ∈ W.

Assume now that A ∈ N_{GL_{6}(ℚ_{7}ˆ )}(W). Then conjugation by A is an automorphism of W that preserves traces. This shows that α(x) = 1 for all x ∈ W such that tr(x) ≠ 0. In particular α(x) = 1 for all pseudoreflections in W, so α = 1. Thus conjugation by A agrees with an inner automorphism, and we are done by Theorem 5.5.

(4) In the cases G_{36}, p = 5, 7 and G_{37}, p = 7 the group in question is the Weyl group of the compact connected Lie groups E_{7} and E_{8}, and the normalizer may be calculated using results of Møller [46]. In these cases the Coxeter-Dynkin diagram have no graph automorphisms so the result follows from [46, 1.2, 1.3]. □

6. Classification of lattices

In this section we review Notbohm’s classification of lattices [57] at odd primes, which we will be using in the next section to compute cohomology of pseudoreflection groups with coefficients being the associated lattice with the natural action. First however we sum up the situation in the case of irreducible representations.

Example 6.1. Concerning the symmetric groups Σ_{n} we have an (n − 1)-dimensional irreducible pseudoreflection representation defined over ℚ. The conjugacy classes of ℤ-representations equivalent to this were found by Craig [18]. The result is that they are given exactly by the lattices of the central quotients SU(n)/C_{d} of SU(n), where d runs through the divisors of n. By exactly the same proofs we get a local version of this saying that the equivalence classes of ℤ_{p}ˆ -representations corresponding to this representation, are exactly the ones corresponding to lattices of the quotients SU(n)/p^{k}, where p^{k} runs through the divisors of n. As a consequence we see that Σ_{n} has a unique lattice at the prime p unless p|n.

Theorem 6.2. Let W ↪ GL_{n}(ℚ_{p}ˆ ) be an irreducible pseudoreflection group with lattice L. Then L/p is irreducible if (W, p) does not belong to the following list:

(1) W ≅ Σ_{n} belonging to the family 1, n ≥ 3, 2 ≤ p ≤ n.

(2) W ≅ G(2, 1, n) belonging to the family 2a with n ≥ 2 and p = 2 or W ≅ G(2, 2, n) belonging to the family 2a with n ≥ 3 and p = 2.

(3) W ≅ G(3, 3, 2) ≅ D_{6}, p = 3, W ≅ G(4, 4, 2) ≅ D_{8}, p = 2 or W ≅ G(6, 6, 2) ≅ D_{12}, p = 3 all belonging to the family 2b.

(4) W ≅ W (F_{4}) (number 28) and p = 2, W ≅ W (E_{6}) (number 35) and p = 3, or W ≅ W (E_{7}) (number 36) and p = 2.

As a consequence, if (W, p) does not belong to the excluded list, then the lattice L is unique. This also holds in the cases W ≅ Σ_{n}, p ∤ n and in the case W ≅ G(6, 6, 2) ≅ D_{12}, p = 3.

Remark 6.3. There is some overlap between the excluded cases, i.e. the group Σ_{3} ≅ G(3, 3, 2) ≅ D_{6} is excluded at p = 3 both by (1) and (3). Similarly Σ_{4} ≅ G(2, 2, 3) is excluded at p = 2 by (1) and (2). Finally G(2, 1, 2) ≅ G(4, 4, 2) ≅ D_{8} is excluded at p = 2 by (2) and (3). These repetitions are made only to make the proof and statement of the theorem clearer.

It should be noted that the theorem is similar to [57, 2.4] but has weaker hypotheses. Due to this, however, some cases have to be excluded.

Proof: By Proposition 4.4 we may assume that p divides |W |.

Assume first that W belongs to the family 2a and is not one of the excluded cases. The representation G(m, r, n) = A(m, r, n) ⋊ Σ_{n} is already defined over ℤ_{p}ˆ . Let the lattice be L.

If m > 2 we have p ≡ 1 (mod m) and in particular p ∤ m. Since we are also assuming p | |W | we have 3 ≤ p ≤ n. We can choose a primitive m’th root of unity θ in 𝔽_{p}^{∗}.

Consider now L/p, and assume 0 ≠ M ⊆ L/p is some submodule. Choose x ∈ M with x ≠ 0. Since W contains the permutation matrices and the diagonal matrix diag(θ, θ^{−1}, 1, …, 1) we see that M contains an element of the form x′ = (x_{1}, x_{2}, 0, …, 0)^{t} with x_{1} ≠ 0. Since n ≥ 3 we also have the diagonal matrix diag(θ, 1, θ^{−1}, 1, …, 1) in W and so W contains ((1 − θ)x_{1}, 0, …, 0)^{t}. Thus M contains all the unit vectors, so M = L/p. This shows that L/p is irreducible and we are done in this case.

Note that exactly the same argument applies also in the case m = 2, n ≥ 3 and p ≠ 2, the key point is just that we can find a primitive m’th root of unity θ and that we have ‘enough’ room to move around, i.e. n ≥ 3.

Thus only the cases where m = 2 and either n = 2 or p = 2 are left. Observe that since we are assuming p | |W | we get p = 2 if n = 2. Thus we have m = 2 and p = 2. These are exactly the cases excluded in (2) and (3). Thus we are done with the groups of type 2a.

In case W belongs to the family 3 the rank is 1 and we are done.

The cases (G_{12}, p = 3), (G_{29}, p = 5), (G_{31}, p = 5), (G_{34}, p = 7), (G_{36}, p = 5), (G_{36}, p = 7) and (G_{37}, p = 7) were settled in section 5. Thus we are left with considering the cases D_{6}, D_{8} and D_{12} from the family 2b and the sporadic Coxeter cases, W (F_{4}), W (E_{6}), W (E_{7}) and W (E_{8}) at primes dividing the order of the group. All of these groups are Coxeter groups and root systems may be found in [37, p. 41–43]. Direct computation shows that L/p is irreducible for all of them except for the cases listed in (3) and (4).

The last part of the theorem is proved by using Proposition 4.3 and Example 6.1. The case of D_{12} at p = 3 is handled by noting that there is only one subgroup of GL_{2}(𝔽_{3}) isomorphic to D_{12} up to conjugation and by Example 4.10 it lifts to a unique (up to conjugation) subgroup of GL_{2}(ℤ_{3}ˆ ). □

If the representation is not irreducible the situation is more complicated. It is possible however to give some structural theorems which enable us to relate general lattices to ‘nice’ lattices by certain short exact sequences. Due to problems in the proofs arising from the non-vanishing at p = 2 of certain homology and cohomology groups with trivial coefficients (see the next section) we will in the rest of this section only consider odd primes. The following definitions and results all come from Notbohm [57]. We will be rather brief and only state the main results.

Definition 6.4. Let W ↪ GL_{n}(ℚ_{p}ˆ ) = GL(U) be a p-adic pseudoreflection group. Let L be a lattice corresponding to this representation, i.e. L is a free ℤ_{p}ˆ -module of rank n such that L⊗ℚ ≅ U as W -modules. By a sublattice of L we mean a submodule of L having the same rank.

We define the discrete torus Ť = (L ⊗ ℚ)/L ≅ (ℤ/p^{∞})^{n}. The lattice L is said to be fixed point free if L^{W} = H^{0}(W, L) = 0 and centerfree if Ť^{W} = 0. If L_{W} = H_{0}(W, L) = 0 we say that L is simply connected.

These definitions are motivated by topological considerations, cf. Example 6.6. However the precise connection has only been resolved recently by work of Dwyer, and we return to the questions in section 10.

Both simply connected and centerfree lattices are fixed point free, [57, 2.1(1) and 3.1(1)]. For fixed point free lattices we have the following results:

Theorem 6.5. [57, 1.2] Every fixed point free lattice L has a centerfree sublattice P L and a simply connected sublattice SL. These are unique up to weak isomorphism. □

Example 6.6. For an irreducible representation W = Σ_{n} ↪ GL_{n}(ℤ_{p}ˆ ) we saw above that all lattices are given as L = L_{SU(n)/C_{p^{k}}}, where p^{k}|n. The unique simply connected lattice is given by SL = L_{SU(n)} and the unique centerfree lattice is given by P L = L_{P U(n)}.

Theorem 6.7. [57, 1.4] Assume the representation W → GL_{n}(ℚ_{p}ˆ ) = GL(U) is fixed point free, and consider the associated splittings W ≅ W_{1} × … × W_{r} and U ≅ U_{1} ⊕ … ⊕ U_{r}.

(1) Any simply connected lattice S corresponding to U splits as a direct sum S ≅ S_{1} ⊕ … ⊕ S_{r}, where S_{i} is a simply connected lattice corresponding to U_{i}.

(2) Any centerfree lattice P corresponding to U splits as a direct sum P ≅ P_{1} ⊕ … ⊕ P_{r}, where P_{i} is a centerfree lattice corresponding to U_{i}. □

As a consequence, if W ↪ GL_{n}(ℚ_{p}ˆ ) is fixed point free and has a unique lattice L, then L is automatically both simply connected and centerfree. Such lattices only occur as direct summands, [57, 1.7].

Now consider a general pseudoreflection group W → GL_{n}(U). Then we have a splitting U = U^{W} ⊕ U′ of W -modules. Since W acts trivially on U^{W}, all sublattices of U^{W} are isomorphic. Pick any sublattice and denote it by Z.

Theorem 6.8. [57, 1.6] Let L be any lattice corresponding to U.

(1) There is a simply connected lattice S and a short exact sequence 0 → S ⊕ Z → L → F → 0, where F is finite and has trivial W -action.

(2) There is a centerfree lattice P and a short exact sequence 0 → L → P ⊕ Z → F′ → 0 where F′ is finite and has trivial W -action. □

Using these results Notbohm [61] has been able to classify (at odd primes) the lattices L for which the ring of invariants ℤ_{p}ˆ [L]^{W} is polynomial. Such a lattice is called a polynomial lattice. The results are stated in the following theorems.

Theorem 6.9. [61, 1.3] A lattice L is polynomial iff the associated simply connected lattice SL is polynomial and L_{W} is torsion free.

This reduces the question to simply connected lattices, and since these split as direct sums we need only consider simply connected irreducible lattices. For those we have Theorem 6.10. [61, 1.4] If S is the simply connected lattice corresponding to an irreducible pseudoreflection group W ↪ GL_{n}(ℚ_{p}ˆ ), then S is polynomial iff (W, p) is not among the pairs (W_{F_{4}}, 3), (W_{E_{6}}, 3), (W_{E_{7}}, 3) and (W_{E_{8}}, 5).

By combining some of the lemmas proved in [61] we get:

Proposition 6.11. Assume L is a polynomial lattice associated to the pseudoreflection group W ↪ GL_{n}(ℤ_{p}ˆ ) with ℤ_{p}ˆ [L]^{W} = ℤ_{p}ˆ [f_{1}, …, f_{n}]. Then we also have ℚ_{p}ˆ [L ⊗ ℚ]^{W} = ℚ_{p}ˆ [f_{1}, …, f_{n}] and 𝔽_{p}[L/p]^{W} = 𝔽_{p}[f̄_{1}, …, f̄_{n}], where¯ denotes reduction modulo p.

Proof: This follows easily from [61, 3.1, 3.2 and 3.3]. □

We only need the following corollary later:

Corollary 6.12. Assume W ↪ GL_{n}(ℚ_{p}ˆ ) is irreducible. If (W, p) is among the cases (G_{12}, p = 3), (G_{29}, p = 5), (G_{31}, p = 5), (G_{34}, p = 7), (G_{36}, p = 5), (G_{36}, p = 7) or (G_{37}, p = 7), then W has a unique lattice L. The invariant rings ℚ_{p}ˆ [L ⊗ ℚ]^{W}, ℤ_{p}ˆ [L]^{W} and 𝔽_{p}[L/p]^{W} are all polynomial with degrees given by Table 1.

Proof: The fact that the lattice is unique follows from Theorem 6.2. Then L is automatically simply connected and thus polynomial by Theorem 6.10. The result now follows by Proposition 6.11. □

7. Cohomology of Weyl groups

In this section we compute the low-dimensional cohomology groups of the finite irreducible padic pseudoreflection groups with the coefficients ℤ_{p}ˆ and 𝔽_{p} with trivial action, and L and L/p with the natural action.

All through this section we assume that p is an odd prime. This is due to the fact that for p = 2 we can say almost nothing, for example even for trivial actions the lowest cohomology groups do not vanish. This is in fact also where the restriction to odd primes in Notbohm’s classification of lattices in the previous section comes from.

The main results are the following. Let W ↪ GL_{n}(ℤ_{p}ˆ ) be a finite p-adic pseudoreflection group with associated lattice L. Then H^{2}(W, L) = 0. Also H^{3}(W, L) = 0 if W is irreducible (with a single exception) or if L is simply connected. These results have applications to the theory of p-compact groups which we discuss in the following sections.

We start by considering trivial coefficients following Notbohm. It should be noted that our proof is basically identical to Notbohm’s proof but somewhat shorter since we can refer directly to known results about Coxeter groups.

Proposition 7.1. Let W ↪ GL_{n}(ℚ_{p}ˆ ) be a finite p-adic pseudoreflection group. Then there exists a subgroup W′ of W such that W′ is a Coxeter group and the index [W : W′] is coprime to p.

Proof: By Proposition 1.3 it is enough to consider W irreducible. If p does not divide |W | we may just take W′ = 1. Otherwise we are in the modular case and if W is itself a Coxeter group then obviously we can take W′ = W. By comparing with Table 2 we see that the only cases left are the groups of type 2a and the cases (G_{12}, p = 3), (G_{29}, p = 5), (G_{31}, p = 5) and (G_{34}, p = 7). The last four cases are handled by Theorem 5.5, since we have an embedding of the Coxeter group W′ = Σ_{n+1} (here n denotes the rank as usual). For the groups of type 2a recall that G(m, r, n) = A(m, r, n) ⋊ Σ_{n} has order n!\frac{m^{n}}{r}. As p is odd and we have the restriction p ≡ 1 (mod m) for m ≥ 3 we see that p ∤ m in all cases. We may thus set W′ = Σ_{n} since then [W : W′] = \frac{m^{n}}{r} is not divisible by p. □

Theorem 7.2. [58, 3.1] Let W be a finite p-adic pseudoreflection group. Then the homology and cohomology groups H_{1}(W, ℤ_{p}ˆ ), H_{1}(W, ℤ/p^{k}), H_{2}(W, ℤ_{p}ˆ ), H_{2}(W, ℤ/p^{k}), H^{1}(W, ℤ_{p}ˆ ), H^{1}(W, ℤ/p^{k}), H^{2}(W, ℤ_{p}ˆ ), H^{2}(W, ℤ/p^{k}), H^{3}(W, ℤ_{p}ˆ ), k ≥ 1, all vanish where the action on the coefficients is trivial.

Proof: By using Proposition 7.1 and a transfer argument we may suppose that W is a Coxeter group. By using the universal coefficient theorems it suffices to show that the groups H_{1}(W, ℤ) and H_{2}(W, ℤ) do not have p-torsion. In fact by a theorem of Ihara and Yokonuma [38] (see also [35]) both groups are actually elementary abelian 2-groups, so since we only consider p odd we are done. □

We next turn to coefficients being the natural action of W on L or L/p.

Theorem 7.3. Let W ⊆ GL_{n}(ℤ_{p}ˆ ) be a finite irreducible p-adic pseudoreflection group with associated lattice L. If (W, p) does not belong to the following list:

(1) W ≅ Σ_{n} belonging to the family 1, n ≠ 2, 4 and 3 ≤ p ≤ n.

(2) W ≅ D_{6} = G(3, 3, 2) belonging to the family 2b and p = 3.

(3) W ≅ W (E_{6}) (number 35) and p = 3. then H^{∗}(W, L) = 0 and H^{∗}(W, L/p) = 0.

Remark 7.4. The special case in (2) of D_{6} = G(3, 3, 2) is in fact already excluded by (1) since the representation is equivalent to that of Σ_{3} by Theorem 2.6 and Theorem 2.7, but we have chosen to mention it again. We give the full answer in this case later, see Theorem 7.6.

Proof: Notice first of all that H^{0}(W, L) = 0 since W is irreducible.

By Theorem 2.1 and Proposition 2.3 we see that in all the cases 4 − 37 except W (E_{6}) (number 35), W has a nontrivial center Z consisting of scalar matrices. Moreover we see that p |̸ |Z| since we only consider p odd. Thus H^{>0}(Z, L) = H^{>0}(Z, L/p) = 0, and since Z acts without fixed points on L and L/p we also obtain H^{0}(Z, L) = H^{0}(Z, L/p) = 0. Considering now the Lyndon-Hochschild-Serre spectral sequence associated to the normal subgroup Z, we immediately obtain H^{∗}(W, L) = 0 and H^{∗}(W, L/p) = 0.

If p ∤ |W | then obviously H^{>0}(W, L) = H^{>0}(W, L/p) = 0 since |W | is invertible in ℤ_{p}ˆ and 𝔽_{p}. By Proposition 4.4 we also have H^{0}(W, L/p) = 0 since L/p is irreducible.

We now only have to consider the modular cases among the families 1, 2a, 2b and 3 and the case of W (E_{6}) at p = 5. Comparing with Table 2 we see that there are no cases to check coming from the family 3 and that the family 2b gives the two cases D_{6} and D_{12} for p = 3. The first case is excluded and since D_{12} contains −1 we are done in this case as well. Concerning cases from the family 1 all of them are excluded except Σ_{2} and Σ_{4}. Since Σ_{2} has order 2 this case has already been done above. Also by Theorem 2.6 and Theorem 2.7 we have Σ_{4} ≅ G(2, 2, 3) belonging to the family 2a which we handle now.

Thus the only case left among the three families is the family 2a. In this case we have p ≡ 1 (mod m) and p ≤ n for m ≥ 3 and p ≤ n for m = 2. Since p is odd we actually have p ∤ m in both cases. The order of the normal subgroup A(m, r, n), m^{n}/r, is thus prime to p. Using the Lyndon-Hochschild-Serre spectral sequence we thus get E_{2}^{s,t} = 0 for t > 0. Since p is odd and A(m, r, n) contains all diagonal matrices diag(1, …, ζ_{m}, …, ζ_{m}^{−1}, …, 1) with all entries equal to 1 except for two which are respectively ζ_{m} and ζ_{m}^{−1}, and these act without fixed points on L and L/p, we see that also E_{2}^{s,0} = 0. Thus we have H^{∗}(W, L) = 0 and H^{∗}(W, L/p) = 0.

Finally we have to consider W = W (E_{6}) at p = 5. By Theorem 6.2 we have a unique lattice L and L/5 is irreducible. Thus H^{0}(W, L/5) = 0. A construction of a root system is given in [37, p. 43]. Using this description we see that W contains the element

σ = [ \begin{matrix}1 & 1 & 0 & −1 & 0 & 0 \\ 0 & 1 & 1 & −1 & −1 & 1 \\ 1 & 1 & 0 & −1 & −1 & 1 \\ 0 & 2 & 1 & −2 & −1 & 1 \\ 0 & 1 & 1 & −1 & −1 & 0 \\ 0 & 0 & 1 & −1 & 0 & 0 \end{matrix} ]

of order 10. Setting N = 1 + σ + … + σ^{9} it is easily checked using linear algebra that we have Ker(σ − 1) = Im(N) and Ker(N) = Im(σ − 1) as matrices over 𝔽_{5}. Using the standard formula for the cohomology of cyclic groups ([82, 6.2.2]) we get that H^{>0}(< σ >, L/5) = 0. Since |W | = 2^{7} · 3^{4} · 5, the index of < σ > is coprime to 5 and thus we get H^{>0}(W, L/5) = 0 using transfer. Therefore we get H^{∗}(W, L/5) = 0.

Consider now the short exact sequence 0 → L →^{·5} L → L/5 → 0. Since the order of W is 2^{7} · 3^{4} · 5, multiplication by 5 is zero on H^{>0}(W, L) but since also H^{∗}(W, L/5) = 0 it has to be an isomorphism as well. We conclude that H^{∗}(W, L) = 0. □

Concerning the excluded cases in the theorem we are able to compute the low-dimensional cohomology groups. We start by considering the symmetric groups, i.e. the cases (1) and (2). The module ℤ_{p}ˆ with trivial action will be denoted by L_{triv} and the n-dimensional permutation module for Σ_{n} will be denoted L_{perm}.

Theorem 7.5. Let Σ_{n} → GL_{n−1}(ℤ_{p}ˆ ) be any irreducible pseudoreflection group with associated lattice L. Then H^{2}(Σ_{n}, L) = 0 and if n ≥ 4 then H^{3}(Σ_{n}, L) = 0.

Proof: We have a short exact sequence 0 → L → L_{P U(n)} → F → 0, where F is finite with trivial Σ_{n}-action. By Theorem 7.2 we see that H^{1}(Σ_{n}, F) = 0 and H^{2}(Σ_{n}, F) = 0. Thus from the long exact sequence in cohomology we see that it suffices to prove that H^{2}(Σ_{n}, L_{P U(n)}) = 0 and H^{3}(Σ_{n}, L_{P U(n)}) = 0 for n ≥ 4.

To show this consider the short exact sequence 0 → L_{triv} → L_{perm} → L_{P U(n)} → 0 of Σ_{n}- modules. Since L_{perm} = (L_{triv} ↓_{Σ_{n−1}}) ↑^{Σ_{n}} is induced up from the trivial action on ℤ_{p}ˆ of the subgroup Σ_{n−1} we get by Shapiro’s lemma [82, section 6.3] that H^{∗}(Σ_{n}, L_{perm}) ≅ H^{∗}(Σ_{n−1}, ℤ_{p}ˆ ). So by Theorem 7.2 we get H^{2}(Σ_{n}, L_{perm}) = 0 and H^{3}(Σ_{n}, L_{perm}) = 0. From the long exact sequence we thus get H^{2}(Σ_{n}, L_{P U(n)}) = 0. We also get an exact sequence

0 → H^{3}(Σ_{n}, L_{P U(n)}) → H^{4}(Σ_{n}, ℤ_{p}ˆ ) → H^{4}(Σ_{n}, L_{perm}).

It is not hard to see that the map H^{4}(Σ_{n}, ℤ_{p}ˆ ) → H^{4}(Σ_{n}, L_{perm}) →^{≅} H^{4}(Σ_{n−1}, ℤ_{p}ˆ ) is induced by the inclusion Σ_{n−1} ↪ Σ_{n}. The homology and cohomology groups of symmetric groups have been computed by Nakaoka [55]. Using these results we obtain that the map above is an isomorphism for n ≥ 4. Thus we have H^{3}(Σ_{n}, L_{P U(n)}) = 0 in this case. □

In the case of Σ_{3} ≅ D_{6} the proof above yields some information, but in fact we can actually compute all its homology and cohomology groups.

Theorem 7.6. Let p = 3 and W = Σ_{3}. Let L_{triv} be the trivial module and let L_{SU(3)} and L_{P U(3)} be the irreducible lattices corresponding to the two possible irreducible pseudoreflection representations of Σ_{3} in GL_{2}(ℤ_{3}ˆ ). Then W has periodic cohomology with period 4 and we have the following table of cohomology and homology groups where n ∈ ℤ is arbitrary. Here Ĥ denotes Tate homology/cohomology groups, see [11, VI.4] or [82, 6.2.4].

H^{0} H_{0} Ĥ^{4n} ≅ Ĥ_{4n+3} Ĥ^{4n+1} ≅ Ĥ_{4n+2} Ĥ^{4n+2} ≅ Ĥ_{4n+1} Ĥ^{4n+3} ≅ Ĥ_{4n}
L_{triv}  ℤ_{3}ˆ ℤ_{3}ˆ  ℤ/3  0  0  0
L_{triv}/3 ℤ/3 ℤ/3  ℤ/3  0  0  ℤ/3
L_{SU(3)}  0  0  0  ℤ/3  0  0
L_{SU(3)}/3 ℤ/3 0  ℤ/3  ℤ/3  0  0
L_{P U(3)}  0 ℤ/3  0  0  0  ℤ/3
L_{P U(3)}/3 0 ℤ/3  0  0  ℤ/3  ℤ/3

Proof: The fact that W ≅ D_{6} ≅ ℤ/3 ⋊ ℤ/2 has periodic cohomology with period 4 follows from [11, Exercise VI.9.6]. In fact by direct computation using the Lyndon-Hochschild-Serre spectral sequence we have [82, Example 6.8.5]

H_{k}(W, ℤ) = \{ \begin{matrix}ℤ & for k = 0 \\ ℤ/2 & for k ≡ 1 (mod 4) \\ ℤ/6 & for k ≡ 3 (mod 4) \\ 0 & otherwise. \end{matrix}

Thus by using the universal coefficient theorems [82, Theorem 6.1.12, Exercise 6.1.5] we immediately obtain the results for trivial coefficients.

For the lattices L_{SU(3)} and L_{P U(3)} and their mod-3 reductions we have short exact sequences

0 → L_{SU(3)} → L_{perm} → L_{triv} → 0
0 → L_{triv} → L_{perm} → L_{P U(3)} → 0,

and similarly after reduction modulo 3. Since L_{perm} = (L_{triv} ↓_{Σ_{2}}) ↑^{Σ_{3}} is equal to the trivial module of the subgroup Σ_{2} induced up (and similarly modulo 3), we get Ĥ^{∗}(W, L_{perm}) = 0 and Ĥ^{∗}(W, L_{perm}/3) = 0 by Shapiro’s lemma.

By considering the long exact sequences induced from the above short exact sequences we get isomorphisms Ĥ^{n}(W, L_{SU(3)}) ≅ Ĥ^{n−1}(W, L_{triv}) and Ĥ^{n}(W, L_{P U(3)}) ≅ Ĥ^{n+1}(W, L_{triv}) and similarly after reduction modulo 3. Thus the results for Ĥ follow from the calculations for trivial coefficients.

We are left with determining H_{0} and H^{0} for the lattices corresponding to SU(3) and P U(3) and their mod-3 reductions. This is done directly. □

Note that this shows that Theorem 7.2 cannot be improved in general since all the groups H_{3}(Σ_{3}, ℤ_{3}ˆ ), H_{3}(Σ_{3}, ℤ/3^{k}), H^{4}(Σ_{3}, ℤ_{3}ˆ ) and H^{3}(Σ_{3}, ℤ/3^{k}) are isomorphic to ℤ/3.

Theorem 7.7. Let p = 3 and L be the lattice of either E_{6} or its quotient P E_{6} and let W = W (E_{6}) be the Weyl group. Then H^{2}(W, L) = H^{3}(W, L) = 0.

Remark 7.8. The author is grateful to D. Benson for pointing out that this result may be proved directly using results in his paper [4]. The proof below is our original proof though.

Proof: The group W contains the simple group U_{4}(2) ≅ S_{4}(3), see the atlas of finite groups [16, p. 26]. From the information given there we see that W contains a maximal subgroup H which has the structure 3_{+}^{1+2} ⋊ 2Σ_{4}. Here 3_{+}^{1+2} is an extra-special 3-group and 2Σ_{4} denotes the double cover of Σ_{4}. A direct calculation (using computer) reveals that H^{2}(H, L/3) = 0 for both lattices. Since |H| = 2^{4} · 3^{4} and |W | = 2^{7} · 3^{4} · 5, H has index coprime to 3, and thus by using a transfer argument we see that H^{2}(W, L/3) = 0. Consider the short exact sequence 0 → L →^{·3} L → L/3 → 0. The induced long exact cohomology sequence shows that multiplication by 3 is surjective on H^{2}(W, L) and injective on H^{3}(W, L). Since both of these are finite abelian 3-groups they are trivial. □

Combining Theorems 7.3, 7.5, 7.6 and 7.7 we have proved:

Theorem 7.9. Let W ↪ GL_{n}(ℤ_{p}ˆ ) be an irreducible pseudoreflection group with associated lattice L. Then H^{2}(W, L) = 0. We also have H^{3}(W, L) = 0 except for the case W = Σ_{3}, L = L_{P U(3)} and p = 3, where H^{3}(Σ_{3}, L_{P U(3)}) = ℤ/3. □

In the case of general lattices we have the following result:

Theorem 7.10. Let W ⊆ GL_{n}(ℤ_{p}ˆ ) be a finite p-adic pseudoreflection group with associated lattice L. Then H^{2}(W, L) = 0. If L is simply connected then we also have H^{3}(W, L) = 0.

Proof: By Theorem 6.8 we can choose a trivial lattice Z and a simply connected lattice S, such that we have a short exact sequence 0 → Z ⊕ S → L → F → 0, where F is a finite trivial ℤ_{p}ˆ [W]-module. We thus have the exact sequence

H^{1}(W, F) → H^{2}(W, Z) ⊕ H^{2}(W, S) → H^{2}(W, L) → H^{2}(W, F).

By Theorem 7.2 the groups H^{1}(W, F), H^{2}(W, Z) and H^{2}(W, F) vanishes. Thus it suffices to prove that H^{2}(W, S) = H^{3}(W, S) = 0 for simply connected lattices S.

Since S is simply connected we have by Theorem 6.7 a splitting W ≅ W_{1} × … × W_{r} and S ≅ S_{1}⊕…⊕S_{r}, where the W_{i}’s act as irreducible pseudoreflection groups on the simply connected lattice S_{i} and the action on S_{j} is trivial for j ≠ i. Now let 1 ≤ i ≤ r and set W′_{i} = ∏_{j≠i} W_{j}. Then H^{∗}(W, S_{i}) = H^{∗}(W_{i} × W′_{i}, S_{i}) may be computed by using the Lyndon-Hochschild-Serre spectral sequence. The E_{2}-page is given by E_{2}^{s,t} = H^{s}(W_{i}, H^{t}(W′_{i}, S_{i})) which vanishes for t = 1, 2, 3 by Theorem 7.2. Thus we have H^{n}(W, S_{i}) = H^{n}(W_{i}, S_{i}) for n = 1, 2, 3. It thus suffices to show that H^{2}(W_{i}, S_{i}) and H^{3}(W_{i}, S_{i}) vanishes for all i.

We are now reduced to looking at the case where W is irreducible and the associated lattice L is simply connected. In this case we are done by Theorem 7.9. □

The fact that H^{2}(W, L) = 0 has already been shown for Coxeter groups by [39, proof of 3.5], see also [49, 7.2]. Thus we could also prove this part of the theorem by using Proposition 7.1 to embed a Coxeter group of coprime index and then use a transfer argument.

Note finally that the restriction to simply connected lattices cannot be removed since by Theorem 7.6 we have H^{3}(Σ_{3}, L_{P U(3)}) ≅ ℤ/3.

8. p-compact groups

In this section we give a quick review of the theory of p-compact groups and give an overview of the recent resolution of the Steenrod problem (for odd primes). We will be rather brief, so the reader is advised to consult the survey articles [47, 59] where references to the main articles also may be found.

The key idea (introduced by Rector in the early seventies) is to try to understand Lie group theory from the viewpoint of algebraic topology. Instead of directly looking at a compact Lie group G we consider instead its classifying space BG and try to express notions from Lie group theory in terms of that. This approach is justified by the following theorem:

Theorem 8.1. [68, 64, 45, 60] Two compact Lie groups G and H are isomorphic iff their classifying spaces BG and BH are homotopy equivalent. □

This was first proved by Scheerer [68] in the simply connected case and independently by Osse and Møller [64, 45] in the connected case. The general case is due to Notbohm [60].

In fact this approach also allows us to consider a more general type of spaces, namely finite loop spaces. A loop space is a connected pointed space BL. Then L := ΩBL defines a loop structure on L. The loop space is called finite if H^{∗}(L, ℤ) is a finitely generated abelian group and π_{0}(L) is finite. If G is a compact Lie group and BG is its classifying space, then by construction of BG we have a fibration G → EG → BG, where EG is contractible. Thus we get a canonical equivalence ΩBG ≃ G. Moreover H^{∗}(G, ℤ) is finite since G is equivalent to a finite CW-complex. Thus BG is a finite loop space.

Rector gave in his papers [67, 66] definitions of subgroups, maximal tori and Weyl groups of finite loop spaces. In particular he showed that S^{3} admits uncountably many loop structures, but there is a unique one which admits a maximal torus.

The most important advance in this subject was the introduction by Dwyer and Wilkerson [30] of p-compact groups.

Definition 8.2. Let p be a prime and BX a loop space. Define X := ΩBX. Then BX is said to be a p-compact group iff one of the following equivalent conditions holds

(1) X is 𝔽_{p}-finite and BX is p-complete.

(2) X is 𝔽_{p}-finite, X is p-complete and π_{0}(X) is a finite p-group.

(3) X is 𝔽_{p}-finite, π_{0}(X) is a finite p-group and π_{i}(X) (i ≥ 1) is a finitely generated module over ℤ_{p}ˆ .

Often we will just be sloppy and say that X is a p-compact group when we really mean that BX is a p-compact group and X = ΩBX.

If G is a compact Lie group, then (BG)_{p}ˆ is a p-compact group iff π_{0}(G) is a finite p-group. Other examples come from spaces with polynomial cohomology. If BX is a pointed connected p-

complete space such that H^{∗}(BX, 𝔽_{p}) is a finitely generated polynomial algebra then the Eilenberg-Moore spectral sequence shows that H^{∗}(ΩBX, 𝔽_{p}) is an exterior algebra on finitely many generators. In particular BX is a p-compact group. We return to the question of spaces with polynomial mod-p cohomology again below.

One should think of p-compact groups as loop spaces with all their homotopy theoretical information concentrated at the prime p. This suggest using a ‘local-to-global’ approach in the study of finite loop spaces. An example is given by the proof by Dwyer and Wilkerson [30] of Moore’s conjecture that if L is a finite loop space then H^{∗}(BL, 𝔽_{p}) is a finitely generated algebra for all primes p.

The study of p-compact groups is facilitated by the basic philosophy that they behave as compact Lie groups. In fact one of the main results in [30] is the existence of a maximal torus T → X, unique up to conjugacy. We will not give the precise definitions here, so the reader should consult [30, 47, 59]. We may assume that BT → BX is a fibration, since otherwise we can just replace it by an equivalent fibration BT′ → BX. We then define the Weyl space W_{T} (X) as the space of self-maps of BT over BX. This is homotopically discrete, and we define the Weyl group W to be the component group π_{0}(W_{T} (X)). This does not depend on the choice of T. The maximal torus normalizer BN is defined as the Borel construction of the action of W_{T} (X) on BT. Thus up to homotopy we have a fibration BT → BN → BW. It should be noted that BN is generally not a p-compact group, since π_{1}(BN) = W is in general not a finite p-group. The action of W on BT gives rise to a representation W → GL(H^{2}(BT, ℤ_{p}ˆ )) = GL_{n}(ℤ_{p}ˆ ), where n is the rank of T. The connection to pseudoreflection groups mentioned in section 4 is due to the fact that if X is connected then W → GL_{n}(ℤ_{p}ˆ ) is a faithful representation of W as a pseudoreflection group. We see that this representation only depends on the homotopy type of BX. Furthermore, for connected X, the maximal torus induces an isomorphism H^{∗}(BX, ℤ_{p}ˆ ) ⊗ ℚ ≅ (H^{∗}(BT, ℤ_{p}ˆ ) ⊗ ℚ)^{W}. We say that X is simple if W → GL_{n}(ℤ_{p}ˆ ) is an irreducible pseudoreflection group.

There are many other notions from classical Lie group theory which have been translated into the theory of p-compact groups. Here we only mention the notion of central monomorphisms and the existence of a terminal central monomorphism Z(X) → X, called the center of X, [31]. If this is trivial we say that X is centerfree. The center Ť^{W} of the lattice is a discrete approximation to the center [31, 7.5, 7.7]. In particular X is centerfree iff it has a centerfree lattice.

Analogously to compact Lie groups there exist product splittings of p-compact groups [32, 58]:

Theorem 8.3. Let p be an odd prime and X a connected p-compact group. If X is centerfree then there exists a splitting X ≅ X_{1} × … × X_{r} of X into simple centerfree connected p-compact groups. We also have corresponding splittings N ≅ N_{1} ×…×N_{r}, W ≅ W_{1} ×…×W_{r} and L ≅ L_{1} ×…×L_{r} of the maximal torus normalizer N, the Weyl group W and the associated lattice L for X, such that N_{i}, W_{i} and L_{i} are respectively the maximal torus normalizer, Weyl group and associated lattice for X_{i}. The same statement holds if centerfree is replaced by simply connected. For p = 2 similar results also hold, but the statements have to be modified somewhat. □

This ends our very brief review of the main properties of p-compact groups.

We end this section by an overview of the recent developments concerning the Steenrod problem. The following question was posed by Steenrod in 1970 [76] : Which finitely generated polynomial algebras over 𝔽_{p} can occur as the mod-p cohomology of a space ?

If G is a compact connected Lie group, then for almost all primes p the cohomology ring H^{∗}(BG, 𝔽_{p}) is polynomial. For example this happens in all cases where p does not divide the order of the Weyl group of G and in many other cases as well.

At the time of Steenrod’s question these were the only known examples. Later examples were found by Clark and Ewing [14], Quillen [65] (using work of Sullivan [77], see also Stasheff [75]) and Zabrodsky [85]. All of these were rather ad-hoc constructions though. In fact most results in that period were non-realization results based on the fact that the polynomial algebra should be an unstable algebra over the Steenrod algebra.

The major breakthrough was the work of Dwyer, Miller and Wilkerson [28] building on earlier results of Adams and Wilkerson [2]. We saw above that if BX is a connected pointed p-complete space with polynomial mod-p cohomology, then X is a p-compact group. In particular it has a maximal torus T and a Weyl group W. It should be noted that the article [28] precedes the theory of p-compact groups, and most of the work in [28] actually goes into proving the existence of maximal tori and Weyl groups. Moreover Dwyer, Miller and Wilkerson [28] showed that for odd primes, H^{∗}(BX, 𝔽_{p}) ≅ H^{∗}(BT, 𝔽_{p})^{W}, so the mod-p invariants has to be a polynomial ring. In fact, combining this with Notbohm’s results [63, 4.9] and [61, 3.3] we see that the W -lattice L = H^{2}(BT, ℤ_{p}ˆ ) is a polynomial lattice (see section 6).

Thus Steenrod’s question for odd primes is reduced to realizing the polynomial lattices. The cases where p is coprime to |W | were already realized by Clark and Ewing [14], and the crystallographic pseudoreflection groups are realized as Weyl groups of the classical compact Lie groups (note that these are not all polynomial though). The only irreducible cases that were open were a few exceptional cases and the cases from the family 2a. The exceptional cases were handled by Aguadé [3] (two of them were already done in [85]). Some of the cases from the family 2a had already been handled by Quillen [65], but the question was only settled by Notbohm [63] who was able to realize all of the cases in this family based on ideas of Oliver.

Recently, building on this and his work on polynomial lattices [61], Notbohm was able to give uniform realizations of all polynomial lattices and prove uniqueness result for the constructed spaces [62], thus settling the Steenrod problem for odd primes. We will not describe the actual construction here but only state the main results.

Theorem 8.4. Let p be an odd prime, and let A be a polynomial algebra over the Steenrod algebra. Then A is the mod-p cohomology of a topological space iff there exists a pseudoreflection group W ⊆ GL_{n}(ℤ_{p}ˆ ) = GL(L) such that L is a polynomial lattice with A ≅ 𝔽_{p}[L/p]^{W} as algebras over the Steenrod algebra. In fact we can choose the realization to be a p-compact group BX with Weyl group W. □

The hardest part of this theorem is proving the existence of the Weyl group W. As already mentioned this was done in Dwyer, Miller and Wilkerson [28] based on Adams and Wilkerson [2]. Concerning uniqueness questions Notbohm proved the following:

Theorem 8.5. Let p be an odd prime, and let Y and Z be p-complete spaces with polynomial mod-p cohomology. Then Y and Z are homotopy equivalent iff they have the same Weyl group data. This is the case if H^{∗}(Y, 𝔽_{p}) ≅ H^{∗}(Z, 𝔽_{p}) as algebras over the Steenrod algebra and one of the following two conditions is satisfied :

(1) Y is 2-connected

(2) The order of the Weyl group of Y is coprime to p. □

It should be noted that the cohomological uniqueness in the case where p does not divide the order of the Weyl group was already proved by Dwyer, Miller and Wilkerson [28].

At the prime p = 2 not much is known. Going through the Clark-Ewing list we see that there is only one 2-adic pseudoreflection group which is not a crystallographic pseudoreflection group, namely group number 24 which has rank 3. It is isomorphic to ℤ/2 × GL_{3}(ℤ/2). A 2-compact group with this Weyl group was constructed by Dwyer and Wilkerson [29] :

Theorem 8.6. There exists a connected 2-compact group DI(4) such that H^{∗}(BDI(4), 𝔽_{2}) is isomorphic as an algebra over the Steenrod algebra to the rank 4, mod-2 Dickson invariants 𝔽_{2}[V]^{GL_{4}(𝔽_{2})}, where V = (𝔽_{2})^{4}. The Weyl group W is the group G_{24} mentioned above. □

Note that this shows that in the result of Dwyer, Miller and Wilkerson that H^{∗}(BX, 𝔽_{p}) = H^{∗}(BT, 𝔽_{p})^{W}, the restriction to odd primes is necessary. This follows since H^{∗}(BT, 𝔽_{2})^{W} has transcendence degree 3, while H^{∗}(BDI(4), 𝔽_{2}) is polynomial on 4 generators.

To conclude, we see that a classification of p-compact groups must be closely linked with the list of examples coming from Steenrod’s problem. To obtain such a classification however, uniqueness problems have to be addressed. This is the topic of the next section.

9. Normalizers, Self-maps and Determinism

In this section we consider homotopy uniqueness questions for p-compact groups. One approach to this is via the maximal torus normalizer. In the case of a compact connected semisimple Lie group G it was shown by Curtis, Wiederhold and Williams [22] that the normalizer N of the maximal torus T determines G. In the general case of compact connected Lie groups, it was shown by Notbohm [60]. They also studied the question of when the extension 0 → T → N → W → 0 is split. This question had also been studied earlier by Tits [79]. Believing in the philosophy that p-compact groups behave as compact Lie groups, the conjecture is the following:

Conjecture 9.1. [59, 5.20] Two connected p-compact groups X and Y are isomorphic if and only if the normalizers of their maximal tori are isomorphic (as loop spaces). At odd primes the normalizer splits and the Weyl group data are sufficient to distinguish between connected p-compact groups.

A more precise version was given by Lannes [44, 5.2]. Let X be a connected p-compact group with maximal torus T, maximal torus normalizer N and Weyl group W. Let L be the lattice of W ⊆ GL_{n}(ℤ_{p}ˆ ). Then BN is up to homotopy the total space of a fibration with base BW and fibre BT. Thus the homotopy type of BN is given by an element γ ∈ H^{3}(W, π_{2}(BT)) = H^{3}(W, L). Denoting the discrete approximations of T and N (see [30, 31]) by respectively Ť and Ň this obstruction γ may also be identified with the cohomology class corresponding to the group extension 0 → Ť → Ň → W → 1. The precise version of the conjecture is then that up to isomorphism X is determined by the triple (W, L, γ) for p = 2 and for odd primes that γ = 0 and X is determined by (W, L). Here we prove the following part of the conjecture :

Theorem 9.2. If p is odd and X is a connected p-compact group then the obstruction γ = γ(X) vanishes, i.e. the normalizer splits.

The results of Tits and Curtis, Wiederhold and Williams [79, 22] shows that this is true in the case of compact connected Lie groups. The general result has also been claimed by Dwyer and Wilkerson [44] but their proof has not appeared. Independently, Notbohm also has a proof which has not appeared either. Proof: Let Z → X be a central monomorphism. By [53, 4.6] and [48, 3.8] we see that the pcompact group X/Z has maximal torus T_{X/Z} = T/Z, Weyl group W_{X/Z} = W and maximal torus normalizer N_{X/Z} = N/Z. This gives a commutative diagram

[figure: T; N; W; T/Z; N/Z; W]

Thus the extension class γ_{X/Z} is the image of the extension class γ_{X}, so by [53, 5.4] it suffices to prove the theorem for X simply connected.

Assume now that this is the case. By Theorem 8.3 we may also assume that X is simple, i.e. that W is an irreducible pseudoreflection group. In that case Theorem 7.9 shows that H^{3}(W, L) = 0 in all cases, except if p = 3 and X has the same Weyl group data as { \atop P U(3)}. By [10] the 3-compact group { \atop P U(3)} is determined by its Weyl group data, so X ≅ { \atop P U(3)}. Since we are assuming that X is simply connected this is a contradiction. Thus for all simple, simply connected X we have H^{3}(W, L) = 0 and in particular γ = 0. □

We now discuss the uniqueness questions involved in the second part of the conjecture. This is the subject of Møller’s theory of N-determinism [49]. We give a list of results summarizing what is currently known. This is followed by a theorem showing how self-maps can be calculated from Weyl group data. We end by showing that all the spaces considered by Aguadé are totally N-determined and we compute their self-maps.

All through this section p will denote an odd prime. First of all Theorem 9.2 has to be modified for p = 2, even in case of compact connected Lie groups. Moreover the theory of N-determinism is not well developed in case p = 2.

The basic question addressed by the theory of N-determinism is to what extent a p-compact group X and its self-maps are determined by the normalizer N of the maximal torus. Assume in the following that X is connected.

Let Out(X) denote the group of invertible elements in [BX, BX]. Then there is a homomorphism N : Out(X) → Out(N) given by restriction, see [49, 3.2]. If this map is injective we say that X has N-determined automorphisms.

Let X′ be another connected p-compact group such that N → X′ is conjugate to the normalizer of a maximal torus. If all such X′ are isomorphic to X we say that X is N-determined. Note that for odd primes this is by Theorem 9.2 equivalent to the condition that X is determined by its Weyl group data (W, L). Thus the remaining part of the conjecture may be stated as saying that all p-compact groups are N-determined.

If X has both N-determined automorphisms and is N-determined, then we say that X is totally N-determined. In this case the map N : Out(X) → Out(N) is an isomorphism ([49, 7.2]). This is the case if for example p does not divide the order of the Weyl group W ([49, 3.11 and 7.7]). The main results of [49] are the reduction of the questions about the determinacy of X to the computation of certain higher limits over the Quillen category of X. The examples below summarizes the current status of the conjecture that all p-compact groups are totally Ndetermined.

In the case of compact connected Lie groups G, the self-maps of BG were computed by Jackowski, McClure and Oliver [39, 40, 41]. Let Ĝ be G viewed as a p-compact group, i.e. let BĜ = (BG)_{p}ˆ . Their results show that N : Out(Ĝ) → Out(N) is an isomorphism, cf. [49, 5.1]. In particular we see that compact connected Lie groups have N-determined automorphisms. The result also further supports the conjecture that they are also N-determined.

Notbohm’s results on the Steenrod problem stated in the previous section, may be reinterpreted as saying that any connected p-compact group X, such that H^{∗}(BX, 𝔽_{p}) is polynomial, is Ndetermined.

Using the centralizer decomposition of P U(p) ([42]) and his obstruction theory, Møller [52] shows that the p-compact group { \atop P U(p)} is totally N-determined. In fact similar methods shows this may be extended to include { \atop P U(n)} for all n < 2p. Using different arguments, Broto and Viruel [10, 9] shows that this result may even be extended to n < p(p − 1). The author it grateful to A. Viruel for informing us about this work.

Møller also considers the first of the spaces on Aguadés list [51] and shows it is totally Ndetermined. Below we shall show that his arguments easily generalize to cover all the Aguadé spaces and compute their self-maps.

First we consider the computation of self-maps in general. Let Aut(W, Ť) denote the subgroup of Aut(W) × Aut(Ť) consisting of the pairs (χ, φ) such that φ is χ-equivariant, i.e. such that φ(w · t) = χ(w) · φ(t) for all w ∈ W, t ∈ Ť. We have a natural homomorphism α : Aut(Ň) → Aut(W, Ť) since Ť is a characteristic subgroup of Ň. The image equals the subgroup consisting of the pairs (χ, φ) that leave the extension class γ ∈ H^{2}(W, Ť) invariant. But since γ = 0 by Theorem 9.2 we see that α is an epimorphism.

Since α is W -equivariant, we obtain an epimorphism Aut(Ň)/W → Aut(W, Ť)/W. The kernel equals H^{1}(W, Ť) which vanishes by Theorem 7.10. Thus we have an isomorphism Aut(Ň)/W ≅ Aut(W, Ť)/W. The left-hand side is equal to Out(N) and it is easy to see that Aut(W, Ť) = N_{GL(L)}(W). This proves:

Theorem 9.3. Let p be odd and X be a connected p-compact group with maximal torus normalizer N and Weyl group W with associated lattice L. Then there is a natural isomorphism Out(N) ≅ N_{GL(L)}(W)/W. In particular if X is totally N-determined then Out(X) ≅ N_{GL(L)}(W)/W. □

We now very quickly show that the arguments used by Møller [51] to construct and analyze the first of the spaces on Aguadé’s list may be generalized to cover all of them. Let W ⊆ GL_{n}(ℚ_{p}ˆ ) be one of the groups in question. We use the notation from section 5.

Theorem 9.4. There exists a connected p-compact group X of rank n with Weyl group W. It is simple, simply connected, centerfree and totally N-determined. The cohomology is given by H^{∗}(BX, R) ≅ H^{∗}(BT, R)^{W} for R = 𝔽_{p}, ℤ_{p}ˆ and ℚ_{p}ˆ , which is polynomial by Corollary 6.12.

Proof: (Sketch) The arguments are almost identical to the ones used in [51]. Let Σ be as in Remark 5.6. Let 𝕀(W, Σ) be the category introduced in [51] and consider the (𝕀(W, Σ))^{op} diagram:

[figure: BSU(n + 1)_{p}; ˆ (N_{W} (Σ)/Σ)^{op}=Z(W)^{op}; BT; W^{op}; Σ^{op}\W^{op}]

where BT is a p-compact torus of rank n. The action of N_{W} (Σ)/Σ = Z (cf. Theorem 5.5) on (BSU(n + 1))_{p}ˆ is given by suitable Adams maps, which is possible since Out((BSU(n + 1))_{p}ˆ ) = (ℤ_{p}ˆ )^{∗} and |Z| divides p − 1 in all cases.

This may be seen to be a centric diagram in the sense of Dwyer and Kan [26]. Using the information collected in Theorem 5.5 combined with [51, 3.1, 3.2] it is easy to see that the higher inverse limits lim_{𝕀(W,Σ)}^{i} π_{j}(BZ(−)), of the functor π_{j}(BZ(−)) applied to the diagram above, all vanish for i, j > 0. This shows that the diagram may be rigidified, i.e. that there exists a diagram in the category of topological spaces inducing the above diagram in the homotopy category. Let BX be the 𝔽_{p}-completion of the homotopy colimit of any such diagram. Arguing as in [51, 3.9] we see that BX is a simply connected p-compact group with maximal torus T → X and Weyl group W. Then the rank is n, and since W is irreducible, X is by definition simple. Since the lattice is unique it is centerfree. This shows that X is centerfree.

The only thing left to prove is that X is totally N-determined. This is done by first identifying the index category of the diagram used above to construct X with a suitable full subcategory of the Quillen category of X, cf. Remark 5.6. Note that all the p-compact groups in the diagram above are totally N-determined. The fact that X is N-determined then follows from the general theory of N-determinism [49, 4.9, 7.15, 7.17] since we have already shown that the appropriate higher inverse limits vanishes. □

It should be noted that since these p-compact groups have polynomial mod-p cohomology, the fact that they are N-determined also follows from Theorem 8.5. The result that they are totally N-determined allows us to compute the self-maps of these spaces.

Corollary 9.5. Let W ⊆ GL_{n}(ℚ_{p}ˆ ) be one of the groups in question and let X be the p-compact group from Theorem 9.4 having W as Weyl group. Then we have Out(X) = (ℤ_{p}ˆ )^{∗}/Z and [BX, BX] = {0} ∪ (ℤ_{p}ˆ )^{∗}/Z, where Z denotes the center of W.

Proof: Note that the formulas make sense since by Proposition 2.3, the center Z consists of scalar matrices. The first part follows immediately by combining Theorems 5.7, 9.3 and 9.4. The statement about [BX, BX] follows from this, since by [50, 5.5 and 5.6] any essential self-map is an equivalence. □

Note finally that Example 1.8 gives the Steenrod algebra structure on H^{∗}(BDI(2), 𝔽_{3}) where BDI(2) is the 3-compact group from Theorem 9.4 having W (DI(2)) = G_{12} ≅ GL_{2}(𝔽_{3}) as Weyl group.

10. Further directions

We end by describing some of the open problems in the theory.

Let p be an odd prime. As mentioned in section 6 the notions centerfreeness and simply connectedness for lattices are inspired by topological considerations. As already mentioned, if X is a connected p-compact group, then the center Ť^{W} of the lattice L is a discrete approximation to the center of X [31, 7.5, 7.7].

For the notion of simply connectedness, the natural conjecture is the following:

Conjecture 10.1. If X is a connected p-compact group, then we have a natural isomorphism π_{1}(X) ≅ H_{0}(W, L) = L_{W}.

In particular simply connected p-compact groups should have simply connected lattices. Dwyer has recently announced a proof of the conjecture, but no details have appeared yet.

The conjecture, if true, has interesting applications to N-determinism. Møller [49] proves that if Z → X is a central monomorphism then X is N-determined if X/Z is (and also that X has N-determined automorphisms if X/Z has). Thus to prove that all p-compact groups are Ndetermined, it is enough to consider centerfree p-compact groups.

It seems likely that, given the conjecture, it is possible to reduce the question to simply connected p-compact groups instead. Using Theorem 8.3 it suffices to consider simple simply connected pcompact groups X. By Conjecture 10.1 these have simply connected lattices. In case the lattice is polynomial then Notbohm’s result Theorem 8.5 shows that X is N-determined. According to Theorem 6.10 the only cases left are then the compact Lie groups F_{4}, E_{6}, E_{7} and E_{8} for p = 3, and E_{8} for p = 5. It seems possible to handle the cases F_{4} at p = 3 and E_{8} at p = 5 via Møllers obstruction theory [49], but at this stage the question remains open.

For p = 2 the situation is much worse, since there are many more technical problems that have to be handled. There is no classification of lattices and Steenrod’s problem is still open. Also the obstruction theory is as yet inadequately developed. Despite this, the conjecture is that any connected 2-compact group is isomorphic to a product of a number of copies of DI(4) (cf. Theorem 8.6) and a connected compact Lie group (regarded as a 2-compact group) [29].

Finally it should be mentioned that there is another approach to uniqueness questions namely cohomological uniqueness. This seems to be a somewhat harder question than N-determinism, since it requires knowledge of the cohomology of the classifying space of the p-compact group. Nonetheless, some results have been obtained. An example is given by Theorem 8.5, but there are many others [27, 28, 56, 81]. Again the prime p = 2 requires more work.

As mentioned, p-compact groups were introduced to study finite loop spaces. Hopefully a classification of p-compact groups would lead to a proof of the following old conjecture [83]:

Conjecture 10.2. If L is a finite loop space with maximal torus, then BL is homotopy equivalent to BG for some Lie group G.

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