On the cohomology of the sporadic simple group J_{4}

David John Green Inst. f. Exp. Math.

Ellernstr. 29 D–45326 Essen

Germany

david@exp-math.uni-essen.de

A version of this paper appeared in

Math. Proc. Cambridge Philos. Soc. 113 (1993), 253–266. \begin{matrix}c \\ ◯ \end{matrix} Cambridge Philosophical Society, 1993.

Introduction

In this paper we calculate part of the integral cohomology ring of the sporadic simple group J_{4}: this group has order 2^{21}.3^{3}.5.7.11^{3}.23.29.31.37.43. More precisely, we obtain all of the cohomology ring except for the 2-primary part. As the cohomology has already been written down [9] at the primes which divide the group order only once, we concentrate here on the primes 3 and 11. In both of these cases the Sylow p-subgroups are extraspecial of order p^{3} and exponent p. We use the method which identifies the p-primary cohomology with the ring of stable classes in the cohomology of a Sylow psubgroup. The stable classes are all invariant under the action of the Sylow pnormaliser; and some time is spent finding invariant classes in the cohomology ring of p_{+}^{1+2}, the extraspecial group. Section 2 studies the prime 11: the invariant classes are the stable classes, because the Sylow 11-subgroups have the Trivial Intersection (T.I.) property. In Section 3 we study the prime 3, and see that all conditions for invariant classes to be stable reduce to one condition.

In Section 4 we obtain the corresponding parts of the Chern subring of J_{4}. We make much use of generalised characters of the Sylow subgroups that are constant on the conjugacy classes of J_{4}. (Strictly, on the intersections of p_{+}^{1+2} with the conjugacy classes of J_{4}.)

1 Preliminaries

In order to calculate the integral cohomology ring of a finite group G, it is enough to calculate the localisation of this ring at each prime p that divides the order of G. After localising at p, restriction from G to any Sylow psubgroup P is an injection, since it is split up to a unit by corestriction. Theorem 1 characterises the image of this restriction map. Recall that a class x in the cohomology of P is stable if for all g in G the restriction of x to the intersection P ∩ P^{g} is equal to the restriction of g^{∗}x from P^{g} to P ∩ P^{g}, where g^{∗} is induced by conjugation by g.

Theorem 1 ([2] XII 10.1) With the above notation, restriction from G to P identifies H^{∗}(G, Z)_{(p)} with the ring of stable classes in H^{∗}(P, Z)_{(p)}.

If g normalises P, then g acts on the cohomology of P. The class x is invariant if it is fixed under this action of the Sylow p-normaliser. The ring H^{inv} of invariant classes clearly contains the stable classes, and is simpler to calculate. In each case studied here, we start by finding the invariant classes and then determine which of them are stable.

The group J_{4} The group J_{4} has order 2^{21}.3^{3}.5.7.11^{3}.23.29.31.37.43, and is the fourth largest sporadic simple group. It was discovered by Janko [4] and

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proved to exist by Norton, Parker, Benson, Conway and Thackray [8], who constructed a representation in GL_{112}(F_{2}). Many properties of J_{4} are listed in the Atlas [3].

For the primes which divide the order of J_{4} only once, the Sylow psubgroups are cyclic and calculating the cohomology is straightforward:

Theorem 2 [9] Let p be a prime that divides the order of J_{4} exactly once. Then H^{∗}(J_{4}, Z)_{(p)} is generated by α_{p}, with degree as shown below:

\begin{matrix}p & 5 & 7 & 23 & 29 & 31 & 37 & 43 \\ deg α_{p} & 8 & 6 & 44 & 56 & 20 & 24 & 28 \\ χ & χ_{2} & χ_{2} & χ_{2} & χ_{2} & χ_{56} & χ_{53} & χ_{46} \end{matrix} .

The only relation is pα_{p} = 0. The generator α_{p} is a Chern class of a representation of J_{4} which affords the irreducible character χ given above. We have used Atlas notation [3], and chosen characters of lowest degree.

The primes that divide the order more than once are 2, 3 and 11. Since the cohomology of the Sylow 2-subgroup is not yet known, there are two primes, 3 and 11, where the calculations are both interesting and feasible. We now list the properties of J_{4} that we use. We are interested in Sylow p-subgroups, their normalisers, and the ways in which Sylow p-subgroups intersect with each other.

Theorem 3 ([3], [5]) The Sylow 11-subgroups of J_{4} are isomorphic to 11_{+}^{1+2}. Distinct Sylow 11-subgroups have trivial intersection. The centraliser of a Sylow 11-subgroup is its centre, and the normaliser is 11_{+}^{1+2}: (5 × 2.S_{4}).

There are two conjugacy classes of elements of order 11 in J_{4}. The central elements in 11_{+}^{1+2} are in class 11A, and the non-central elements are in class 11B.

The Sylow 3-subgroups of J_{4} are isomorphic to 3_{+}^{1+2}. The centraliser of a Sylow 3-subgroup is cyclic of order six, and the normaliser is (2 × 3_{+}^{1+2}: 8): 2.

Since J_{4} contains M_{12}, which in turn contains 3^{2}: 2S_{4}, there are Sylow 3-subgroups of J_{4} that have different centres and intersect with order 3^{2}. All elements of order 3 in J_{4} are in the conjugacy class 3A.

The extraspecial p-group p_{+}^{1+2} The extraspecial p-group of order p^{3} and exponent p is a Sylow p-subgroup of J_{4} when p is 3 or 11. A presentation is

p_{+}^{1+2} = ⟨A, B, C: A^{p} = B^{p} = C^{p} = [A, C] = [B, C] = 1, [A, B] = C⟩,

where [X, Y] denotes XY X^{−1}Y^{−1}. The centre is cyclic of order p, generated by C.

The automorphisms of p_{+}^{1+2} are those endomorphisms which send A to A^{r′}B^{s′}C^{t′} and B to A^{r}B^{s}C^{t}, with r′s − rs′ non-zero. The inner automorphisms are the automorphisms of the form A ↦ AC^{t′} and B ↦ BC^{t}; so we can describe the outer automorphisms of p_{+}^{1+2} thus:

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Lemma 4 The outer automorphism group of p_{+}^{1+2} is isomorphic to GL_{2}F_{p}: the matrix ({r′ \atop s′} {r \atop s}) with determinant j corresponds to the class of the auto-

morphism which sends A to A^{r′}B^{s′}, sends B to A^{r}B^{s}, and C to C^{j}.

(Note that the map we have given here is not a homomorphism from GL_{2}F_{p} to the automorphism group of p_{+}^{1+2}.)

We use this isomorphism to identify outer automorphisms of p_{+}^{1+2} with elements of GL_{2}F_{p}. This group acts on the cohomology of p_{+}^{1+2}, because inner automorphisms of a group pass to the trivial automorphism in cohomology.

For a Sylow p-subgroup P of a group G, let Out_{p} denote the group of outer automorphisms of P which are induced by inner automorphisms of G; that is, the outer automorphisms that occur as conjugation by some element of the Sylow p-normaliser N_{p}.

We now give the cohomology of p_{+}^{1+2} and a subgroup: for proofs see [7].

Proposition 5 Let p be an odd prime: then H^{∗}(C_{p}^{B} × C_{p}^{C}, Z) is generated by β and γ in degree 2, and χ in degree 3. All three generators have additive order p, and χ squares to zero. The automorphism of C_{p} ×C_{p} which switches the two factors sends β ↔ γ and χ ↦ −χ.

Theorem 6 ([7], [6]) The cohomology ring H^{∗}(p_{+}^{1+2}, Z) is generated by

Generator  α_{1}, α_{2} ν_{1}, ν_{2} θ_{j} for 2 ⩽ j ⩽ p − 2  κ  ζ
Degree  2  3  2j  2p − 2 2p
Additive order p  p  p  p  p^{2}

.

The ν_{i} and the θ_{j} square to zero. The remaining relations (including some redundant ones) are:

\begin{matrix}α_{i}κ = & −α_{i}^{p} & α_{1}ν_{2} = α_{2}ν_{1} & α_{1}α_{2}^{p} = & α_{1}^{p}α_{2} \\ ν_{i}κ = & −α_{i}^{p−1}ν_{i} & κ^{2} = α_{1}^{2p−2} − α_{1}^{p−1}α_{2}^{p−1} + α_{2}^{2p−2} & α_{2}^{p}ν_{1} = & α_{1}^{p}ν_{2} \end{matrix}

α_{i}θ_{j} = ν_{i}θ_{j} = θ_{k}θ_{j} = κθ_{j} = 0  ν_{1}ν_{2} = \begin{cases}λθ_{3} & for some λ ∈ F_{p}^{×}, if p > 3 \\ 3λζ & for some λ = ±1, if p = 3. \end{cases}

Under the map of Lemma 4, the matrix ({a_{11} \atop a_{21}} {a_{12} \atop a_{22}} ) with determinant j fixes κ, sends θ_{k} to j^{k}θ_{k}, sends ζ to j^{p}ζ and sends

α_{i} ↦ a_{i1}α_{1} + a_{i2}α_{2} ν_{i} ↦ j(a_{i1}ν_{1} + a_{i2}ν_{2}).

Under restriction to C_{p}^{B} × C_{p}^{C}, the generators α_{1}, ν_{1} and θ_{j} are sent to zero, and

α_{2} ↦ β  ν_{2} ↦ χ  κ ↦ −β^{p−1}  ζ ↦ γ^{p} − γβ^{p−1}.

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We derive two corollaries which help us to calculate inside this cohomology ring. The first looks at the additive structure, and the second relates the odd- and even-degree classes.

Corollary 7 The ring H^{∗}(p_{+}^{1+2}, Z) has the additive structure of the direct sum of three modules: the free Z-module generated by 1, the free Z/p^{2}-module on the ζ^{ℓ} for ℓ > 0, and the F_{p}-vector space with basis

α_{1}^{i}α_{2}^{j}ζ^{ℓ} for i > 0 and j < p  α_{2}^{j}ζ^{ℓ} for j > 0  κζ^{ℓ} α_{1}^{i}α_{2}^{j}ν_{1}ζ^{ℓ} for j < p  α_{2}^{j}ν_{2}ζ^{ℓ}  θ_{k}ζ^{ℓ}.

Let V_{1} be the submodule spanned by the α_{1}^{i}α_{2}^{j}ζ^{ℓ} and α_{2}^{j}ζ^{ℓ}, let V_{2} be the submodule spanned by the α_{1}^{i}α_{2}^{j}ν_{1}ζ^{ℓ} and α_{2}^{j}ν_{2}ζ^{ℓ}, and V_{3} the submodule spanned by 1, ζ^{ℓ}, θ_{k}ζ^{ℓ} and κζ^{ℓ}. Then H^{∗}(p_{+}^{1+2}, Z) is the direct sum V_{1} ⊕ V_{2} ⊕ V_{3}, and this decomposition is respected by the action of the automorphism group.

Proof: Using the relations in Theorem 6, these classes are seen to span the cohomology ring. To see that they are linearly independent (in the obvious sense), consider the orders of the groups H^{n}(p_{+}^{1+2}) as obtained in [7].

Corollary 8 Let E be the ring of even-degree classes in H^{∗}(p_{+}^{1+2}, Z). So E is V_{1} ⊕ V_{3}, and V_{2} is the set of odd-degree classes. Also, V_{1} is the E-module generated by the α_{i}, and V_{2} is the E-module generated by the ν_{i}. Then the map Φ: V_{1} → V_{2} defined on the generators by α_{i} ↦ ν_{i} is a well-defined E-module isomorphism.

If x and y are classes in V_{1}, then x(Φy) = Φ(xy) = (Φx)y. If ψ is an automorphism of p_{+}^{1+2} that sends C to C^{j}, then ψ^{∗}(Φx) = jΦ(ψ^{∗}x).

Proof: This is simple to check: for example, observe that Φ(α_{1}α_{2}) is welldefined.

2 The 11-local cohomology

In this section we prove:

Theorem 9 The ring H^{∗}(J_{4}, Z)_{(11)} has generators as follows:

Generator  α  β  μ  ν  ϕ_{j} for 2 ⩽ j ⩽ 9 κ  η
Degree  56 148 35 127  220 − 20j  20 220
Additive order 11 11 11 11  11  11 11^{2}

.

The classes μ, ν and ϕ_{j} square to zero, and the other relations are:

\begin{aligned}\begin{matrix}αν = & βμ \\ μν = & 0 \end{matrix} & \begin{matrix}α^{3} = & −βκ \\ α^{2}μ = & −νκ \end{matrix} & α^{2}β = κ^{2}η & \begin{matrix}β^{2} = & −ακη \\ βν = & −μκη \end{matrix} \\ & αϕ_{j} = βϕ_{j} = κϕ_{j} = μϕ_{j} = νϕ_{j} = ϕ_{i}ϕ_{j} = 0 . \end{aligned}

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Since J_{4} is T.I. at 11, all the invariant classes are stable. Hence the 11-local cohomology of J_{4} is the ring of invariant classes in the cohomology of 11_{+}^{1+2}; that is, the ring of classes fixed by the subgroup Out_{11} of GL_{2}F_{11}. From Theorem 3, we see that Out_{11} is isomorphic to 5 × 2.S_{4}. There is only one conjugacy class of such sugroups in GL_{2}F_{11}, because GL_{2}F_{11} is 5×2.L_{2}(11).2, and from the Atlas we see that L_{2}(11).2 contains only one conjugacy class of subgroups isomorphic to S_{4}. So, choosing new generators for 11_{+}^{1+2} if necessary, any subgroup of GL_{2}F_{11} which is isomorphic to 5 × 2.S_{4} will do. Let A = ({−1 \atop 0} {0 \atop −1}), B = ({0 \atop −1} {1 \atop 0}), C = ({−1 \atop −3} {−3 \atop 1}), D = ({3 \atop 4} {5 \atop −4}), E = ({4 \atop −3} {5 \atop −4}) and

F = ({3 \atop 0} {0 \atop 3}). Then the subgroup of GL_{2}F_{11} generated by these six matrices is isomorphic to 5 × 2.S_{4}: for F has order 5 and is central, A is central, and the other relations are:

\begin{matrix}A^{2} = D^{3} = E^{2} = I \\ B^{2} = C^{2} = A \end{matrix}  \begin{aligned}& & CB = & ABC \\ & DB = & BCD & DC = & BD \\ EB = & ACE & EC = & ABE & ED = & D^{−1}E . \end{aligned}

This corresponds to taking generators (1 2)(3 4), (1 3)(2 4), (2 3 4) and (2 3) for S_{4}.

Let H be EF. We will use the normal series

1 ◁ ⟨B⟩ ◁ ⟨B, C⟩ ◁ ⟨B, D⟩ ◁ ⟨B, D, H⟩ = Out_{11}

in order to calculate the invariant classes. In the following results we obtain the ring of classes fixed by each successive term in the normal series. Since the series is normal, each group in the series acts on the fixed ring of the previous group. Theorem 9 is a trivial corollary of Proposition 13 below. We begin by concentrating on the ring generated by α_{1} and α_{2}.

Proposition 10 Let S be F_{11}[α_{1}, α_{2}]/(α_{1}α_{2}^{11} − α_{1}^{11}α_{2}). Then the subring of elements which are fixed by B, C and D is generated by

E = α_{1}^{6} − 2α_{1}^{5}α_{2} − 5α_{1}^{4}α_{2}^{2} − 5α_{1}^{2}α_{2}^{4} + 2α_{1}α_{2}^{5} + α_{2}^{6}
G = α_{1}^{8} + α_{1}^{7}α_{2} − 4α_{1}^{6}α_{2}^{2} + 4α_{1}^{5}α_{2}^{3} − 4α_{1}^{3}α_{2}^{5} − 4α_{1}^{2}α_{2}^{6} − α_{1}α_{2}^{7} + α_{2}^{8},

and is F_{11}[E, G]/(E^{4} − G^{3}).

In the following lemma we start to prove this proposition by finding the subring fixed by B. The proof of this lemma is typical of the invariant calculations to come, and we give the proof in some detail.

Lemma 11 In Proposition 10, the subring S^{1} of elements which are fixed by B is generated by

A = α_{1}^{2} + α_{2}^{2} B = α_{1}^{4} + 5α_{1}^{2}α_{2}^{2} + α_{2}^{4} C = 4(α_{1}α_{2}^{3} − α_{1}^{3}α_{2}),

and is F_{11}[A, B, C]/(3B^{2}C − C^{3}, A^{4} − B^{2} − C^{2}).

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Proof: Let V be a two-dimensional F_{11}-vector space, and let α_{1} and α_{2} be a basis for its dual V^{∗}. Let Σ be the symmetric algebra F_{11}[α_{1}, α_{2}] of V^{∗}. Then S is the quotient Σ/(α_{1}α_{2}^{11} − α_{1}^{11}α_{2}). The natural action of GL(V) on V^{∗} induces an action on Σ. This induces in turn an action on S, because matrices act on α_{1}α_{2}^{11} − α_{1}^{11}α_{2} as multiplication by their determinant. This is the action described in Theorem 6.

We want the elements in S that are fixed by B = ({0 \atop −1} {1 \atop 0}), which sends α_{1} to α_{2} and α_{2} to −α_{1}. We start by choosing new generators for S which are eigenvectors for B. The characteristic polynomial of B is X^{2} + 1, and since F_{11} does not contain a square root of −1, we extend the field to F_{121}. Let i be a square root of −1 in F_{121}, and denote by V̂^{∗}, Σ̂ and Ŝ the obvious extensions. Setting β_{1} = α_{1} − iα_{2} and β_{2} = α_{1} + iα_{2}, we have a new basis for V̂^{∗} with Bβ_{1} = iβ_{1} and Bβ_{2} = −iβ_{2}. Then Σ̂ = F_{121}[β_{1}, β_{2}], and every monomial β_{1}^{n}β_{2}^{m} is an eigenvector with eigenvalue i^{n−m}.

The fixed elements in Σ̂ are spanned by the β_{1}^{n}β_{2}^{m} for which n ≡ m mod 4, and are therefore generated by γ_{1} = β_{1}^{4}, γ_{2} = β_{1}β_{2} and γ_{3} = β_{2}^{4}. The relation γ_{2}^{4} = γ_{1}γ_{3} is clear from the definitions. Since the β_{i} are algebraically independent in Σ̂, this is the only relation. Now turn to Ŝ: certainly (the images of) the β_{i} generate Ŝ, and the fixed subring Ŝ^{1} contains the γ_{i}. The γ_{i} generate Ŝ^{1} too: for β_{1}^{12} − β_{2}^{12} = 2i(α_{1}α_{2}^{11} − α_{1}^{11}α_{2}), and so the quotient map from Σ̂ to Ŝ is a bijection when restricted to the span of those β_{1}^{n}β_{2}^{m} with m < 12.

So Ŝ^{1} is generated by γ_{1}, γ_{2} and γ_{3}, which satisfy the relations γ_{1}^{3} = γ_{3}^{3} and γ_{2}^{4} = γ_{1}γ_{3}. We show that these two relations are sufficient. Let J be the ideal in Σ̂ consisting of the relations in Ŝ: then J is generated by γ_{1}^{3} − γ_{3}^{3}. This generator lies in Σ̂^{1}, the subring of fixed elements in Σ̂. To show that our relations are sufficient we have to show that γ_{1}^{3} − γ_{3}^{3} generates the ideal Σ̂^{1} ∩ J in Σ̂^{1}. This means that for every x in Σ̂ such that (γ_{1}^{3} − γ_{3}^{3})x is fixed, we must prove that there is a fixed y such that (γ_{1}^{3} − γ_{3}^{3})x = (γ_{1}^{3} − γ_{3}^{3})y.

To prove this, let T be the subspace of Σ̂ with basis those monomials β_{1}^{n}β_{2}^{m} such that n ≢ m mod 4. Then T is additively a complement of Σ̂^{1} in Σ̂, and if x ∈ T is such that (β_{1}^{12} − β_{2}^{12})x ∈ Σ̂^{1}, then (β_{1}^{12} − β_{2}^{12})x = 0.

The intersection of S and Ŝ^{1} is the fixed points of Ŝ^{1} under the action of the Galois group of F_{121} over F_{11}. The non-trivial automorphism switches β_{1} and β_{2}, fixes γ_{2} and switches γ_{1} and γ_{3}. So if we define A = γ_{2}, B = \frac{1}{2}(γ_{1} + γ_{3}) and C = \frac{1}{2i}(γ_{1} − γ_{3}) then A, B and C all lie in S^{1}. Since Ŝ^{1} is generated over F_{121} by A, B and C subject to the relations C^{3} = 3B^{2}C and A^{4} = B^{2} + C^{2}, it follows that S^{1} is generated over F_{11} with the same generators and relations.

We now continue with the proof of Proposition 10; having worked through one step in detail, we now only give the highlights.

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Proof of Proposition 10: The matrix C = ({−1 \atop −3} {−3 \atop 1}) sends α_{1} to −α_{1} − 3α_{2} and α_{2} to −3α_{1} + α_{2}, and so sends A to −A, sends B to −5B − 3C, and C to −3B + 5C. The eigenvectors are β_{1} = B − 2C, β_{2} = B − 5C and β_{3} = A, with eigenvalues 1, −1 and −1 respectively. Then S^{1} is the quotient of F_{11}[β_{1}, β_{2}, β_{3}] by the relations β_{2}^{2} = −3β_{1}^{2} + 4β_{3}^{4} and β_{1}β_{2}^{2} = β_{1}^{3}. The subring S^{2} of elements in S^{1} fixed by C is therefore spanned by the β_{1}^{n}β_{2}^{m}β_{3}^{ℓ} for m ≡ ℓ mod 2, and generated by γ_{1} = β_{1}, γ_{2} = β_{2}^{2}, γ_{3} = β_{2}β_{3} and γ_{4} = β_{3}^{2}. The relation γ_{3}^{2} = γ_{2}γ_{4} follows from the definition, and the relations in S^{1} become γ_{2} = −3γ_{1}^{2} + 4γ_{4}^{2} and γ_{1}γ_{2} = γ_{1}^{3}. These relations are sufficient: consider the complementary subspace spanned by the β_{1}^{n}β_{2}^{m}β_{3}^{ℓ} with m ≢ ℓ mod 2.

Generator γ_{2} is redundant, so we eliminate it and put D = γ_{1}, put E = γ_{3} and F = γ_{4} to get S^{2} isomorphic to the quotient of F_{11}[D, E, F] by the relations E^{2} = 4F^{3} − 3D^{2}F and D^{3} = DF^{2}. In terms of α_{1} and α_{2}, we have F = α_{1}^{4} + 2α_{1}^{2}α_{2}^{2} + α_{2}^{4} and

D = α_{1}^{4} − 3α_{1}^{3}α_{2} + 5α_{1}^{2}α_{2}^{2} + 3α_{1}α_{2}^{3} + α_{2}^{4}
E = α_{1}^{6} − 2α_{1}^{5}α_{2} − 5α_{1}^{4}α_{2}^{2} − 5α_{1}^{2}α_{2}^{4} + 2α_{1}α_{2}^{5} + α_{2}^{6}.

We now look for the elements also fixed by D = ({3 \atop 4} {5 \atop −4}): this sends α_{1} to 3α_{1} + 5α_{2} and α_{2} to 4α_{1} − 4α_{2}. So D sends D to 5D − 5F, fixes E and sends F to 4D + 5F. The characteristic polynomial X^{2} + X + 1 splits over F_{121} with roots 5 ± 3i. Eigenvectors are β_{1} = E, β_{2} = D − 2iF and β_{3} = D + 2iF, with eigenvalues 1, 5 + 3i and 5 − 3i respectively. Then Ŝ^{2} is the quotient of F_{121}[β_{1}, β_{2}, β_{3}] by the relations β_{3}^{3} = −β_{2}^{3} and β_{2}^{3} − β_{3}^{3} = 4iβ_{1}^{2}. The fixed elements are spanned by the β_{1}^{n}β_{2}^{m}β_{3}^{ℓ} with m ≡ ℓ mod 3 and so generated by γ_{1} = β_{1}, γ_{2} = β_{2}^{3}, γ_{3} = β_{2}β_{3} and γ_{4} = β_{3}^{3}. The relation γ_{3}^{3} = γ_{2}γ_{4} is immediate from these definitions, and the relations in Ŝ^{2} become γ_{4} = −γ_{2} and γ_{2} − γ_{4} = 4iγ_{1}^{2}. Sufficiency: consider the subspace spanned by the β_{1}^{n}β_{2}^{m}β_{3}^{ℓ} with m ≢ ℓ.

We eliminate the redundant generators γ_{2} and γ_{4}. Set G = −2γ_{3} and recall that γ_{1} = E. These are both defined over F_{11}, and so the result is proved.

We now use the result of Proposition 10, together with the decomposition of Corollary 7 and the isomorphism Φ of Corollary 8 to find the classes in the cohomology of 11_{+}^{1+2} which are fixed by B, C and D.

Proposition 12 The subring R^{3} of H^{∗}(11_{+}^{1+2}, Z) of classes which are fixed by the actions of B, C and D is generated by E, G, e = ΦE, g = ΦG, κ, ζ and the θ_{j}. All generators have additive order 11, apart from ζ, which has 11^{2}. The classes e, g and θ_{j} square to zero. The other relations are:

\begin{aligned}\begin{matrix}Eg = & eG \\ eg = & 0 \end{matrix} & \begin{matrix}E^{3} = & −Gκ \\ E^{2}e = & −gκ \end{matrix} & E^{2}G = κ^{2} & \begin{matrix}G^{2} = & −Eκ \\ Gg = & −eκ \end{matrix} \\ & Eθ_{j} = Gθ_{j} = κθ_{j} = eθ_{j} = gθ_{j} = θ_{k}θ_{j} = 0. \end{aligned}

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Proof: The matrices B, C and D all have determinant 1, and so fix κ, ζ and the θ_{j}. Hence V_{3} is contained in R^{3}. Since V_{1} is graded by powers of ζ, Proposition 10 implies that the fixed classes in V_{1} are the polynomials in E, G and ζ. Since B, C and D all commute with Φ, the fixed classes in V_{2} are precisely the images under Φ of the fixed classes in V_{1}. Hence R^{3} is generated by E, G, e, g, κ, ζ and the θ_{j}.

Relations: From Proposition 10 we get E^{4} = G^{3}, and it follows from Corollary 8 that Eg = Ge and E^{3}e = G^{2}g. Since α_{i}ν_{1}ν_{2} = 0, we have eg = 0. We know that κ^{2}, Eκ and Gκ must be polynomials in E and G: comparing degrees, we see that they are scalar multiples of E^{2}G, G^{2} and E^{3} respectively. Comparing coefficients of highest powers of α_{1} we have κ^{2} = E^{2}G, Eκ = −G^{2} and Gκ = −E^{3}. Applying Φ we obtain eκ = −Gg and gκ = −E^{2}e. The relations E^{4} = G^{3} and E^{3}e = G^{2}g are redundant, because they follow from the two ways of evaluating EGκ, Geκ respectively.

Sufficiency: Any polynomial in the generators can be decomposed into a sum of classes in V_{1}, V_{2} and V_{3}, and so we must show that we have all relations inside these three submodules. All elements in V_{3} are fixed, so we are safe there. Since degree of ζ grades V_{1}, all the relations there are accounted for by E^{4} = G^{3} and 11^{2}ζ = 11E = 11G = 0. Since Φ is an isomorphism from V_{1} to V_{2}, extra relations inside V_{2} would imply extra relations inside V_{1}.

Proposition 13 The ring H^{inv} of classes in H^{∗}(11_{+}^{1+2}, Z) which are fixed by B, D and H is generated by κ, η = ζ^{10}, ϕ_{i} = θ_{i}ζ^{10−i}, and

α = (α_{1}^{6} − 2α_{1}^{5}α_{2} − 5α_{1}^{4}α_{2}^{2} − 5α_{1}^{2}α_{2}^{4} + 2α_{1}α_{2}^{5} + α_{2}^{6}) ζ^{2}
μ = ((α_{1}^{5} − 2α_{1}^{4}α_{2} − 5α_{1}^{3}α_{2}^{2} − 5α_{1}α_{2}^{4} + 2α_{2}^{5})ν_{1} + α_{2}^{5}ν_{2}) ζ
β = (α_{1}^{8} + α_{1}^{7}α_{2} − 4α_{1}^{6}α_{2}^{2} + 4α_{1}^{5}α_{2}^{3} − 4α_{1}^{3}α_{2}^{5} − 4α_{1}^{2}α_{2}^{6} − α_{1}α_{2}^{7} + α_{2}^{8}) ζ^{6}
ν = ((α_{1}^{7} + α_{1}^{6}α_{2} − 4α_{1}^{5}α_{2}^{2} + 4α_{1}^{4}α_{2}^{3} − 4α_{1}^{2}α_{2}^{5} − 4α_{1}α_{2}^{6} − α_{2}^{7})ν_{1} + α_{2}^{7}ν_{2}) ζ^{5}.

All generators have additive order 11, apart from η, which has 11^{2}; the generators μ, ν and ϕ_{j} square to zero; the other relations are:

\begin{aligned}\begin{matrix}αν = & βμ \\ μν = & 0 \end{matrix} & \begin{matrix}α^{3} = & −βκ \\ α^{2}μ = & −νκ \end{matrix} & α^{2}β = κ^{2}η & \begin{matrix}β^{2} = & −ακη \\ βν = & −μκη \end{matrix} \\ & αϕ_{i} = βϕ_{i} = κϕ_{i} = μϕ_{i} = νϕ_{i} = ϕ_{j}ϕ_{i} = 0. \end{aligned}

Proof: We need the classes in R^{3} which are fixed by H. Recall that H is EF = ({1 \atop 2} {4 \atop −1}). This has determinant 2, which has multiplicative order 10 in Z/11 and 110 in Z/121. So κ is fixed, θ_{j} is sent to 2^{j}θ_{j}, and ζ to 2^{11}ζ. For degree reasons, E and G must be eigenvectors; comparing coefficients of highest powers of α_{1}, we get E ↦ 2^{−2}E and G ↦ 2^{4}E. Hence e ↦ 2^{−1}e and g ↦ 2^{5}g. Therefore all generators of R^{3} are eigenvectors.

The fixed classes in V_{3} are spanned by 1, ζ^{10ℓ}, κζ^{10ℓ} and θ_{j}ζ^{10ℓ−j}. Hence they are generated by κ, η = ζ^{10} and ϕ_{j} = θ_{j}ζ^{10−j}. From Proposition 12,

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the intersection V_{1} ∩ R^{3} is the module on E, G, EG, E^{2} and κ^{2} over the ring generated by κ and ζ. Hence the invariant classes in V_{1} are the module on Eζ^{2}, Gζ^{6}, EGζ^{8}, E^{2}ζ^{4} and κ^{2} over the ring generated by κ and ζ^{10}. Therefore we need two extra generators α = Eζ^{2} and β = Gζ^{6} to account for V_{1}. Turning to V_{3}, we have HΦ(x) = 2Φ(Hx), and so HΦ(x) = Φ(H(xζ)). Hence Φx is fixed if and only if xζ is, and so the invariant classes in V_{2} are the module on eζ, gζ^{5}, Geζ^{7}, Eeζ^{3} and Φ(κ^{2})ζ^{9} = EGeζ^{9} over the ring generated by κ and ζ^{10}. We set μ = eζ and ν = gζ^{5}, and this completes our set of generators: note that Φ(κ^{2})ζ^{9} = αβμ. The relations are clearly true and sufficient, by comparison with the relations in Proposition 12.

This completes the proof of Theorem 9. We round off this section with a result that expresses the 11-primary cohomology of J_{4} as a simply-defined subring of a ring with three generators.

Theorem 14 Let R be a commutative graded Z_{(11)}-algebra, generated by A and Z in degree 2 and M in degree 3, subject to the relations M^{2} = 0 and 11^{2}Z = 11A = 11M = 0. Then there is a monomorphism which embeds H^{∗}(J_{4}, Z)_{(11)} in R as follows:

\begin{matrix}α ↦ & A^{6}Z^{22} \\ μ ↦ & A^{5}MZ^{11} \end{matrix}  \begin{matrix}β ↦ & A^{8}Z^{66} \\ ν ↦ & A^{7}MZ^{55} \end{matrix}  ϕ_{j} ↦ 11Z^{110−10j}  \begin{matrix}κ ↦ & −A^{10} \\ η ↦ & Z^{110} & . \end{matrix}

Proof: This map is easily seen to be well-defined, so we only have to show that it is injective. The ring H^{∗}(J_{4}, Z)_{(11)} has the additive structure of the direct sum of three modules: the free Z_{(11)}-module generated by 1; the free Z/11^{2}-module on the η^{ℓ} for ℓ > 0; and the F_{11}-vector space with basis as follows:

\begin{matrix}κ^{k}η^{ℓ} & for k > 0 & ακ^{k}η^{ℓ} & βκ^{k}η^{ℓ} & αβκ^{k}η^{ℓ} & α^{2}κ^{k}η^{ℓ} \\ αβμκ^{k}η^{ℓ} & & μκ^{k}η^{ℓ} & νκ^{k}η^{ℓ} & βμκ^{k}η^{ℓ} & αμκ^{k}η^{ℓ} \end{matrix}  ϕ_{j}η^{ℓ}.

It is straightforward to check that these generators all map to different monomials of the correct additive order, and the result is proved.

Notice that R is isomorphic to the Z_{(11)}-cohomology ring of C_{11} × C_{112}. It should also be noted that there does not exist such a nice embedding for H^{∗}(p_{+}^{1+2}, Z)_{(p)}.

3 The 3-local cohomology

We now apply the same methods to the 3-primary cohomology of J_{4}. Calculating the invariant classes is much more straightforward than in the p = 11 case, but this time not all invariant classes are stable. We prove:

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Theorem 15 The ring H^{∗}(J_{4}, Z)_{(3)} is generated by α in degree 16, ξ in degree 12 and ϖ in degree 27. The only relations are 3^{2}ξ = 3α = 3ϖ = 0 and ϖ^{2} = 0.

By Theorem 1, the 3-local cohomology is the subring of stable classes in the cohomology of P = 3_{+}^{1+2}. Since all stable classes are invariant, we start by calculating H^{inv}, the ring of classes fixed by Out_{3}.

From Theorem 3, the order of Out_{3} is 2^{4}, and it is therefore a Sylow 2-subgroup of GL_{2}F_{3}. As Sylow subgroups are unique up to conjugacy, any one will do. Write J = ({1 \atop −1} {−1 \atop −1}), K = ({0 \atop −1} {1 \atop 0}) and L = ({0 \atop 1} {1 \atop −1}). Then J = L^{2}, and the subgroup generated by K and L has order 2^{4}: for K has order 4, L has order 8, L^{4} = K^{2}, and LK = KL^{3}. Hence Out_{3} is generated by K and L (up to choice of generators for P), and a normal series for Out_{3} is

1 ◁ ⟨J ⟩ ◁ ⟨J , K⟩ ◁ ⟨K, L⟩ = Out_{3}.

We now obtain the ring of classes fixed by each successive term in this normal series in a sequence of lemmas leading up to Proposition 17, where H^{∗}(3_{+}^{1+2}, Z)^{inv} is obtained. After this we address the problem of finding the stable classes.

Recall from Theorem 6 that the integral cohomology of 3_{+}^{1+2} is generated by α_{1}, α_{2}, ν_{1}, ν_{2}, κ and ζ.

Lemma 16 The ring R^{2} of classes which are fixed by J and K is generated by κ, ζ and c = α_{1}^{3}ν_{1} − α_{1}α_{2}^{2}ν_{1} + α_{2}^{3}ν_{2}. The only relations are 3^{2}ζ = 3κ = 3c = 0 and c^{2} = 0.

Proof: Since J and K have determinant 1, they fix κ and ζ, and commute with the map Φ of Corollary 8. Let T be the ring generated by κ and ζ, and S the ring generated by α_{1} and α_{2}. Every class in T is fixed, and we show below that the fixed classes in S are generated by κ^{2} and κ^{3}. Hence, in the decomposition of Corollary 7, the intersection R^{2} ∩ V_{1} is the T -module on κ^{2}. Every class in V_{3} is fixed and by Corollary 8, R^{2} ∩ V_{2} is the T -module on c = Φκ^{2}: so we have a generating set. The relations follow immediately, and allow us to decompose any polynomial in the generators into a sum of classes in V_{1}, V_{2} and V_{3}. We certainly have all relations in V_{1} and V_{3}, therefore also in V_{2} by applying the isomorphism Φ: so the result is established.

It remains to prove that, in the ring S generated by α_{1} and α_{2}, the fixed classes are generated by κ^{2} and κ^{3}. We use the method of Lemma 11. Effectively, S is F_{3}[α_{1}, α_{2}]/(α_{1}α_{2}^{3} − α_{1}^{3}α_{2}). Eigenvectors for J over F_{9} are β_{1} = α_{1} + α_{2} − iα_{2} and β_{2} = α_{1} + α_{2} + iα_{2} with eigenvalues i and −i respectively. The ring S^{1} of classes in S fixed by J is generated by A = α_{1}^{2} − α_{1}α_{2} − α_{2}^{2} and B = α_{1}^{4} − α_{1}^{3}α_{2} − α_{2}^{4} with the relation B^{2} = A^{4}. Then Aκ = −B and κ^{2} = A^{2}. Hence S^{1} has a basis consisting of 1, Aκ^{k} and κ^{ℓ}, with ℓ > 1. As K fixes κ and sends A to −A, the classes in S fixed by J and K are spanned by 1 and κ^{ℓ} for ℓ > 1, as claimed.

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Proposition 17 The ring H^{∗}(3_{+}^{1+2}, Z)^{inv} is generated by κ, η = ζ^{2} and d = (α_{1}^{3}ν_{1} − α_{1}α_{2}^{2}ν_{1} + α_{2}^{3}ν_{2})ζ. The relations are 3^{2}η = 3κ = 3d = 0 and d^{2} = 0.

Proof: κ is fixed as usual. L has determinant −1, and so ζ ↦ −ζ. We also have c ↦ −c, because c = Φκ^{2} and LΦ = −ΦL. The fixed classes are therefore generated by κ, ζ^{2} and cζ, and the relations are obvious.

We must now determine which of the invariant classes are stable. As P has order 3^{3}, distinct conjugates of P can intersect with order 3^{2}, 3 or 1.

Proposition 18 Let G be a finite group with Sylow p-subgroup P ≅ p_{+}^{1+2}, and suppose that x ∈ H^{∗}(P, Z) satisfies the stability condition for all g such that the intersection P ∩ P^{g} has order at least p^{2}: then x is stable.

Proof: Trivial if the intersection has order 1. If order p, recall that x satisfies the stability condition for g when a certain pair of cohomology classes of P ∩ P^{g} are equal. This certainly implies equality after corestriction to P. Conversely, if x is “stable after corestriction” for all g ∈ G, then x is stable: inspect the proof in [2] of our Theorem 1. So x is stable if it is “stable after corestriction” for the g such that P ∩P^{g} is cyclic of order p. But corestriction from C_{p} to p_{+}^{1+2} factors through corestriction from C_{p} to C_{p}×C_{p}, which is the zero map (away from degree zero, where all stability conditions are trivial). It is the zero map because restriction from C_{p} × C_{p} to C_{p} is surjective (split by inflation), restriction followed by corestriction is multiplication by the index p, and cohomology classes of C_{p} × C_{p} in positive degree have additive order p.

Before studying intersections of order 3^{2}, we prove an important lemma which means that choosing new generators for P has no effect on the invariant classes.

Lemma 19 Let ψ be an automorphism of 3_{+}^{1+2}. The induced cohomology automorphism ψ^{∗} fixes every class in H^{∗}(3_{+}^{1+2}, Z)^{inv}.

Proof: From Proposition 17, the ring H^{inv} is generated by κ, ζ^{2} and Φ(κ^{2}ζ). Define j by ψC = C^{j}. The automorphism ψ^{∗} fixes κ, multiplies ζ^{2} by j^{6} and multiplies Φ(κ^{2}ζ) by j^{2}. Since j ≡ ±1 mod 3, we have j^{2} ≡ 1 mod 3 and j^{6} ≡ 1 mod 9. Hence ζ^{2} and Φ(κ^{2}ζ) are also fixed.

All subgroups of P ≅ 3_{+}^{1+2} of order 3^{2} have the structure C_{3} × C_{3}, and contain the centre. They are all normal, but are permuted transitively by the automorphism group. Recall that P has generators A, B and C, with C central.

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Lemma 20 Let g ∈ J_{4} be such that P and P^{g} intersect with order 3^{2}, and let x be a class in H^{∗}(P, Z)^{inv}. If P and P^{g} have the same centre then x is stable with respect to g. If they have different centres then x is stable with respect to g if and only if Res_{⟨B,C⟩}^{P} x is fixed by the automorphism f of ⟨B, C⟩ which sends B ↔ C.

Proof: Same centre: Since Lemma 19 allows us to choose new generators for 3_{+}^{1+2} without affecting the invariant classes, we may assume that P ∩P^{g} is the subgroup ⟨B, C⟩. Then gBg^{−1} is in P and gCg^{−1} is a power of C. Let ψ be an automorphism of P that sends B to gBg^{−1} and C to gCg^{−1}. Therefore including ⟨B, C⟩ in P^{g} and then conjugating by g is the same map to P as including in P and then applying ψ. In cohomology, this is the stability condition twisted by ψ^{∗}. Since ψ^{∗} fixes invariant classes, the result follows. Different centres: Again, we may assume that P ∩ P^{g} is ⟨B, C⟩, with gBg^{−1} now a power of C and gCg^{−1} an element of P. For ψ, pick an automorphism of P which sends B to gCg^{−1} and C to gBg^{−1}. Inclusion in P^{g} followed by conjugation is now the same map as f followed by inclusion in P followed by ψ, and the result follows as before.

The importance of Lemma 19 here is that it allows us to choose automorphisms of 3_{+}^{1+2} without having to consider whether they are induced by inner automorphisms of J_{4}. We now finish the proof of Theorem 15.

Proposition 21 The stable classes in H^{∗}(3_{+}^{1+2}, Z) are generated by α = κζ^{2}, ξ = ζ^{2} −κ^{3} and ϖ = (α_{1}^{3}ν_{1} −α_{1}α_{2}^{2}ν_{1} +α_{2}^{3}ν_{2})ζ^{3}. The relations are 3^{2}ξ = 3α = 3ϖ = 0 and ϖ^{2} = 0.

Proof: We have seen that the only g ∈ J_{4} that impose non-trivial conditions for invariant classes to be stable are the g for which P and P^{g} have different centres and intersect with order 3^{2}. Such g do exist, because J_{4} contains the Mathieu group M_{12}, which in turn contains 3^{2}: 2S_{4}. So an invariant class is stable when its restriction to ⟨B, C⟩ is fixed by the map f^{∗}.

From Theorem 6, restriction to ⟨B, C⟩ sends κ to −β^{2}, sends η to (γ^{3} − β^{2}γ)^{2} and d to β^{3}(γ^{3} − β^{2}γ)χ. From Proposition 5 we see that f^{∗} sends β ↔ γ and χ to −χ. Hence the classes κη, η − κ^{3}, and dη are stable: we claim that they generate all the stable classes.

Consider first an even-dimensional invariant class π whose restriction to ⟨B, C⟩ is fixed by f^{∗}; then π is a polynomial in κ and η. We prove that π is a polynomial in κη and η − κ^{3} by induction on degree. Subtracting powers of η − κ^{3} if necessary, we assume π = κηπ′ + ℓκ^{j} for some ℓ ∈ F_{3}. Then the restriction of κηπ′ is divisible by β^{2}γ^{2}, whereas Res_{⟨B,C⟩}^{P} ℓκ^{j} = (−1)^{j}ℓβ^{2j}. Hence ℓ is zero, for otherwise Res_{⟨B,C⟩}^{P} π would contain a monomial β^{2j} without the corresponding γ^{2j} required to be fixed by f^{∗}. Therefore π = κηπ′, and so Res_{⟨B,C⟩}^{P} π′ is fixed by f^{∗}. Since π′ has lower degree than π, it is a

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polynomial in κη and η−κ^{3} by the inductive hypothesis, and so the inductive step is proved. The result is trivial in degree zero, and so κη and η − κ^{3} do generate the even-dimensional stable classes. The odd-dimensional case is similar: no invariant class which includes a monomial of the form dκ^{j} can be stable. We have a generating set for the stable classes, and the relations follow from Proposition 17.

4 The Chern subring

Chern classes are a link between the ordinary representation theory and the integral cohomology of G. They are defined in the appendix of [1]. A representation ρ of a finite group G has Chern classes c_{i}(ρ) ∈ H^{2i}(G, Z) for i ⩾ 0, with c_{0}ρ = 1, and c_{i}ρ is zero if i exceeds the degree of ρ. Two important properties are naturality, c_{i}(f^{!}ρ) = f^{∗}(c_{i}ρ), and the Whitney sum formula c(ρ_{1} ⊕ ρ_{2}) = c(ρ_{1})c(ρ_{2}). Here c(ρ) = ∑_{i⩾0}c_{i}ρ is the total Chern class

of ρ.

The Chern subring of G, denoted Ch(G), is the subring of H^{∗}(G, Z) generated by all Chern classes in H^{∗}(G, Z). In this section we prove:

Theorem 22 The 11-primary part Ch(J_{4})_{(11)} of the Chern subring is generated by κ, η and the ϕ_{i}. It is also generated by the Chern classes of a representation affording the irreducible character χ_{4} of degree 299, 367 (Atlas notation).

The 3-primary part Ch(J_{4})_{(3)} of the Chern subring is generated by 3ξ, 3ξ^{2}, ξ^{3} and α^{3}. It is also generated by the Chern classes of a representation affording the irreducible character χ_{2} of degree 1, 333.

Restriction commutes with taking Chern classes. So we restrict each representation of J_{4} to a Sylow p-subgroup, decompose this restriction, and apply the Whitney formula.

The irreducible representations of p_{+}^{1+2} are ρ^{xy} for 0 ⩽ x, y ⩽ p − 1, and ρ^{z} for 1 ⩽ z ⩽ p − 1. They have characters

χ^{xy}: A^{r}B^{s}C^{t} ⟼ exp {2πi(rx + sy)/p}
χ^{z}: A^{r}B^{s}C^{t} ⟼ \{ \begin{matrix}p exp \{2πizt/p\} & r = s = 0 \\ 0 & otherwise. \end{matrix}

The ρ^{xy} are 1-dimensional, and the ρ^{z} are induced from ⟨B, C⟩.

Theorem 23 ([6]) The total Chern classes of the irreducible representations

of p_{+}^{1+2} are c(ρ^{xy}) = 1+xα_{1}+yα_{2} and c(ρ^{z}) = 1+ ∑_{j=2}^{p−2} \frac{1}{p} \binom{p}{j}z^{j}θ_{j} +κ+z^{p}ζ.

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We call the intersection of P with a conjugacy class of G a G-conjugacy class in P, and define a G-class function on P to be a function on P which is constant on the G-conjugacy classes. Then characters of G restricted to P are G-class functions.

Lemma 24 Let P a Sylow p-subgroup of a finite group G. Then Ch(G)_{(p)} is the subring of H^{∗}(P, Z) generated by the Chern classes of those representations of P whose characters are G-class functions.

Proof: It is enough to show that every character ϕ of P which is a G-class function extends to a generalised character of G. From Brauer’s Induction Theorem, a class function on G is a generalised character if its restriction to every elementary subgroup of G is a generalised character.

Extend ϕ to a class function ψ on G by defining ψ in the obvious way on p-elements (including the identity). If g has order mp^{r} where p | m̸, then pick an s such that m | s and p^{r} ≡ 1 mod s, and define ψ(g) = ψ(g^{s}). Clearly ψ is a character after restriction to any subgroup which is the direct product of a p-group and a p′-group; but all elementary groups are of this form.

We obtain a spanning set for those characters of P which are G-class functions. In p_{+}^{1+2}, let pC be the set of non-identity central elements, let pD be the non-central elements and pE all non-identity elements. From Theorem 3, the J_{4}-conjugacy classes in 11_{+}^{1+2} are 1, 11C and 11D. In 3_{+}^{1+2} they are 1 and 3E.

Lemma 25 Let χ be a generalised character of p_{+}^{1+2} which is constant on pC

and on pD. Then χ is in the Z-span of χ^{00}, χ^{D} = ∑_{x,y=0}^{p−1} χ^{xy} and χ^{C} = ∑_{z=1}^{p−1} χ^{z}.

If χ is constant on pE then χ is in the Z-span of χ^{00} and χ^{E} = χ^{D} + pχ^{C}.

So the J_{4}-class functions in 11_{+}^{1+2} are spanned by χ^{00}, χ^{D} and χ^{C}, and in 3_{+}^{1+2} by χ^{00} and χ^{E}. We have character tables

\begin{aligned}& 1 & 11C & 11D \\ χ^{00} & 1 & 1 & 1 \\ χ^{D} & 121 & 121 & 0 \\ χ^{C} & 110 & −11 & 0 \end{aligned}  \begin{aligned}& 1 & 3E \\ χ^{00} & 1 & 1 \\ χ^{E} & 27 & 0 \end{aligned} .

Proof: First part: the automorphism group of p_{+}^{1+2} acts transitively on pC and on pD. Hence it fixes χ and χ^{00}, and acts transitively on the χ^{xy} (x and y not both zero) and on the χ^{z}. But χ is in the Z-span of the χ^{xy} and the χ^{z}. The rest is obvious.

Proposition 26 Representations affording the characters χ^{D} and χ^{C} have total Chern classes c(ρ^{D}) = 1 + κ^{p} and

c(ρ^{C}) = (1 + κ)^{p−1} − ζ^{p−1} + ∑_{i=2}^{p−2}(−1)^{i+1} \frac{1}{p}\binom{p}{i}θ_{i}ζ^{p−i−1}.

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Proof: We shall make repeated use of the identity ∏_{r=0}^{p−1}(X − r) ≡ X^{p} − X

mod p. The Chern classes of ρ^{xy} and ρ^{z} are in Theorem 23. By the Whitney formula,

c(ρ^{D}) = ∏_{x,y=0}^{p−1}(1 + xα_{1} + yα_{2})

= ∏_{x=0}^{p−1} ((1 + xα_{1})^{p} − (1 + xα_{1})α_{2}^{p−1})
= (1 − α_{2}^{p−1})^{p} − (1 − α_{2}^{p−1})(α_{1}^{p} − α_{1}α_{2}^{p−1})^{p−1}.

Now, α_{2}(α_{1}^{p} − α_{1}α_{2}^{p−1}) = α_{1}^{p}α_{2} − α_{1}α_{2}^{p}, which is zero. Hence

(1 − α_{2}^{p−1})(α_{1}^{p} − α_{1}α_{2}^{p−1})^{p−1} = α_{1}^{p(p−1)} − α_{1}^{(p−1)^{2}}α_{2}^{p−1}.

Therefore

c(ρ^{D}) = (1 − α_{2}^{p−1})^{p} − α_{1}^{p(p−1)} + α_{1}^{(p−1)^{2}}α_{2}^{p−1}
= 1 + κ^{p},

since, by induction on r,

κ^{r} = (−1)^{r}(α_{1}^{r(p−1)} − α_{1}^{(r−1)(p−1)}α_{2}^{p−1} + α_{2}^{r(p−1)}) for r ⩾ 2.

For ρ^{C}, the Whitney formula gives

c(ρ^{C}) = ∏_{z=1}^{p−1} (1 + ∑_{i=2}^{p−2} \frac{1}{p}\binom{p}{i}z^{i}θ_{i} + κ + z^{p}ζ)

= (1 + κ)^{p−1} − 1 + ∏_{z=1}^{p−1}(1 + z^{p}ζ) + ∑_{i=2}^{p−2} \frac{1}{p}\binom{p}{i} ∑_{j=0}^{p−2} a_{ij}θ_{i}ζ^{j},

where a_{ij} is the coefficient of XY^{j} in ∏_{z=1}^{p−1}(1 + z^{i}X + zY). There are no

terms κ^{i}ζ^{j}, with i and j both positive, because

∏_{z=1}^{p−1} (1 + κ + z^{p}ζ) ≡ (1 + κ)^{p−1} − ζ^{p−1} (mod p).

Since X^{p} ≡ X mod p, we see that 1^{p}, 2^{p}, …, (p − 1)^{p} are p − 1 distinct (p − 1)-roots of 1 in Z/p^{2}, and these two facts imply that

∏_{z=1}^{p−1}(X − z^{p}) ≡ X^{p−1} − 1 (mod p^{2}).

Therefore ∏_{z=1}^{p−1}(1+z^{p}ζ) = 1−ζ^{p−1}. Lemma 27 below says that a_{ij} ≡ 0 (mod p)

unless i + j = p − 1, in which case a_{ij} ≡ (−1)^{i−1}. The result is proved.

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Lemma 27 Let a_{ij} be the coefficient of XY^{j} in ∏_{z=1}^{p−1}(1 + z^{i}X + zY), where

1 ⩽ i ⩽ p − 1 and 0 ⩽ j ⩽ p − 2. Then

a_{ij} ≡ \{ \begin{matrix}(−1)^{i−1} & if i + j = p − 1 \\ 0 & otherwise \end{matrix} (mod p).

Proof: The result is certainly true for i = 1, because a_{1j} is the coefficient of XY^{j} in 1 − (X + Y)^{p−1}. It is also true for j = p − 2, since a_{i,p−2} is

the coefficient of XY^{p−2} in ∏_{z=1}^{p−1}(z^{i}X + zY), which means that a_{i,p−2} = (p −

1)! ∑_{z=1}^{p−1} z^{i−1}.

Otherwise, if j ⩽ p − 3 and i ⩾ 2, then a_{ij} + a_{i−1,j+1} = b_{j+1} ∑_{z=1}^{p−1} z^{i−1}.

Here b_{j} is the coefficient of Y^{j} in ∏_{z=1}^{p−1}(1 + zY), with a_{ij} corresponding to the

terms z_{1}^{i}z_{2} … z_{j+1}, and a_{i−1,j+1} to the z_{1}^{i−1}z_{2} … z_{j+2}. But for these i, the

sum ∑_{z=1}^{p−1} z^{i−1} is zero mod p. Therefore a_{ij} + a_{i−1,j+1} ≡ 0 (mod p).

Proof of Theorem 22: We know the Chern classes of ρ^{D} and ρ^{C} from Proposition 26. For p = 3, a representation affording χ^{E} has total Chern class (1+κ^{3}) ((1 + κ)^{2} − ζ^{2})^{3} = 1−3ξ +3ξ^{2} −ξ^{3} −α^{3}. Therefore by Lemma 24, we have the claimed generators for the Chern subring. It may easily be shown that the Chern classes of the stated representations of J_{4} also generate.

The author would like to thank his research supervisor, Dr C.B. Thomas, for all his advice and encouragement, and Prof. R. Knörr for the proof of Lemma 24. This research was carried out whilst the author was supported by the Science and Engineering Research Council.

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