CHERN CLASSES AND EXTRASPECIAL GROUPS OF ORDER p^{5}

DAVID J. GREEN

Abstract A presentation is obtained for the Chern subring modulo nilradical of both extraspecial p-groups of order p^{5}, for p an odd prime. Moreover, it is proved that, for every extraspecial p-group of exponent p, the top Chern classes of the irreducible representations do not generate the Chern subring modulo nilradical. Finally, a related question about symplectic invariants is discussed, and solved for Sp_{4}(𝔽_{p}). The main innovation in this work is to consider extraspecial groups as central products, and to partition the maximal elementary abelian subgroups of the central product into those which lift to abelian subgroups of the corresponding direct product, and those which do not.

Introduction

The author and I. J. Leary argued in [5] that, when trying to determine the mod-p cohomology ring of a p-group, the subquotient obtained by factoring out the nilradical and then taking the subring generated by Chern classes is a natural and worthy object of study. In contrast to the partial results of a general nature obtained in that paper, the current article presents a complete determination of this subquotient for the extraspecial p-group p_{+}^{1+4} of order p^{5} and exponent p, for p an odd prime.

Once more, the theorem of Quillen that characterizes nilpotent elements by their restrictions to elementary abelian subgroups will play a fundamental role. The key insight of the current work is that, by employing a suitable partition of the set of maximal elementary abelian subgroups, the search for

Submitted 7 June 1995.

1991 Mathematics Subject Classification 20J06.

The author gratefully acknowledges the support of the Max Kade Foundation.

relations in the Chern subring may be pursued in a methodical fashion. This partition is related to the fact that the extraspecial p-group in question is the central product of two copies of the extraspecial group p_{+}^{1+2} of order p^{3} and exponent p.

This technique lends itself to a number of generalisations that could be used to launch an attack by induction on the Chern subrings of all extraspecial groups. It is to be hoped that further research will identify the correct generalisation for this task.

Much of the paper is in fact taken up in solving a related problem in pure algebra.

Problem. For an odd prime p and a positive integer n, let E_{n} be a 2n-dimensional 𝔽_{p}-vector space, carrying a nondegenerate symplectic bilinear form. Let K_{0}, …, K_{n−1} be indeterminates, and let F_{n} denote the polynomial algebra S(E_{n}^{∗}) ⊗_{𝔽p} 𝔽_{p}[K_{0}, …, K_{n−1}].

For each maximal totally isotropic subspace I of E_{n}, there is then a unique algebra homomorphism q_{I}: F_{n} → S(I^{∗}) which behaves on E_{n}^{∗} as the restriction map E_{n}^{∗} → I^{∗}, and sends K_{r} to D_{r}(I^{∗}), the Dickson invariant in the (p^{n} − p^{r})th symmetric power of I^{∗}.

The intersection of the kernels of all the q_{I} is an ideal in F_{n}; define Q_{n} to be the corresponding quotient algebra. Give a presentation for Q_{n}.

A presentation for Q_{2} is achieved in Theorem 17. After that, we shall prove in Theorem 23 that the Chern subring modulo nilradical for p_{+}^{1+4} is isomorphic to Q_{2} ⊗_{𝔽p} 𝔽_{p}[Z] for an indeterminate Z in degree 2n. Together, these constitute the main result of the paper.

Afterwards, we will proceed with some further applications of this inflation technique. Tezuka and Yagita studied the ring generated by the top Chern classes of the irreducible representations of p_{+}^{1+2n}. This is contained in the Chern subring modulo nilradical, and it is shown in Theorem 20 that this containment is strict. After this, we discuss a problem about invariants of the symplectic group which is related to the cohomology of extraspecial groups. Last of all, the Chern subring modulo nilradical is obtained for the extraspecial p-group p_{−}^{1+4} of order p^{5} and exponent p^{2}.

It is obtaining the presentation for Q_{2} which requires the most work. Define T_{2} to be the image in Q_{2} of S(E_{2}^{∗}). The structure of T_{2} is known from the work of Tezuka and Yagita. We need to understand the ring generated by T_{2} and the images κ_{0}, κ_{1} of K_{0}, K_{1} respectively. Consider E_{2} as the orthogonal direct sum E′ ⊥ E′′ of two nondegenerate 2-dimensional symplectic spaces. Partition the set of maximal totally isotropic subspaces I of E into Φ, the set of those I which decompose as I = (I ∩ E′) ⊕ (I ∩ E′′); and Ψ, the set of those I which do not. Associated to this partition, an “inflation” map Q_{2} → Q_{1} ⊗ Q_{1} is constructed in Lemma 3. In Proposition 4 it is shown how to reduce questions about Q_{2} to questions about Q_{1} ⊗ Q_{1}. A key step in the proof of this result is Lemma 6, which is in some sense the core of the paper. It describes how T_{2} would look if Q_{2} had been defined using only the maximal totally isotropic subspaces from Φ, or only the maximal totally isotropic subspaces from Ψ. After this, obtaining a presentation for Q_{2} is merely a matter of hammering the relations out, although some of these are rather complicated.

The author wishes to thank J. L. Alperin and I. J. Leary for encouragement and helpful conversations.

1. A regular sequence

Let b denote the symplectic form E_{n} ⊗_{𝔽p} E_{n} → 𝔽_{p}, and denote by q the quotient map F_{n} → Q_{n}. Then, for each maximal totally isotropic subspace I of E_{n}, there is a unique map q̂_{I}: Q_{n} → S(I^{∗}) such that q̂_{I}q = q_{I}.

Write T_{n} for the image of S(E_{n}^{∗}) under q. Pick a symplectic basis A_{1}, …, A_{n}, B_{1}, …, B_{n} for E_{n}: so A_{i} ⊥ A_{j}, B_{i} ⊥ B_{j} and b(A_{i}, B_{j}) = δ_{ij}. Take the corresponding dual basis A_{1}^{∗}, …, A_{n}^{∗}, B_{1}^{∗}, …, B_{n}^{∗} for E_{n}^{∗}. Define elements of T_{n} by α_{i} = q(A_{i}^{∗}) and β_{i} = q(B_{i}^{∗}). Then α_{1}, …, α_{n}, β_{1}, …, β_{n} generate T_{n} as an 𝔽_{p}-algebra. Denote by R_{r}(E_{n}) the element A_{1}^{∗}B_{1}^{∗}^{p^{r}} − A_{1}^{∗}^{p^{r}}B_{1}^{∗} + ⋯ + A_{n}^{∗}B_{n}^{∗}^{p^{r}} − A_{n}^{∗}^{p^{r}}B_{n}^{∗} of S(E_{n}^{∗}). If the vector space E_{n} is clearly determined by the context, this notation will be shortened to R_{r}. Note that R_{r} equals the b(v, F^{r}(v)) of [3].

Theorem 1. (Tezuka–Yagita) The sequence R_{1}, …, R_{n} in S(E_{n}^{∗}) is a regular sequence. The ideal generated by these elements contains R_{r} for all r ⩾ 1, and is the kernel of the surjection S(E_{n}^{∗}) → T_{n}.

Proof. See Proposition 8.2 of [3]. The h of that paper is defined to be the codimension in E_{n} of a maximal totally isotropic subspace, and so takes the value n here. The first two parts are explicitly stated. The last part follows, using the Nullstellensatz, from the fact that the ideal is radical and from the description of the variety in terms of isotropic subspaces.

2. Dickson invariants

We now recall the salient facts about the Dickson invariants. See Chapter 8 of Benson’s book [2] for proofs.

Let V be an m-dimensional 𝔽_{p}-vector space. For each 0 ⩽ r ⩽ m − 1, there is a Dickson invariant D_{r}(V) in the (p^{m} − p^{r})th symmetric power of V such that

∏_{v∈V}(X − v) = X^{p^{m}} + ∑_{r=0}^{m−1}(−1)^{m−r}D_{r}(V)X^{p^{r}} in S(V)[X].  (1)

The natural action of GL(V) on S(V) has as ring of invariants the polynomial algebra 𝔽_{p}[D_{0}, …, D_{m−1}].

In several papers, D_{r}(V) is denoted c_{m,r}. There are two reasons for using nonstandard notation: to avoid a clash with the standard notation for Chern classes, and to specify V explicitly.

Each Dickson invariant is a polynomial in the elements of any basis for V. These polynomials are independent of the choice of basis, and in fact depend only on p, r and m, since the D_{r} are invariant under GL(V). If w_{1}, …, w_{m} are elements of an 𝔽_{p}-vector space W, and r < m, then D_{r}(w_{1}, …, w_{m}) shall denote the evaluation at (w_{1}, …, w_{m}) of the polynomial for D_{r}(𝔽_{p}^{m}). The following lemma describes how this will be related to the Dickson invariants of the space spanned by the w_{i}. It takes a particularily elegant form when we work with dual spaces.

Lemma 2. Let V be an m-dimensional 𝔽_{p}-vector space, and U a subspace of codimension ℓ. Then, for every 0 ⩽ r ⩽ m − 1,

Res_{U} (D_{r}(V^{∗})) = \{ \begin{matrix}D_{r−ℓ}(U^{∗})^{p^{ℓ}} & if ℓ ⩽ r, and \\ 0 & otherwise. \end{matrix}

Proof. Obvious from the definition of D_{r}.

This paper is particularily concerned with small cases. The first three Dickson invariants are easily calculated: D_{0}(w) = w^{p−1} and

D_{0}(w_{1}, w_{2}) = (w_{1}w_{2}^{p} − w_{1}^{p}w_{2})^{p−1} D_{1}(w_{1}, w_{2}) = \frac{w_{1}^{p^{2}−1} − w_{2}^{p^{2}−1}}{w_{1}^{p−1} − w_{2}^{p−1}} . (2)

Note in particular that if we take for any m-dimensional V a non-zero element of each 1-dimensional subspace, and multiply these together, we get an element of S(V) which is well-defined up to a scalar, and has (p−1)th power D_{0}(V). If m = 2 and V has basis v_{1}, v_{2}, then this element is v_{1}v_{2}^{p} − v_{1}^{p}v_{2}.

In Q_{n}, define κ_{n,r} = q(K_{r}) for 0 ⩽ r ⩽ n − 1. Then the α_{i}, β_{j} and κ_{n,r} together generate the 𝔽_{p}-algebra Q_{n}. Note that the algebraic independence of the Dickson invariants, together with the definition of q, ensures that the κ_{n,r} not only are algebraically independent over 𝔽_{p} in Q_{n}, but also, no polynomial in them over 𝔽_{p} is a zero divisor in Q_{n}. For small values of n we will abbreviate κ_{n,r}, denoting κ_{1,0}, κ_{2,0}, κ_{2,1} by κ, κ_{0}, κ_{1} respectively.

3. Partition

The object of this section is to prove a result (Proposition 4) which allows the search for relations in Q_{2} to be carried out in Q_{1} ⊗ Q_{1}. This is achieved by partitioning the set of maximal totally isotropic subspaces of E_{2} into two families, and for each family, determining which elements of Q_{2} it fails to detect. This partition can in fact be performed in E_{n}, and so we will only restrict ourselves to E_{2} when this becomes necessary.

Suppose that n = ℓ + m. Then E_{n} is the orthogonal direct sum E_{n} = E_{ℓ} ⊥ E_{m} of nondegenerate symplectic spaces E_{ℓ}, E_{m}. Partition the set of maximal totally isotropic subspaces I of E_{n} as Φ ∐ Ψ, where I ∈ Φ if and only if I is the direct sum of (necessarily maximal) totally isotropic subspaces of E_{ℓ} and E_{m}.

Lemma 3. Let n = ℓ + m. Then the isomorphism S(E_{n}^{∗}) ≅ S(E_{ℓ}^{∗}) ⊗ S(E_{m}^{∗}) induces an inflation homomorphism π^{∗}: Q_{n} → Q_{ℓ} ⊗ Q_{m} such that, for any x ∈ Q_{n}, we have π^{∗}(x) = 0 if and only if q̂_{I}(x) = 0 for all I ∈ Φ.

Proof. As 𝔽_{p}^{n} ≅ 𝔽_{p}^{ℓ}⊕𝔽_{p}^{m} and GL_{ℓ}(𝔽_{p})×GL_{m}(𝔽_{p}) ⩽ GL_{n}(𝔽_{p}), each D_{r}(𝔽_{p}^{n}) is a polynomial in the D_{s}(𝔽_{p}^{ℓ}) and the D_{t}(𝔽_{p}^{m}). Define π^{∗}(κ_{n,r}) to be the corresponding polynomial in the κ_{ℓ,s}⊗1 and the 1⊗κ_{m,t}. The rest is obvious.

Remark. The above partition is related to the fact that the extraspecial group P_{n} is the central product P_{ℓ} ∗ P_{m}. The inflation map corresponds to the cohomology inflation from P_{n} to P_{ℓ} × P_{m}.

Definition. For a polynomial f ∈ T_{2}[x_{0}, x_{1}] with coefficients in T_{2}, define e_{κ}(f) ∈ Q_{2} by

e_{κ}(f) = f(κ_{0}, κ_{1}) − f(D_{0}(α_{1}, β_{1}), D_{1}(α_{1}, β_{1})) ,

and define γ_{2} to be the element D_{1}(α_{1}, β_{1}) − D_{1}(α_{2}, β_{2}) of T_{2}.

Proposition 4. Let f ∈ T_{2}[x_{0}, x_{1}] be a polynomial with coefficients in T_{2}. Then the element f(κ_{0}, κ_{1}) of Q_{2} belongs to T_{2} if and only if there exists t ∈ T_{2} such that the equation e_{κ}(f) = tγ_{2} holds after inflation to Q_{1}⊗Q_{1}. If such t does exist, then e_{κ}(f) = tγ_{2} holds in Q_{2}.

Before proving this proposition, we shall establish two auxilliary results. We shall work with the partition associated to the orthogonal direct sum decomposition E_{n} = E_{1} ⊥ E_{n−1}, where n ⩾ 2 and E_{1} has basis A_{1}, B_{1}.

Lemma 5. For n ⩾ 2, let I be a maximal totally isotropic subspace of E_{n}. Then I ∈ Ψ if and only if the restrictions of α_{1} and β_{1} are linearly independent in I^{∗}.

Proof. Obvious.

By the definition of Q_{n}, the maximal totally isotropic subspaces in Φ and Ψ combined detect every non-zero element of Q_{n}. We shall now determine the ideals of elements which Φ and Ψ individually fail to detect. Recall that T_{n} ≅ S(E_{n}^{∗})/(R_{1}, …, R_{n}).

Lemma 6. 1. In S(E_{n}^{∗})/(R_{1}, …, R_{n−1}), there is a unique γ_{n} such that (α_{1}β_{1}^{p} − α_{1}^{p}β_{1})γ_{n} = R_{n}. In particular, γ_{2} = D_{1}(α_{1}, β_{1}) − D_{1}(α_{2}, β_{2}), as in the definition preceding Proposition 4.

2. Consider the ideal in T_{n} of classes whose image under q̂_{I} is zero for every I ∈ Φ. It is the principal ideal generated by α_{1}β_{1}^{p} − α_{1}^{p}β_{1}.

3. The corresponding ideal for Ψ is also principal; it is generated by γ_{n} (considered as an element of T_{n}).

Proof. Write T̂ for S(E_{n}^{∗})/(R_{1}, …, R_{n−1}). For part 1, we have R_{r}(E_{n}) = R_{r}(E_{1}) + R_{r}(E_{n−1}). By the Tezuka–Yagita theorem for E_{n−1}, it follows that R_{n}(E_{n}) lies in the ideal in S(E_{n}^{∗}) generated by the R_{r}(E_{n}) for r < n, and the R_{r}(E_{1}) for r ⩽ n. Now apply the Tezuka–Yagita theorem again, this time for E_{1}. Hence (the image of) R_{n}(E_{n}) lies in the principal ideal of T̂ generated by α_{1}β_{1}^{p} −α_{1}^{p}β_{1}. Therefore γ_{n} exists; it is unique since R_{n}(E_{n}) is a non-zero divisor in T̂. Observing that (α_{1}β_{1}^{p} − α_{1}^{p}β_{1})D_{1}(α_{1}, β_{1}) = α_{1}β_{1}^{p^{2}} − α_{1}^{p^{2}}β_{1}, we can verify the equation for γ_{2}.

For part 2, observe first that q̂_{I}(α_{1}β_{1}^{p} − α_{1}^{p}β_{1}) is zero in S(I^{∗}) for every I ∈ Φ, and non-zero for every I ∈ Ψ. Since S(I^{∗}) is an integral domain, it follows that q̂_{I}(γ_{n}) = 0 in S(I^{∗}) for every I ∈ Ψ. Therefore, if t ∈ T_{n} satisfies q̂_{I}(t) = 0 for every I ∈ Φ, then tγ_{n} = 0 in T_{n}.

Now pick t̂ ∈ T̂ lying above t. Then for some s ∈ T̂, t̂γ_{n} = sR_{n}(E_{n}). Multiplying both sides by α_{1}β_{1}^{p} − α_{1}^{p}β_{1} and rearranging then yields (t̂ − s(α_{1}β_{1}^{p} − α_{1}^{p}β_{1}))R_{n} = 0. Since R_{n} is a non-zero divisor, t̂ lies in the ideal of T̂ generated by α_{1}β_{1}^{p} − α_{1}^{p}β_{1}, proving part 2. The same method works for part 3.

Remark. It would appear that the main obstacle to obtaining a presentation for Q_{n} is the current lack of a version of Lemma 6 for general n.

We can now proceed with the proof of Proposition 4.

Proof of Proposition 4. For r = 0 or 1, and I ∈ Ψ, the images under q̂_{I} of κ_{r} and D_{r}(α_{1}, β_{1}) both equal D_{r}(I^{∗}). So for every I ∈ Ψ, the images under q̂_{I} of f(κ_{0}, κ_{1}) and f(D_{0}(α_{1}, β_{1}), D_{1}(α_{1}, β_{1})) must be equal. Therefore if f(κ_{0}, κ_{1}) ∈ T_{2} then there exists t ∈ T_{2} such that

e_{κ}(f) = tγ_{2} .  (3)

Conversely, suppose that there is a t ∈ T_{2} such that Eqn. (3) holds after q̂_{I} for every I ∈ Φ. Then Eqn. (3) holds in Q_{2}, since each side of the equation is in ker q̂_{I} for every I ∈ Ψ. Hence f(κ_{0}, κ_{1}) ∈ T_{2}.

4. Technical results

In light of Proposition 4, we want a presentation for Q_{1}. This has generators α_{1}, β_{1} and κ_{1,0}. We shall drop the subscripts from these generators.

Proposition 7. The 𝔽_{p}-algebra Q_{1} is generated by α, β and κ; a sufficient set of relations is κ^{2} = α^{2(p−1)} − α^{p−1}β^{p−1} + β^{2(p−1)}, ακ − α^{p} = 0, and βκ − β^{p} = 0.

Proof. These relations are easily verified after every q̂_{I}. Note that they imply the relation αβ^{p} − α^{p}β. Therefore by Theorem 1, all relations in T_{1} are present. It only remains to prove that κ does not lie in T_{1}. To see this, note that κ^{p} = α^{p(p−1)} − α^{(p−1)^{2}}β^{p−1} + β^{p(p−1)}. This is not the p-th power of any polynomial in α and β.

In Q_{1} ⊗ Q_{1}, we need to be able to distinguish the κ of the first factor from that of the second. Write κ′, κ′′ for κ ⊗ 1, 1 ⊗ κ respectively. Let T_{1,1} denote the subring T_{1} ⊗ T_{1} of Q_{1} ⊗ Q_{1}. Both T_{2} and T_{1,1} have generators α_{1}, α_{2}, β_{1}, β_{2}. In T_{1,1} however, the relations are generated by α_{1}β_{1}^{p} − α_{1}^{p}β_{1} and α_{2}β_{2}^{p} − α_{2}^{p}β_{2}.

Lemma 8. Let f ∈ T_{1,1}[y_{1}, y_{2}] be a polynomial without constant term. Then f(κ′, κ′′) lies in the sub T_{1,1}-module of Q_{1} ⊗ Q_{1} generated by κ′ and κ′′ if and only if the coefficients in f of both y_{1}y_{2} and y_{1}^{2}y_{2}^{2} are without zero degree term.

Proof. We have κ′^{r}κ′′^{s} ∈ T_{1,1} if and only if neither r nor s is one. If we remove, for all such (r, s), both y_{1}^{r+1}y_{2}^{s} and y_{1}^{r}y_{2}^{s+1} from the set of monomials in y_{1}, y_{2}, then we are left with 1, y_{1}y_{2} and y_{1}^{2}y_{2}^{2}. If δ is one of the generators of T_{1,1}, then δκ′κ′′ and δκ′^{2}κ′′^{2} lie in the submodule in question. So it remains to show that κ′κ′′ and its square do not. As κ′κ′′ lying there would imply that κ′^{2}κ′′^{2} did too, it is enough to show that κ′^{2}κ′′^{2} is not in the T_{1,1}-module generated by κ′ and κ′′.

Suppose that κ′^{2}κ′′^{2} = gκ′ + hκ′′, with g, h ∈ T_{1,1}. Now κ′^{2}κ′′^{2} involves α_{1}^{p−1}α_{2}^{p−1}β_{1}^{p−1}β_{2}^{p−1}, and so gκ′ + hκ′′ must too. But every term in gκ′ must involve either α_{1}^{p} or β_{1}^{p}, and every term in hκ′′ must involve either α_{2}^{p} or β_{2}^{p}. Since the only relations in T_{1,1} are α_{1}β_{1}^{p} = α_{1}^{p}β_{1} and α_{2}β_{2}^{p} = α_{2}^{p}β_{2}, we have derived a contradiction.

Lemma 9. Suppose that u, v ∈ T_{1,1} satisfy uκ′ = vκ′′. Then u − v lies in the ideal in T_{1,1} generated by α_{1}α_{2}^{p} − α_{1}^{p}α_{2}, α_{1}β_{2}^{p} − α_{1}^{p}β_{2}, β_{1}α_{2}^{p} − β_{1}^{p}α_{2} and β_{1}β_{2}^{p} − β_{1}^{p}β_{2}.

Proof. Each of the four elements u − v listed in the statement satisfies uκ′ = vκ′′, and so lies in the ideal in question. Conversely, define a monomial α_{1}^{r_{1}}α_{2}^{r_{2}}β_{1}^{s_{1}}β_{2}^{s_{2}} to be admissible if both s_{1} < p unless r_{1} = 0, and s_{2} < p unless r_{2} = 0. Then the admissible monomials form a basis for the 𝔽_{p}-vector space T_{1,1}. In particular, u and v may be expressed in terms of this basis.

Since uκ′ = vκ′′ with both u and v ∈ T_{1,1}, it follows that uκ′ and vκ′′ lie in T_{1,1}. Hence every admissible monomial in u has either r_{1} or s_{1} positive, and every admissible monomial in v has either r_{2} or s_{2} positive. Then uκ′ is obtained from u as follows: the coefficients remain the same, and each monomial has r_{1} increased by p − 1, unless r_{1} = 0, in which case s_{1} is increased by p − 1. Similarily, vκ′′ is obtained from v by increasing r_{2} or s_{2}. Since uκ′ = vκ′′, there is an induced bijection between the admissible monomials in u and in v, and the difference between any admissible monomial in u and the corresponding monomial in v is divisible by one of the four elements listed in the statement.

Lemma 10. Let f ∈ T_{2}[x_{0}, x_{1}]; then, in Q_{1} ⊗ Q_{1},

π^{∗}(e_{κ}(f)) = f (κ′κ′′(κ′ − κ′′)^{p−1}, κ′′(κ′ − κ′′)^{p−1} + κ′^{p}) − f(0, κ′^{p}) .

Moreover, π^{∗}(γ_{2}) = (κ′ − κ′′)^{p}.

Proof. The maps Q_{1} ⊗ Q_{1} → S(I^{∗}), for all I ∈ Φ, detect the elements of Q_{1} ⊗ Q_{1}. Using Lemma 6 and Eqn. (2), it is straightforward to verify that the equations π^{∗}(γ_{2}) = (κ′ − κ′′)^{p}, π^{∗}(κ_{0}) = κ′κ′′(κ′ − κ′′)^{p−1}, π^{∗}(κ_{1}) = (κ′^{p+1} − κ′′^{p+1})/(κ′ − κ′′), π^{∗}(D_{0}(α_{1}, β_{1})) = 0 and π^{∗}(D_{1}(α_{1}, β_{1})) = κ′^{p} hold after mapping to any such S(I^{∗}).

5. Deriving the relations

We are now in a position to derive the relations in Q_{2}.

Lemma 11. The generator κ_{1} does not belong to T_{2}, but its p^{th} power does:

κ_{1}^{p} = ∑_{i=0}^{p} D_{1}(α_{1}, β_{1})^{p−i}D_{1}(α_{2}, β_{2})^{i} .  (4)

Proof. Inflate to Q_{1} ⊗ Q_{1}. Now, κ^{s} ⊗ κ^{t} belongs to T_{1,1} if and only if neither s nor t equals 1. Then π^{∗}(κ_{1}) equals κ^{p−1} ⊗ κ + κ ⊗ κ^{p−1} modulo T_{1,1}, by Lemma 10. So κ_{1} ∉ T_{2}.

If I ∈ Ψ then q̂_{I}(D_{1}(α_{i}, β_{i})) = D_{1}(I^{∗}) for i = 1, 2. Also, π^{∗}(D_{1}(α_{i}, β_{i})) is κ′^{p}, κ′′^{p} for i = 1, 2 respectively. This establishes Eqn. (4).

We now turn our attention to the T_{2}-submodule of Q_{2} generated by 1 and κ_{1}.

Proposition 12. Let f ∈ T_{2}[x_{0}, x_{1}] be a polynomial with coefficients in T_{2}. The element f(κ_{0}, κ_{1}) of Q_{2} belongs to T_{2}κ_{1} + T_{2} if and only if the coefficients in f of x_{0} and, if p = 3, of x_{1}^{2} have no degree zero part.

Proof. By Proposition 4, f(κ_{0}, κ_{1}) belongs to T_{2}κ_{1}+T_{2} if and only if there exist a, b ∈ T_{2} such that the equation

e_{κ}(f) = aγ_{2} + be_{κ}(x_{1})

holds after inflation to Q_{1} ⊗ Q_{1}. Lemma 10 says, that this is the case if and only if there exist a′, b′ ∈ T_{1,1} such that, in Q_{1} ⊗ Q_{1},

f (κ′κ′′(κ′ − κ′′)^{p−1}, κ′′(κ′ − κ′′)^{p−1} + κ′^{p}) − f(0, κ′^{p}) = a′(κ′ − κ′′)^{p} + b′κ′′(κ′ − κ′′)^{p−1} .

Observe that both sides of the equation are divisible by (κ′−κ′′)^{p−1}. Performing this division on the right hand side yields a′κ′ + (b′ − a′)κ′′. By Lemma 8, this means that such a′, b′ exist if and only if the coefficients of both y_{1}y_{2} and y_{1}^{2}y_{2}^{2} in (f(y_{1}y_{2}(y_{1} − y_{2})^{p−1}, y_{2}(y_{1} − y_{2})^{p−1} + y_{2}^{p}) − f(0, y_{1}^{p})) /(y_{1}−y_{2})^{p−1} have no zero degree term. This happens exactly when the coefficients in f(x_{0}, x_{1}) of x_{0} and, if p = 3, of x_{1}^{2}, have no zero degree part.

Remark. There now remain so few elements outside T_{2}κ_{1}+T_{2}, that their degrees distinguish them.

Lemma 13. Let δ ∈ q(E_{2}^{∗}). Then δκ_{0} = δ^{p}κ_{1} − δ^{p^{2}}.

Proof. Apply q̂_{I} for any maximal totally isotropic subspace I. This sends δ to some element of I^{∗}, and κ_{r} to D_{r}(I^{∗}). Observe that the left hand side of Eqn. (1) vanishes whenever X is an element of V.

Proposition 14. Let δ_{1}, δ_{2} ∈ E^{∗}. Then

(δ_{1}δ_{2}^{p} − δ_{1}^{p}δ_{2}) (κ_{1} − D_{1}(δ_{1}, δ_{2})) = 0 .

Proof. Let I be a maximal totally isotropic subspace of E_{2}. If q̂_{I}(δ_{1}), q̂_{I}(δ_{2}) are linearly independent in I^{∗}, then κ_{1} and D_{1}(δ_{1}, δ_{2}) both map to D_{1}(I^{∗}) under q̂_{I}. If they are linearly dependent, then q̂_{I}(δ_{1}δ_{2}^{p} − δ_{1}^{p}δ_{2}) = 0.

We can now derive a presentation for the T_{2}-module generated by 1 and κ_{1}.

Lemma 15. The following ideals in T_{2} are equal:

• The ideal J_{1} of all u ∈ T_{2} such that uκ_{1} ∈ T_{2}.

• The ideal J_{2} in T_{2} generated by α_{1}α_{2}^{p} − α_{1}^{p}α_{2}, α_{1}β_{2}^{p} − α_{1}^{p}β_{2}, β_{1}α_{2}^{p} − β_{1}^{p}α_{2}, β_{1}β_{2}^{p} − β_{1}^{p}β_{2} and α_{1}β_{1}^{p} − α_{1}^{p}β_{1}. Note that the last generator equals −(α_{2}β_{2}^{p} − α_{2}^{p}β_{2}).

Proof. By Proposition 14, J_{2} ⊆ J_{1}. We shall show that J_{1} ⊆ J_{2}. First we reduce this to a problem in Q_{1} ⊗ Q_{1}. Let u ∈ J_{1}. Then uκ_{1} ∈ T_{2}, so by Proposition 4, u(κ_{1} − D_{1}(α_{1}, β_{1})) = vγ_{2} for some v ∈ T_{2}. Now inflate to Q_{1} ⊗ Q_{1}. We get π^{∗}(u)κ′′(κ′ − κ′′)^{p−1} = π^{∗}(v)(κ′ − κ′′)^{p}. Since (κ′ − κ′′)^{p−1} is a non-zero divisor in Q_{1} ⊗ Q_{1}, we cancel and rearrange to get π^{∗}(u + v)κ′′ = π^{∗}(v)κ′. By Lemma 9, it follows that π^{∗}(u) lies in the ideal in T_{1,1} generated by the images under π^{∗} of the first four generators of J_{2}. Since the kernel in T_{2} of inflation is principal, generated by the fifth generator of J_{2}, we are done.

The following lemma will help us to describe some elements of T_{2} involved in relations in Q_{2}.

Lemma 16. Suppose that u ∈ Q_{2} satisfies q̂_{I}(u) = 0 for all I ∈ Ψ. Then for each t ∈ T_{1,1}, there is a unique v ∈ Q_{2} such that q̂_{I}(v) = 0 for all I ∈ Ψ and π^{∗}(v) = tπ^{∗}(u). It therefore makes sense to refer to v as tu. In particular, this result holds for u = γ_{2}, and for u = e_{κ}(f) for any f ∈ T_{2}[x_{0}, x_{1}].

Proof. The inflation map T_{2} → T_{1,1} is surjective. Pick any t̂ ∈ T_{2} such that π^{∗}(t̂) = t. Then t̂u satisfies the requirements on v. The uniqueness part follows from Lemma 3 and the definition of Q_{n}.

We can now put the above results together to obtain a presentation for Q_{2}. Define polynomials f_{1}, f_{2} ∈ 𝔽_{p}[y_{1}, y_{2}] to be y_{1}^{2}(y_{1}^{p+1} −y_{2}^{p+1})/(y_{1}^{2} −y_{2}^{2}), respectively y_{1}^{2}y_{2}^{2}(y_{1}^{p−3} − y_{2}^{p−3})/(y_{1} − y_{2}) + 2y_{1}^{p} + y_{1}^{p−2}y_{2}^{2} + 2y_{2}^{p}.

Theorem 17. A presentation for the commutative 𝔽_{p}-algebra Q_{2} consists of six generators α_{1}, α_{2}, β_{1}, β_{2}, κ_{0}, κ_{1}, together with relations as follows:

α_{1}β_{1}^{p} − α_{1}^{p}β_{1} + α_{2}β_{2}^{p} − α_{2}^{p}β_{2} = 0  (5)
α_{1}β_{1}^{p^{2}} − α_{1}^{p^{2}}β_{1} + α_{2}β_{2}^{p^{2}} − α_{2}^{p^{2}}β_{2} = 0  (6)
(δ_{1}δ_{2}^{p} − δ_{1}^{p}δ_{2}) (κ_{1} − D_{1}(δ_{1}, δ_{2})) = 0 for δ_{1}, δ_{2} ∈ {α_{1}, β_{1}, α_{2}, β_{2}} (7)
δκ_{0} − δ^{p}κ_{1} + δ^{p^{2}} = 0 for δ ∈ {α_{1}, β_{1}, α_{2}, β_{2}}  (8)
κ_{1}^{p} = ∑_{i=0}^{p} D_{1}(α_{1}, β_{1})^{p−i}D_{1}(α_{2}, β_{2})^{i}  (9)
e_{κ}(x_{0}^{2}) = κ′^{2}κ′′^{2}(κ′ − κ′′)^{p−2}γ_{2}  (10)
e_{κ}(x_{0}x_{1}) = (f_{1}(κ′, κ′′) + f_{1}(κ′′, κ′)) e_{κ}(x_{1}) + f_{1}(κ′′, κ′)γ_{2} .  (11)

If p > 3 then

e_{κ}(x_{1}^{2}) = f_{2}(κ′, κ′′)e_{κ}(x_{1}) + (κ′^{p−2}κ′′^{2} + κ′′^{p})γ_{2}  (12)

If p = 3, then Eqn. (12) is replaced by the relations

e_{κ}(ε_{1}x_{1}^{2}) = (ε_{1}κ′′^{3} + 2ε_{1}^{3}κ′′^{2} + 2ε_{1}^{7})e_{κ}(x_{1}) + ε_{1}^{3}κ′′^{2}γ_{2} for ε_{1} = α_{1}, β_{1} (13)
e_{κ}(ε_{2}x_{1}^{2}) = (2ε_{2}κ′^{3} + ε_{2}^{3}κ′^{2} + 2ε_{2}^{7})e_{κ}(x_{1}) + ε_{2}^{7}γ_{2} for ε_{2} = α_{2}, β_{2}. (14)

In fact, Eqn. (9) is a consequence of the other relations if p > 3, though this would be hard to verify directly. Note that Eqn. (6) is redundant too: this can be seen from above.

The only relations in T_{2} are the first two relations above; the first three relations above carry all information about T_{2}κ_{1} + T_{2}.

Proof. It only remains to establish the last five relations. All are proved using the method of the proof of Proposition 12. We give one example, the case of e_{κ}(x_{0}x_{1}). We have

π^{∗}(e_{κ}(x_{0}x_{1})) = κ′κ′′(κ′ − κ′′)^{p−1}(κ′^{p+1} − κ′′^{p+1})/(κ′ − κ′′) ,

and require a′, b′ ∈ T_{1,1} such that

a′κ′ + (b′ − a′)κ′′ = (κ′^{p+2}κ′′ − κ′κ′′^{p+2})/(κ′ − κ′′)

= (κ′^{p+3}κ′′ − κ′^{2}κ′′^{p+2} − κ′κ′′^{p+3} + κ′^{p+2}κ′′)/(κ′^{2} − κ′′^{2})

= f_{1}(κ′′, κ′)κ′ + f_{1}(κ′, κ′′)κ′′ .

As both f_{1}(κ′′, κ′) and f_{1}(κ′, κ′′) lie in T_{1,1}, we are done.

6. Extraspecial p-groups

The 2n-dimensional 𝔽_{p}-vector space E_{n} may be viewed as an elementary abelian p-group of rank 2n. Let N be a cyclic group of order p. The nondegenerate symplectic form b on E may be viewed as a map E×E → N. Denote by P_{n} the extraspecial p-group p_{+}^{1+2n} of order p^{2n+1} and exponent p. There is a central extension 1 → N → P_{n} →^{ψ} E → 1, such that, for g_{1}, g_{2} ∈ P_{n}, the commutator [g_{1}, g_{2}] equals b (ψ(g_{1}), ψ(g_{2})). The maximal elementary abelian subgroups of P_{n} have p-rank n+1, and are exactly the inverse images under ψ of the maximal totally isotropic subspaces of E.

To determine the irreducible characters of P_{n}, pick an embedding of the additive group of 𝔽_{p} in ℂ^{×}. There are p^{2n} linear characters of P_{n}, all of which factor through ψ. These may be identified with the elements of the dual space E_{n}^{∗}. The p − 1 remaining irreducible characters all have degree p^{n} and are induced from any maximal elementary abelian subgroup of P_{n}. Let χ̂ be a nontrivial linear character of N. Then for each 1 ⩽ i ⩽ p − 1 there is an irreducible character χ_{i} of P_{n} whose restriction to any maximal elementary abelian subgroup M is the sum of all linear characters of M whose restriction to N is χ̂^{⊗i}.

For each ϕ ∈ E_{n}^{∗}, pick a representation ρ_{ϕ} of P_{n} whose character is linear, corresponding to ϕ. Let ρ_{1} be a representation of P_{n} affording the character χ_{1}.

Definition. For any finite group G, define h^{∗}(G) to be the quotient of the graded commutative ring H^{∗}(G, 𝔽_{p}) by its nilradical. Define ch(G) to be the subring of h^{∗}(G) generated by the images under the homomorphism H^{∗}(G, ℤ) → H^{∗}(G, 𝔽_{p}) → h^{∗}(G) of the Chern classes of the representations of G.

The reader is referred to the appendix of Atiyah’s paper [1] for a concise introduction to Chern classes of group representations. A proof of the following theorem may be found in Chapter 8 of Evens’ book [4].

Theorem 18. (Quillen) Let G be a finite group, and let ξ be a class in h^{∗}(G). Then ξ is zero if and only if Res_{E} ξ = 0 in h^{∗}(E) for every elementary abelian p-subgroup E of G.

Recall that Q_{n} is defined in terms of the polynomial algebra S(E_{n}^{∗}) ⊗_{𝔽p} 𝔽_{p}[K_{0}, …, K_{n−1}], denoted F_{n}.

Theorem 19. Let Z be an indeterminate. There is a unique 𝔽_{p}-algebra homomorphism f: F_{n} ⊗_{𝔽p} 𝔽_{p}[Z] → H^{∗}(p_{+}^{1+2n}, ℤ) ⊗_{ℤ} 𝔽_{p} which sends ϕ ∈ E_{n}^{∗} to c_{1}(ρ_{ϕ}), sends K_{r} to (−1)^{n−r}c_{p^{n}−p^{r}}(ρ_{1}), and sends Z to c_{p^{n}}(ρ_{1}). This homomorphism induces an isomorphism f̄: Q_{n} ⊗_{𝔽p} 𝔽_{p}[Z] → ch(p_{+}^{1+2n}).

Proof. Let ρ, σ be degree one representations of P_{n}. Since c_{1}(ρ ⊗ σ) = c_{1}(ρ)+c_{1}(σ), the algebra homomorphism f is well-defined; clearly it is unique.

Any elementary abelian p-group A may be viewed as an 𝔽_{p}-vector space. There is an isomorphism ch(A) → S(A^{∗}) which sends the first Chern class of any degree one representation to the corresponding element of A^{∗}. Moreover, ch(A) = h^{∗}(A).

For 1 ⩽ j ⩽ p − 1, let ρ_{j} be a representation of P_{n} which affords the character χ_{j}. Let I be any maximal totally isotropic subspace of E_{n}, and let M be the corresponding maximal elementary abelian subgroup of P_{n}. Then I^{∗} is the subspace of M^{∗} which annihilates N. Pick some γ ∈ h^{2}(M) such that Res_{N}(γ) = c_{1}(χ̂). If we restrict the total Chern class of ρ_{j} to M and apply the Whitney sum formula, we have

Res_{M} c(ρ_{j}) = ∏_{v∈I^{∗}}(1 + v + jγ) ,

= 1 + ∑_{r=0}^{n−1}(−1)^{n−r}D_{r}(I^{∗}) + j (γ^{p^{n}} + ∑_{r=0}^{n−1}(−1)^{n−r}D_{r}(I^{∗})γ^{p^{r}}) .

Quillen’s Theorem then says that c_{p^{n}−p^{r}}(ρ_{j}) = c_{p^{n}−p^{r}}(ρ_{1}) and c_{p^{n}}(ρ_{j}) = jc_{p^{n}}(ρ_{1}) in ch(P_{n}). Moreover, these are the only non-zero Chern classes of the induced representations. Hence the map from H^{∗}(P_{n}, ℤ) ⊗ 𝔽_{p} down to h^{∗}(G) maps Im(f) onto ch(P_{n}). Observe that Z is the only generator of F_{n} ⊗ 𝔽_{p}[Z] whose image under Res_{M} ∘f involves γ, and that Res_{M} f(Z) is transcendental over S(I^{∗}). Therefore, we only have to show that the induced map Q_{n} → ch(P_{n}) is both injective and well-defined. But, for every y ∈ F_{n} and for every I, the elements q_{I}(y) and Res_{M} f(y) of S(I^{∗}) are equal. The result then follows by Quillen’s Theorem and the definition of Q_{n}.

Remark. Observe that the maps f and f̄ double the degree of homogeneous elements.

7. A general inequality

It is the business of this section to prove that Q_{n} always strictly contains T_{n}. Specifically, we prove the following theorem.

Theorem 20. For every n ⩾ 1, we have κ_{n,0} ∉ T_{n}. Therefore c_{p^{n}−1}(ρ_{1}) lies outside the subring of ch(p_{+}^{1+2n}) generated by the first Chern classes and c_{p^{n}}(ρ_{1}).

Proof. Proposition 7 gives us the case n = 1. We shall prove the rest of the result by considering the inflation map π^{∗}: Q_{n+1} → Q_{1} ⊗_{𝔽p} Q_{n}, and showing that π^{∗}(κ_{n+1,0}) lies outside T_{1} ⊗ Q_{n} for all n ⩾ 1.

The inflation map is associated to the orthogonal direct sum decomposition E_{n+1} = E_{1} ⊥ E_{n}. For each maximal elementary abelian subgroup I_{n+1} of E_{n+1} this induces the decomposition I_{n+1} = I_{1} ⊕ I_{n}. Using this decomposition and Eqn. (1), we can express the Dickson invariants of I_{n+1} in terms of the Dickson invariants for I_{1} and I_{n}. In particular, if we define D_{n}(I_{n}) = 1, then

D_{0}(I_{n+1}^{∗}) = (D_{0}(I_{1}^{∗}) ⊗ D_{0}(I_{n}^{∗})) (∑_{j=0}^{n}(−1)^{j}D_{0}(I_{1}^{∗})^{\frac{p^{n−j} −1}{p−1}} ⊗ D_{n−j}(I_{n}^{∗}))^{p−1} ,

whence, also defining κ_{n,n} = 1, we have

π^{∗}(κ_{n+1,0}) = (κ ⊗ κ_{n,0}) (∑_{j=0}^{n}(−1)^{j}κ^{\frac{p^{n−j} −1}{p−1}} ⊗ κ_{n,n−j})^{p−1} .  (15)

Since κ^{r} ∈ T_{1} if and only if r ≠ 1, the right hand side of this equation equals κ ⊗ κ_{n,0}^{p} modulo T_{1} ⊗ Q_{n}.

8. Symplectic invariants

Closely related to the work of this paper is a question about symplectic invariants. The symplectic group Sp_{2n}(𝔽_{p}) is by definition the group of those linear transformations of E_{n} which preserve the nondegenerate symplectic form b. The invariants of the corresponding action of Sp_{2n}(𝔽_{p}) on S(E_{n}^{∗}) were determined by Carlisle and Kropholler, and are described in Section 8.3 of Benson’s book [2].

The ring of invariants in S(E_{n}^{∗}) is generated by R_{1}(E_{n}^{∗}), …, R_{2n−1}(E_{n}^{∗}), D_{n}(E_{n}^{∗}), …, D_{2n−1}(E_{n}^{∗}). Recall from Theorem 1 that the quotient of S(E_{n}^{∗}) by the ideal generated by the regular sequence R_{1}(E_{n}^{∗}), …, R_{n}(E_{n}^{∗}) is T_{n}. There is therefore an induced action of Sp_{2n}(𝔽_{p}) on T_{n}. It is natural to ask what is the ring of invariants of this action.

By the Tezuka–Yagita Theorem, every R_{r}(E_{n}^{∗}) is zero in T_{n}. Certainly every D_{r}(E_{n}^{∗}) is still invariant. But now there are other invariants as well.

Proposition 21. The natural action of Sp_{2n}(𝔽_{p}) on T_{n} has as ring of invariants the intersection of T_{n} with 𝔽_{p}[κ_{n,0}, …, κ_{n,n−1}].

Proof. The symplectic group permutes the maximal totally isotropic subspaces I of E_{n} transitively. In addition, for any I, every automorphism of I may be extended to a symplectic transformation on E_{n}. Hence, for every I and for every symplectic invariant x ∈ T_{n}, the element q̂_{I}(x) of S(I^{∗}) is invariant under GL(I), and this invariant is independent of I. Since the Dickson invariants in S(I^{∗}) generate the invariants under GL(I), it follows that x equals some polynomial over 𝔽_{p} in κ_{n,0}, …, κ_{n,n−1}. Conversely, any such polynomial is invariant under the action of Sp_{2n}(𝔽_{p}) on Q_{n}.

Theorem 22. The ring of invariants under the natural action of Sp_{4}(𝔽_{p}) on T_{2} is the subring of the polynomial algebra 𝔽_{p}[κ_{0}, κ_{1}] of polynomials whose support contains neither any κ_{0}κ_{1}^{r} with r ⩾ 0 nor any κ_{1}^{r} with p ∤ r. Over the polynomial algebra 𝔽_{p}[κ_{0}^{2}, κ_{1}^{p}], the ring of invariants is the free module generated by 1, κ_{0}^{2}κ_{1}^{s} for 1 ⩽ s ⩽ p − 1, and κ_{0}^{3}κ_{1}^{s} for 0 ⩽ s ⩽ p − 1.

Proof. Let f(x_{0}, x_{1}) be any polynomial in 𝔽_{p}[x_{0}, x_{1}]. By Proposition 4, f(κ_{0}, κ_{1}) belongs to T_{2} if and only if there exists a ∈ T_{2} such that

f(κ_{0}, κ_{1}) − f (D_{0}(α_{1}, β_{1}), D_{1}(α_{1}, β_{1})) = aγ_{2}

holds after inflation to Q_{1} ⊗ Q_{1}. That is, if and only if there exists a′ ∈ T_{1,1} such that, in Q_{1} ⊗ Q_{1},

f (κ′κ′′(κ′ − κ′′)^{p−1}, κ′′(κ′ − κ′′)^{p−1} + κ′^{p}) − f(0, κ′^{p}) = a′(κ′ − κ′′)^{p} . (16)

Observe that both sides of Eqn. (16) are divisible by (κ′ − κ′′)^{p−1}. Doing this to the right hand side yields a′(κ′ − κ′′). Suppose f is the monomial x_{0}^{r}x_{1}^{s}. If r ⩾ 2, then f(κ_{0}, κ_{1}) ∈ T_{2}. If r = 1, then the left hand side of Eqn. (16) is κ′κ′′(κ′−κ′′)^{p−1} (κ′′(κ′ − κ′′)^{p−1} + κ′^{p})^{s}, which is not divisible by (κ′−κ′′)^{p}. If r = 0, then get (κ′′(κ′ − κ′′)^{p−1} + κ′^{p})^{s} − κ′^{sp}, which is divisible by (κ′ − κ′′)^{p} if and only if p | s. The monomials x_{0}^{r}x_{1}^{s} such that κ_{0}^{r}κ_{1}^{s} ∉ T_{2} all have distinct degrees when evaluated at (κ_{0}, κ_{1}). Hence 𝔽_{p}[κ_{0}, κ_{1}] ∩ T_{2} is the subring of 𝔽_{p}[κ_{0}, κ_{1}] described in the statement. By Proposition 21 it follows that the ring of invariants is as claimed.

Remark. Using Theorem 17, we could in principle give expressions in terms of the α_{i} and β_{j} for each generator of this ring of invariants: however, these expressions would be very complicated. The current form of the result is likely to be the more illuminating.

9. The other extraspecial group

In this section we shall study the extraspecial group p_{−}^{1+4} of order p^{5} and exponent p^{2}, and determine the Chern ring ch(p_{−}^{1+4}).

The groups p_{+}^{1+2n} and p_{−}^{1+2n} are very similar. There is a central extension

1 → N → p_{−}^{1+2n} →^{ψ} E_{n} → 1, such that, for g_{1}, g_{2} ∈ p_{−}^{1+2n}, the commut-

ator [g_{1}, g_{2}] equals b (ψ(g_{1}), ψ(g_{2})). The element g of p_{−}^{1+2n} has exponent p if A_{1}^{∗}(ψ(g)) = 0. If not, then g has exponent p^{2}, and g^{p} lies in N, the centre of p_{−}^{1+2n}. The maximal elementary abelian subgroups of p_{−}^{1+2n} have p-rank n + 1, and are exactly the inverse images under ψ of those maximal totally isotropic subspaces of E which contain B_{1}. Once more, the irreducible representations consist of a one-dimensional representation ρ_{ϕ} for each ϕ ∈ E_{n}^{∗}, together with representations ρ_{1}, …, ρ_{p−1} induced from a maximal abelian subgroup, such that the restriction of ρ_{j} to any maximal elementary abelian subgroup M is the sum of all one-dimensional representations of M whose characters restrict to N as χ̂^{⊗j}.

Define Q_{n}^{−} to be the quotient of F_{n} by the intersection of the kernels of the maps q_{I} for all maximal totally isotropic subspaces I of E_{n} containing B_{1}. These are precisely those I such that q_{I}(A_{1}^{∗}) = 0. Observe that Q_{n}^{−} is a quotient of Q_{n}.

Theorem 23. Let Z be an indeterminate. There is a unique 𝔽_{p}-algebra homomorphism f: F_{n} ⊗_{𝔽p} 𝔽_{p}[Z] → H^{∗}(p_{−}^{1+2n}, ℤ) ⊗_{ℤ} 𝔽_{p} which sends ϕ ∈ E_{n}^{∗} to c_{1}(ρ_{ϕ}), sends K_{r} to (−1)^{n−r}c_{p^{n}−p^{r}}(ρ_{1}), and sends Z to c_{p^{n}}(ρ_{1}). This homomorphism induces an isomorphism f̄: Q_{n}^{−} ⊗_{𝔽p} 𝔽_{p}[Z] → ch(p_{−}^{1+2n}).

Proof. The proof of Theorem 19 is easily adapted to this case.

Remark. Once more, the maps f and f̄ double the degree of homogeneous elements.

Now we determine the structure of Q_{2}^{−}.

Theorem 24. The algebra Q_{2}^{−} has a presentation consisting of the generators α_{1}, α_{2}, β_{1}, β_{2}, κ_{0}, κ_{1} of Q_{2} together with the relations

α_{1} = 0  (17)
α_{2}β_{2}^{p} − α_{2}^{p}β_{2} = 0  (18)
κ_{0} = β_{1}^{p−1}κ_{1} − β_{1}^{p^{2}−1}  (19)
ε_{2}(κ_{1} − D_{1}(β_{1}, ε_{2})) = 0 for ε_{2} = α_{2}, β_{2}  (20)
(κ_{1} − D_{1}(β_{1}, α_{2}))(κ_{1} − D_{1}(β_{1}, β_{2})) = 0 .  (21)

In particular, (the image of) β_{1} is a non-zero divisor.

Proof. Denote by q_{−} the quotient map F_{n} → Q_{n}^{−}. Then, for any x ∈ F_{n} we have q_{−}(x) = 0 if and only if q_{I}(x) = 0 for all I containing B_{1}. Also, q_{−} factors through q: there is a unique algebra homomorphism q̂_{−}: Q_{n} → Q_{n}^{−} such that q_{−} = q̂_{−}q.

If I contains B_{1}, then q_{I}(B_{1}^{∗}) is a non-zero divisor and q_{I}(A_{1}^{∗}) is zero in S(I^{∗}). Hence q̂_{−}(β_{1}) is a non-zero divisor in Q_{n}^{−}, and q̂_{−}(α_{1}) = 0. Equation (18) is now an immediate consequence of Eqn. (5). Putting δ = β_{1} in Eqn. (8) and then cancelling yields Eqn. (19). Equation (18) implies that, for any I, the restrictions of either B_{1}^{∗}, A_{2}^{∗} or B_{1}^{∗}, B_{2}^{∗} must span I^{∗}; therefore Eqn. (20) and Eqn. (21) hold. The necessity of the relations is established.

By the Tezuka–Yagita Theorem for p_{−}^{1+4} (see Proposition 8.2 of [3]), all the relations in q̂_{−}(T_{2}) are generated by Eqn. (17) and Eqn. (18). It therefore only remains to prove that q̂_{−}(κ_{1}) ∉ q̂_{−}(T_{2}). Equation (4) implies that

κ_{1}^{p} = ∑_{i=0}^{p} β_{1}^{(p−i)(p^{2}−p)}(α_{2}^{p^{2}−p} − α_{2}^{(p−1)^{2}}β_{2}^{p−1} + β_{2}^{p^{2}−p})^{i} in Q_{2}^{−}.

But the right hand side is not the p^{th} power of any element of q̂_{−}(T_{2}).

References

[1] M. F. Atiyah. Characters and cohomology of finite groups. Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64.

[2] D. J. Benson. Polynomial Invariants of Finite Groups. London Math. Soc. Lecture Note Ser. no. 190 (Cambridge Univ. Press, 1993).

[3] D. J. Benson and J. F. Carlson. The cohomology of extraspecial groups. Bull. London Math. Soc. 24 (1992), 209–235. Erratum: Bull. London Math. Soc. 25 (1993), 498.

[4] L. Evens. The Cohomology of Groups. (Oxford Univ. Press, 1991).

[5] D. J. Green and I. J. Leary. Chern classes and extraspecial groups. Preprint No. 16 (1995), Inst. f. Exp. Math., Univ. GH Essen.

Department of Mathematics University of Chicago 5734 S. University Chicago, IL 60637 U.S.A.