The mod-3 Cohomology Ring of the O’Nan Sporadic Simple Group

Nicola Whitley McLallen

Author address: University of Illinois, 1409 W. Green, Urbana, Illinois 61801

E-mail address: whitley@math.uiuc.edu

THESIS

submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

in the Graduate College of the

University of Illinois at Urbana-Champaign, 2000

.

Research supported in part by NSF grant number DMS 9600106.

Contents

Chapter 1. Introduction 1

Chapter 2. Preliminary Definitions and Results 2 2.1. Definition of Cohomology 2 2.2. Relations to Cohomology of Subgroups 3 2.3. Cohomology of Elementary Abelian Groups 4

Chapter 3. The O’Nan Group 7 3.1. Action of G on E 7

Chapter 4. Invariants of the Polynomial Subalgebra 10 4.1. Invariant Theory 10 4.2. Generators For 𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}]^{G} 13

Chapter 5. The Cohomology Algebra H^{∗}(ON, 𝔽_{3}) 21 5.1. Polynomial Tensor Exterior Algebras 21 5.2. Generators for H^{∗}(ON, 𝔽_{3}) 24

Bibliography 32

Appendix A. Computer Output 33 A.1. MAGMA: Construction of Invariants 33 A.2. Relations among Secondary Invariants of R^{G} 34 A.3. MAGMA: Check Generators for R^{G} 39 A.4. MAGMA: Check Generators for (R ⊗ S_{1})^{G} 40 A.5. Macaulay 2: Check Generators for (R ⊗ S_{1})^{G} 44 A.6. Macaulay 2: Check Generators for (R ⊗ S_{2})^{G} 48

CHAPTER 1 Introduction

In recent years, there has been interest in computing the mod-p (for p a prime) cohomology rings of finite groups and, in particular, the finite simple groups. Many of the mod-2 cohomology rings have been computed, including those of the Mathieu groups M_{12} and M_{22} [1, 2], the Janko groups J_{2} and J_{3} [7] and the McLaughlin Group [4]. For p odd, less work has been done. In [13], Milgram computed the cohomology of the Mathieu group M_{23} for all primes p dividing the order of the group, and John Green computed the mod-3 cohomology ring of M_{24} in [9].

For the O’Nan sporadic simple group ON, the mod-2 cohomology was computed by Adem and Milgram in [3] and the mod-7 cohomology was computed by Tezuka and Yagita in [20]. In this thesis, the mod-3 cohomology ring H^{∗}(ON, 𝔽_{3}) of ON is computed.

Chapter 2 gives a brief summary of the definitions and results from cohomology theory that will be needed. These results imply that the cohomology ring H(ON, 𝔽_{3}) can be computed as the algebra of invariants of the action of a certain group G on a polynomial algebra tensored with an exterior algebra. In Chapter 3, a representation of that group G is given. This representation determines the action of G on the “polynomial-tensor-exterior algebra”. The invariants of the polynomial algebra are computed in Chapter 4, and in Chapter 5 the invariants of the entire polynomial-tensor-exterior algebra are computed. The main result is Theorem 5.2.7 which gives a complete list of generators for the cohomology algebra.

CHAPTER 2 Preliminary Definitions and Results

2.1. Definition of Cohomology

Let G be a finite group, and R a commutative ring. Let RG denote the group ring of G over R, and let M be a left RG-module. We regard R as a trivial RGmodule. The cohomology of G with coefficients in M is

H^{∗}(G, M) = Ext_{RG}^{∗}(R, M).

This is defined as follows. Take an RG-projective resolution (P_{∗}, ∂) of R, i.e. an exact sequence

⋯ ⟶ P_{n} ⟶^{∂_{n}} P_{n−1} ⟶ ⋯ ⟶ P_{1} ⟶^{∂_{1}} P_{0} ⟶^{ϵ} R ⟶ 0 (2.1.1)

where each P_{i} is a projective left RG-module and each map is an RG-homomorphism. Applying the functor Hom_{RG}(−, M) gives a cochain complex

⋯ ⟵ Hom_{RG}(P_{n}, M) ⟵^{∂_{n}^{∗}} Hom_{RG}(P_{n−1}, M) ⟵ ⋯ ⟵^{∂_{1}^{∗}} Hom_{RG}(P_{0}, M) ⟵^{∂_{0}^{∗}} 0 (2.1.2)

of R-modules, where ∂_{n}^{∗}, for n ≥ 1, is the map induced from ∂_{n}, i.e. ∂_{n}^{∗}(ϕ) = ϕ ∘ ∂_{n} for ϕ ∈ Hom_{RG}(P_{n−1}, M). Then the cohomology of G is the cohomology of this complex, i.e., the R-module

H^{∗}(G, M) = ⨁_{n=0}^{∞} H^{n}(G, M),

where H^{n}(G, M) = ker ∂_{n+1}^{∗}/im ∂_{n}^{∗} for all n = 0, 1, . . .. The low-dimensional cohomology groups can be identified by using the bar resolution (see [8, 2.3]). If M is a trivial RG-module, then H^{0}(G, M) ≃ M and H^{1}(G, M) ≃ Hom(G, M^{+}) as R-modules, where M^{+} denotes the additive group of M.

The cohomology H^{∗}(G, M) is a contravariant functor in G and a covariant functor in M. So a group homomorphism ϕ : G′ → G induces

ϕ^{∗} : H^{∗}(G, M) → H^{∗}(G′, M),

and an RG-homomorphism f : M → M′ induces

f^{∗} : H^{∗}(G, M) → H^{∗}(G, M′).

If

0 → M′ ⟶^{i} M ⟶^{j} M′′ → 0

is a short exact sequence of RG-modules then there is a long exact sequence of R-modules in cohomology,

⋯ → H^{n}(G, M) ⟶^{j^{∗}} H^{n}(G, M′′) ⟶^{δ} H^{n+1}(G, M′) ⟶^{i^{∗}} H^{n+1}(G, M) → · · · .

The connecting homomorphism δ is a natural map.

For any prime p, let 𝔽_{p} be a field of p elements. Then the Bockstein homomorphism β : H^{n}(G, 𝔽_{p}) → H^{n+1}(G, 𝔽_{p}) is the connecting homomorphism arising from the short exact sequence of trivial ℤG-modules,

0 → 𝔽_{p} → ℤ/p^{2}ℤ → 𝔽_{p} → 0.

From now on we shall consider the situation where R = k, a field, and M is the trivial kG-module k. Then H^{∗}(G, k) is a “graded-commutative” k-algebra, i.e., there is a product

H^{m}(G, k) × H^{n}(G, k) → H^{m+n}(G, k)

which satisfies αβ = (−1)^{mn}βα for α ∈ H^{m}(G, k) and β ∈ H^{n}(G, k). See [8, Chapter 3] for constructions and properties of this product, called the cup product. This product is functorial, in the sense that induced maps in cohomology are algebra homomorphisms. Note that the set of all elements of even degree forms a commutative k-subalgebra H^{ev}(G, k) of the algebra H^{∗}(G, k).

2.2. Relations to Cohomology of Subgroups

Let H be a subgroup of G. If (2.1.1) is a projective kG-resolution, it is also a projective kH-resolution. Then the inclusions

Hom_{kG}(P_{n}, k) ↪ Hom_{kH}(P_{n}, k), n ≥ 0

induce a map on cohomology, the restriction,

res_{G,H} : H^{∗}(G, k) → H^{∗}(H, k).

Now let σ ∈ G. For each n ≥ 0 there is a map

σ̃ : Hom_{kH}(P_{n}, k) → Hom_{k(σHσ^{−1})}(P_{n}, k)

defined by

σ̃(ϕ)(x) = σϕ(σ^{−1}x) for all x ∈ P_{n}.

These induce a conjugation map

σ^{∗} : H^{∗}(H, k) → H^{∗}(σHσ^{−1}, k).

The restriction and conjugation maps are algebra homomorphisms, because they are induced from group homomorphisms. (See [8, 4.1]). If H is normal in G, this conjugation map defines an action of G as automorphisms of the graded algebra H^{∗}(H, k). In fact, under this action H acts trivially on H^{∗}(H, k). So there is an induced action of G/H on H^{∗}(H, k).

For H any (not necessarily normal) subgroup of G, let S be a set of left coset representatives of H in G. Considering the projective resolution (2.1.1), define, for each n ≥ 0,

T_{G/H} : Hom_{kH}(P_{n}, k) → Hom_{kG}(P_{n}, k)

by

T_{G/H}(ϕ) = ∑_{σ∈S} σ̃(ϕ).

Then these maps T_{G/H} are independent of the choice of coset representatives and commute with the differential ∂. So they induce a homomorphism

tr_{H,G} : H^{∗}(H, k) → H^{∗}(G, k)

of graded k-vector spaces, called the transfer. Unlike the restriction and conjugation maps, the transfer is not an algebra homomorphism. The restriction and transfer maps are related by Proposition 2.2.1. If H is a subgroup of G, then

tr_{H,G} ∘ res_{G,H} = |G : H| · Id,

where Id is the identity automorphism of H^{∗}(G, k).

Applying this proposition in the case where H is the trivial subgroup {1}, and using the fact that H^{n}({1}, k) = 0 for n > 0, we get

Corollary 2.2.2. |G| · H^{n}(G, k) = {0} for n > 0.

We are interested in computing the cohomology of G over a field k of characteristic p, where p is a prime that divides the order of G. In this case, we can restrict our attention to a Sylow p-subgroup of G.

Proposition 2.2.3. Let P be a Sylow p-subgroup of G for a prime p, and k a field of characteristic p. Then res_{G,P} : H^{n}(G, k) → H^{n}(P, k) is injective for n ≥ 0. Furthermore, α ∈ H^{n}(P, k) is in the image of res_{G,P} if and only if

res_{P,P ∩σP σ^{−1}}(α) = res_{σP σ^{−1},P ∩σP σ^{−1}}(σ^{∗}α) (2.2.4)

for every σ ∈ G.

Proof. Note that for each n ≥ 0, the cohomology module H^{n}(G, k) is equal to its p-primary component H^{n}(G, k)_{p}. So the Proposition follows from [8, Theorem 4.2.5 and Corollary 4.2.7].

Elements α ∈ H^{n}(P, k) satisfying (2.2.4) are called stable elements. In the special case where the Sylow p-subgroups of G are abelian, the next result, due to Swan [19], gives a characterization of the set of these stable elements as a ring of invariants.

Proposition 2.2.5. Let p be a prime such that the Sylow p-subgroups of G are abelian. Let P be one such Sylow p-subgroup and let N be the normalizer of P in G. Then for n ≥ 0, the set of stable elements of H^{n}(P, k) is the set H^{n}(P, k)^{N/P} of elements of H^{n}(P, k) that are fixed under the conjugation action of N/P.

Proof. See [8, Theorem 4.2.8].

Now, putting Propositions 2.2.3 and 2.2.5 together, we obtain

Theorem 2.2.6. Let G be a finite group and k a field of characteristic p. If a Sylow p-subgroup P of G is abelian, then restriction is an isomorphism of kalgebras:

H^{∗}(G, k) ≃ H^{∗}(P, k)^{N_{G}(P)/P} .

2.3. Cohomology of Elementary Abelian Groups

We will be computing the cohomology of the O’Nan simple group over the field 𝔽_{3} of 3 elements. The Sylow 3-subgroups of this group are elementary abelian. So we now discuss the cohomology of such groups.

Let k be a field of characteristic p, where p is an odd prime. If C is a cyclic group of order p, then the cohomology of C over k is well known. (See, for example, [8, 3.2].) For each n ≥ 0, we have H^{n}(C, k) ≃ k as vector spaces. If α ∈ H^{m}(C, k) for m odd, then α^{2} = 0. Let x generate H^{2}(C, k) and y generate H^{1}(C, k). Then x^{n} is a generator of H^{2n}(C, k). Hence H^{ev}(C, k) is a polynomial algebra k[x] in x, and

H^{∗}(C, k) = H^{ev}(C, k) ⊕ H^{ev}(C, k) · y.

Here y commutes with x and y^{2} = 0. Note that x^{n}y is a basis element of H^{2n+1}(C, k) for any integer n ≥ 0. Since y^{2} = 0, we can write

H^{∗}(C, k) ≃ k[x] ⊗ Λ(y), (2.3.1)

the tensor product over k of a polynomial algebra in x and an exterior algebra in y.

Let E be an elementary abelian group of order p^{d} for an arbitrary d ≥ 0. Then E ≃ C^{d} where C is cyclic of order p. The Künneth Theorem [8, 2.5] gives the cohomology of any direct product G_{1} × G_{2} of two groups G_{1} and G_{2} over a field k as

H^{∗}(G_{1} × G_{2}, k) ≃ H^{∗}(G_{1}, k) ⊗ H^{∗}(G_{2}, k), (2.3.2)

with grading given by

H^{i}(G_{1} × G_{2}, k) = ⨁_{j=0}^{i} H^{j}(G_{1}, k) ⊗ H^{j−i}(G_{2}, k).

The tensor product over k on the right of (2.3.2) is a k-algebra with product given by

(α_{1} ⊗ α_{2})(β_{1} ⊗ β_{2}) = (−1)^{(deg α_{2})(deg β_{1})}α_{1}β_{1} ⊗ α_{2}β_{2}

for homogeneous elements α_{1}, α_{2} ∈ H^{∗}(G_{1}, k) and β_{1}, β_{2} ∈ H^{∗}(G_{2}, k). Then the isomorphism (2.3.2) is an isomorphism of k-algebras. Applying this to the group E ≃ C^{d}, we conclude that the cohomology of E is isomorphic to a tensor product of d copies of H^{∗}(C, k),

H^{∗}(E, k) ≃ (H^{∗}(C, k))^{⊗d}.

It follows (see [8, 3.5]) that

H^{∗}(E, k) ≃ k[x_{1}, …, x_{d}] ⊗ Λ(y_{1}, …, y_{d}), (2.3.3)

the tensor product over k of a polynomial algebra in commuting variables x_{1}, …, x_{d} and an exterior algebra in y_{1}, …, y_{d}. The generators x_{i} and y_{i} are obtained as follows. For i = 1, …, d, let π_{i} : E → C_{i} be the projection of E onto its ith cyclic factor C_{i}. This projection induces a monomorphism of graded cohomology k-algebras,

π_{i}^{∗} : H^{∗}(C_{i}, k) → H^{∗}(E, k).

Let x̃_{i} and ỹ_{i} be generators of H^{∗}(C_{i}, k) in degrees 2 and 1 respectively, so that H^{∗}(C_{i}, k) ≃ k[x̃_{i}] ⊗ Λ(ỹ_{i}) (as in (2.3.1)). We may take x_{i} = π_{i}^{∗}(x̃_{i}) and y_{i} = π_{i}^{∗}(ỹ_{i}). Then {x_{1}, …, x_{d}, y_{1}, …, y_{d}} generate H^{∗}(E, k). Note that {y_{1}, …, y_{d}} is a k-basis for H^{1}(E, k).

Now let us consider the case k ≃ 𝔽_{p}, where we can apply the Bockstein homomorphism β defined in Chapter 2.1.

Theorem 2.3.4. If E is an elementary abelian group of order p^{d}, then

H^{∗}(E, 𝔽_{p}) ≃ 𝔽_{p}[x_{1}, …, x_{d}] ⊗ Λ(y_{1}, …, y_{d}),

where {y_{1}, …, y_{d}} is an 𝔽_{p}-basis of H^{1}(E, 𝔽_{p}) and x_{i} = β(y_{i}) ∈ H^{2}(E, 𝔽_{p}) for i = 1, …, d.

Proof. First, consider a cyclic factor C_{i} of E. By (2.3.1),

H^{∗}(C_{i}, 𝔽_{p}) ≃ 𝔽_{p}[x′_{i}] ⊗ Λ(y′_{i}). (2.3.5)

Consider the Bockstein homomorphism β : H^{1}(C_{i}, 𝔽_{p}) → H^{2}(C_{i}, 𝔽_{p}). By definition of β, the sequence

⋯ → H^{1}(C_{i}, ℤ/p^{2}ℤ) ⟶^{j^{∗}} H^{1}(C_{i}, 𝔽_{p}) ⟶^{β} H^{2}(C_{i}, 𝔽_{p}) ⟶^{i^{∗}} H^{2}(C_{i}, ℤ/p^{2}ℤ) → ⋯

is exact. The maps i^{∗} and j^{∗} are the maps induced from i and j respectively in the short exact sequence

0 → 𝔽_{p} ⟶^{i} ℤ/p^{2}ℤ ⟶^{j} 𝔽_{p} → 0.

Under the natural isomorphisms H^{1}(C_{i}, ℤ/p^{2}ℤ) ≃ Hom(C_{i}, ℤ/p^{2}ℤ) and H^{1}(C_{i}, 𝔽_{p}) ≃ Hom(C_{i}, 𝔽_{p}), the induced map j^{∗} corresponds to composition with j. Since C_{i} is cyclic of order p, this map j^{∗} is the zero map. Therefore β is injective. But H^{2}(C_{i}, 𝔽_{p}) ≃ 𝔽_{p} ≃ H^{1}(C_{i}, 𝔽_{p}). So β is an isomorphism

β : H^{1}(C_{i}, 𝔽_{p}) ≃ H^{2}(C_{i}, 𝔽_{p}).

Therefore we may choose x′_{i} = β(y′_{i}) in (2.3.5).

Let x_{i} and y_{i} be the images of x′_{i} and y′_{i}, respectively, under the monomorphism

π_{i}^{∗} : H^{∗}(C_{i}, 𝔽_{p}) → H^{∗}(E, 𝔽_{p}),

where π_{i} is the projection of E = C_{1} × ⋯ × C_{d} onto C_{i}. Then H^{∗}(E, 𝔽_{p}) ≃ 𝔽_{p}[x_{1}, …, x_{d}] ⊗ Λ(y_{1}, …, y_{d}) by (2.3.3). Since β is natural, it commutes with the maps induced from the projections. So we have x_{i} = β(y_{i}) for i = 1 …, d.

Suppose that the preceding E is a Sylow p-subgroup of a group H. Then G = N_{H}(E)/E acts on H^{∗}(E, 𝔽_{p}). Let Y = H^{1}(E, 𝔽_{p}), an 𝔽_{p}-vector space with basis {y_{1}, …, y_{d}}. The vector space X with basis {x_{1}, …, x_{d}} is isomorphic to Y under the Bockstein β. Since β commutes with the action of G, the space X is G-invariant. So the action of G on H^{∗}(E, 𝔽_{p}) is determined by its action on the 𝔽_{p}G-module Y. We also view the vector space E ≃ (ℤ_{p})^{d} as an 𝔽_{p}G-module, with the action of G given by conjugation in the group H. As an 𝔽_{p}G-module, Y is isomorphic to Hom(E, 𝔽_{p}). This isomorphism determines a G-invariant duality

ϕ : Y × E → 𝔽_{p}.

If R : G → Aut(Y) and \overline{R} : G → Aut(E) are the representations of G on Y and E repectively, then R and \overline{R} are dual representations. In terms of matrices, R and \overline{R} are related as follows. Let \{e_{1}, …, e_{d}\} be an 𝔽_{p}-basis for E, dual to the basis {y_{1}, …, y_{d}} of Y with respect to ϕ. The basis {e_{i}} for E determines a matrix representation of \overline{R}, say \overline{ρ} : G → GL_{d}(𝔽_{p}). Similarly, \{y_{i}\} determines a representation ρ : G → GL_{d}(𝔽_{p}) of R. Then ρ(σ) is the transpose inverse (\overline{ρ}(σ))^{−t} of the matrix \overline{ρ}(σ), for each σ ∈ G. The space X is isomorphic to Y as 𝔽_{p}G-modules under β. So with respect to the basis {x_{i}} = {β(y_{i})} of X, the action of G on X is also given by ρ.

CHAPTER 3 The O’Nan Group

Let ON denote the O’Nan simple group. This group was defined by O’Nan in [14]. We are going to compute the cohomology of ON over the field 𝔽_{3}. This cohomology is determined by the 3-local structure of ON, which is described in [14, 5.13-5.14]. A Sylow 3-subgroup of ON is an elementary abelian group E ≅ (ℤ_{3})^{4} which is self-centralizing in ON. So G = N_{ON}(E)/E acts faithfully on E. This group G has a normal subgroup L ≅ Q_{8} ∗ D_{8}, an extraspecial group of order 2^{1+4} which is a central product of a quaternion group Q_{8} and a dihedral group D_{8}, both of order eight. G is a non-split extension of L with factor group G/L ≅ D_{10}, a dihedral group of order 10 acting faithfully on L/Z(L).

3.1. Action of G on E

We construct an explicit representation for G on E. Here we identify E with the vector space of all 1 × 4 row vectors over 𝔽_{3} with basis {e_{1}, e_{2}, e_{3}, e_{4}}, where

e_{1} = (1, 0, 0, 0), e_{2} = (0, 1, 0, 0), e_{3} = (0, 0, 1, 0), e_{4} = (0, 0, 0, 1).

So the automorphism group of E can be identified with the general linear group GL_{4}(3) acting on E from the right.

Lemma 3.1.1. In GL_{4}(3), the matrices

A_{1} = ( \begin{matrix}& & 1 & 0 \\ & & 0 & 1 \\ −1 & 0 \\ 0 & −1 \end{matrix} ) ,  A_{2} = ( \begin{matrix}0 & 1 \\ −1 & 0 \\ & & 0 & −1 \\ & & 1 & 0 \end{matrix} ) ,
B_{1} = ( \begin{matrix}& & 0 & 1 \\ & & 1 & 0 \\ 0 & −1 \\ −1 & 0 \end{matrix} ) and B_{2} = ( \begin{matrix}0 & 1 & 1 & 0 \\ −1 & 0 & 0 & −1 \\ −1 & 0 & 0 & 1 \\ 0 & 1 & −1 & 0 \end{matrix} )

generate a subgroup ℒ that is isomorphic to Q_{8} ∗ D_{8}. Up to equivalence, this is the only such subgroup of GL_{4}(3).

Proof. The matrices A_{1} and A_{2} satisfy A_{1}^{2} = A_{2}^{2} = −I and A_{1}A_{2} = −A_{2}A_{1}. So they generate a subgroup ℒ_{1} isomorphic to Q_{8} with center generated by A_{1}^{2} = A_{2}^{2} = −I. This subgroup is centralized by B_{1} and B_{2}, which satisfy B_{1}^{4} = B_{2}^{2} = I and B_{2}B_{1}B_{2} = B_{1}^{−1}. Therefore ℒ_{2} = ⟨B_{1}, B_{2}⟩ is isomorphic to D_{8}, with center generated by B_{1}^{2} = −I. So ℒ = ⟨A_{1}, A_{2}, B_{1}, B_{2}⟩ is a central product of ℒ_{1} and ℒ_{2} with center ⟨−I⟩.

The group ℒ ≃ Q_{8} ∗ D_{8} has 16 complex linear characters and one irreducible character of degree 4. None of the linear characters is faithful, as they are all trivial on the center of ℒ. Therefore, since ℒ is a 2-group, it has, up to equivalence, only one faithful representation in GL_{4}(3).

Lemma 3.1.2. The subgroup ℒ defined in Lemma 3.1.1 preserves the symplectic form 𝔉 on E defined by

𝔉(X, Y) = XB_{1}Y^{t} for X, Y ∈ E, (3.1.3)

where B_{1} is the matrix defined in Lemma 3.1.1. Up to sign, this is the unique symplectic form preserved by ℒ.

Proof. To show that ℒ preserves the form 𝔉, it suffices to show that MB_{1}M^{t} = B_{1} for each M ∈ {A_{1}, A_{2}, B_{1}, B_{2}}. Each M ∈ {A_{1}, A_{2}, B_{1}} centralizes B_{1} and satisfies M^{t} = −M = M^{−1}. So MB_{1}M^{t} = B_{1}. Also, B_{2}^{t} = −B_{2}. So B_{2}B_{1}B_{2}^{t} = −B_{2}B_{1}B_{2} = −B_{1}^{−1} = B_{1}, as required. The linear group ℒ is absolutely irreducible over 𝔽_{3}. So the only forms it preserves are the scalar multiples of 𝔉, that is, 𝔉 and −𝔉.

Lemma 3.1.4. Let ℒ and 𝔉 be as defined in Lemmas 3.1.1 and 3.1.2. If 𝒢 is a subgroup of GL_{4}(3) containing ℒ as a normal subgroup with 𝒢/ℒ ≃ D_{10}, then 𝒢 also preserves the form 𝔉. Furthermore, 𝒢 is unique up to conjugation in 𝒩, the normalizer of ℒ in GL_{4}(3).

Proof. The linear group ℒ is absolutely irreducible and contains ±I. So ℒ is self-centralizing in GL_{4}(3). Any automorphism of ℒ must leave invariant (up to equivalence) the unique faithful irreducible 𝔽_{3}-representation of ℒ on E and therefore is induced by a conjugation in GL_{4}(3). Hence 𝒩/ℒ ≃ Out(ℒ) ≃ S_{5}. Here Out(ℒ) denotes the outer automorphism group of ℒ. Now 𝒩/ℒ acts on the form 𝔉, with image {±𝔉}. The subgroup that preserves 𝔉 is a subgroup of index at most 2 in 𝒩/ℒ ≃ S_{5}, so it contains the copy of A_{5} in 𝒩/ℒ. The group 𝒢 is a subgroup of 𝒩 with 𝒢/ℒ ≃ D_{10}. But the only subgroups of S_{5} isomorphic to D_{10} are contained in A_{5}. In fact, such a subgroup is unique up to conjugation in A_{5} because it is the normalizer of a Sylow 5-subgroup in A_{5}. Therefore 𝒢 preserves 𝔉 and is unique up to conjugation in 𝒩.

Conjugation in ON induces a monomorphism \overline{ρ} of G into GL_{4}(3). The normal subgroup L ≃ Q_{8} ∗ D_{8} of G has, to within equivalence, a unique faithful representation on E. So we may assume that \overline{ρ}(L) is the subgroup ℒ = ⟨A_{1}, A_{2}, B_{1}, B_{2}⟩ of Lemma 3.1.1. Lemma 3.1.2 shows that ℒ leaves the form 𝔉 invariant. Then so does \overline{ρ}(G), by Lemma 3.1.4. Hence \overline{ρ}(G) is contained in a symplectic group SP_{4}(3). Using GAP to compute the normalizer of ℒ, we check that

T = ( \begin{matrix}1 & 1 & −1 & 1 \\ −1 & 0 & −1 & 0 \\ 0 & −1 & 0 & 1 \\ −1 & 1 & 1 & 1 \end{matrix} ) and W = ( \begin{matrix}−1 & 1 & 1 & 0 \\ 0 & 1 & −1 & 1 \\ 1 & 0 & −1 & −1 \\ 1 & 1 & 0 & 1 \end{matrix} )

normalize ℒ and leave the form 𝔉 invariant. Since these matrices satisfy T^{5} = I, W^{2} = −I, T^{W} = T^{4}, they generate a subgroup isomorphic to 2.D_{10}. By Lemma 3.1.4 this subgroup is unique up to conjugation, and we can therefore take \overline{ρ}(G) to be generated by {A_{1}, A_{2}, B_{1}, B_{2}, T, W }.

Theorem 3.1.5. The cohomology algebra of ON over 𝔽_{3} is isomorphic to the invariant algebra

(𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}] ⊗ Λ(y_{1}, y_{2}, y_{3}, y_{4}))^{G}

of the tensor product of a polynomial algebra 𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}] in four commuting variables x_{1}, …, x_{4} with the exterior algebra Λ(y_{1}, y_{2}, y_{3}, y_{4}) in y_{1}, …, y_{4}. Here G acts on the vector spaces X with basis {x_{1}, …, x_{4}} and Y with basis {y_{1}, …, y_{4}} via the representation ρ : G → GL_{4}(3) as follows. For σ ∈ G,

x_{i}σ = ∑_{j=1}^{4} ρ(σ)_{ij}x_{j} and y_{i}σ = ∑_{j=1}^{4} ρ(σ)_{ij}y_{j}.

The image ρ(G) of G is generated by

α_{1} = ρ(A_{1}) = ( \begin{matrix}& & 1 & 0 \\ & & 0 & 1 \\ −1 & 0 \\ 0 & −1 \end{matrix} ) ,  α_{2} = ρ(A_{2}) = ( \begin{matrix}0 & 1 \\ −1 & 0 \\ & & 0 & −1 \\ & & 1 & 0 \end{matrix} ) ,
β_{1} = ρ(B_{1}) = ( \begin{matrix}& & 0 & 1 \\ & & 1 & 0 \\ 0 & −1 \\ −1 & 0 \end{matrix} ) ,  β_{2} = ρ(B_{2}) = ( \begin{matrix}0 & −1 & −1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & −1 \\ 0 & −1 & 1 & 0 \end{matrix} ) ,
τ = ρ(T) = ( \begin{matrix}1 & 1 & −1 & 1 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & −1 \\ −1 & 1 & 1 & 1 \end{matrix} ) and ω = ρ(W) = ( \begin{matrix}1 & 0 & −1 & −1 \\ −1 & −1 & 0 & −1 \\ −1 & 1 & 1 & 0 \\ 0 & −1 & 1 & −1 \end{matrix} ) .

Proof. Since a Sylow 3-subgroup E of G is abelian, Theorem 2.2.6 gives

H^{∗}(ON, 𝔽_{3}) ≃ H^{∗}(E, 𝔽_{3})^{G}.

By Theorem 2.3.4,

H^{∗}(E, 𝔽_{3}) = 𝔽_{3}[x_{1}, …, x_{d}] ⊗ Λ(y_{1}, …, y_{d}).

The representation \overline{ρ} : G → GL_{4}(3) above gives the action of G on E with respect to the basis \{e_{1}, …, e_{d}\} of E. Then G acts on X and Y via the dual ρ of \overline{ρ}. Generators for this action are α_{1}, α_{2}, β_{1}, β_{2}, τ, ω, which are the transpose inverses of A_{1}, A_{2}, B_{1}, B_{2}, T, W, respectively.

CHAPTER 4 Invariants of the Polynomial Subalgebra

In this chapter we find generators for the commutative subring 𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}]^{G} of the cohomology ring.

4.1. Invariant Theory

We begin with standard results on the polynomial invariants of finite groups. This material is covered in the books by Smith [15] and Sturmfels [18], and the article by Stanley [17]. Here X is an n-dimensional vector space over a field K with basis {x_{1}, …, x_{n}}. Let G be a finite group which acts on X via the representation ρ : G → GL_{n}(K). Then G also acts on the polynomial ring R = K[x_{1}, …, x_{n}] as follows. If x denotes the column vector (x_{1}, x_{2}, x_{3}, x_{4})^{t}, then for σ ∈ G and f ∈ R,

(fσ)(x) = f(ρ(σ) · x).

The ring R is graded, and each component R_{j} is a KG-module with K-basis consisting of all monomials in x_{1}, …, x_{n} of degree j. We are interested in computing the invariant subalgebra

R^{G} = {f ∈ R | fσ = f for all σ ∈ G}.

The Poincaré Series. The ring of invariants R^{G} is graded with jth homogeneous component R_{j}^{G} consisting of the invariant homogeneous polynomials in x_{1}, …, x_{n} of degree j. The dimensions of the homogeneous components are the coefficients in the Poincaré series

P (R^{G}, t) = ∑_{j=0}^{∞}(dim_{K} R_{j}^{G})t^{j},

a formal power series in the variable t. In the case where K is a field of characteristic zero, a formula for this series is given by the following classical result.

Theorem 4.1.1 (Molien’s Theorem). If K is a field of characteristic zero and G is a finite group acting on R = K[x_{1}, … x_{n}] via ρ : G → GL_{n}(K), then the Poincaré series of the invariant ring R^{G} is

P (R^{G}, t) = \frac{1}{|G|} ∑_{σ∈G} \frac{1}{det(I − ρ(σ)t)}.

Proof. We assume that K contains all the exp(G)th-roots of unity, by replacing K with a larger extension field if necessary. Each R_{j} is a KG-module, say with character χ_{j}. Then by character theory,

dim_{K} R_{j}^{G} = \frac{1}{|G|} ∑_{σ∈G} χ_{j}(σ).

Now for σ ∈ G, let λ_{1}, …, λ_{n} be the eigenvalues of ρ(σ). Then the eigenvalues for the action of σ on R_{j} are the products λ_{1}^{a_{1}} ⋯ λ_{n}^{a_{n}} for integers a_{i} ≥ 0 with a_{1} + ⋯ + a_{n} = j. So

χ_{j}(σ) = ∑_{\begin{matrix}a_{1}+···+a_{n}=j \\ a_{1},...,a_{n}≥0 \end{matrix}} λ_{1}^{a_{1}} ⋯ λ_{n}^{a_{n}},

and

∑_{j=0}^{∞} χ_{j}(σ)t^{j} = ∑_{j=0}^{∞} ( ∑_{a_{1}+···+a_{n}=j} λ_{1}^{a_{1}} ⋯ λ_{n}^{a_{n}} ) t^{j}

= ∏_{i=1}^{n} \frac{1}{1 − λ_{i}t}

= \frac{1}{det(I − ρ(σ)t)}.

Therefore

∑_{j=0}^{∞}(dim_{K} R_{j}^{G})t^{j} = ∑_{j=0}^{∞} \frac{1}{|G|} ∑_{σ∈G} χ_{j}(σ)

= \frac{1}{|G|} ∑_{σ∈G} ∑_{j=0}^{∞} χ_{j}(σ)

= \frac{1}{|G|} ∑_{σ∈G} \frac{1}{det(I − ρ(σ)t)}.

Remark 4.1.2. Molien’s Theorem also applies if K is a field of non-zero characteristic p prime to the order of G, by lifting eigenvalues to characteristic zero as follows. Suppose, by extending K if necessary, that K contains a primitive exp(G)- th root of unity z′. Let ℤ[z] be the extension ring of ℤ generated by a primitive exp(G)- th root of unity z ∈ ℂ. Then there is a unique homomorphism γ of the ring ℤ[z] into K sending z to z′. Since the exponent exp(G) of G is relatively prime to p, this homomorphism sends the cyclic group < z > isomorphically onto < z′ >. So there is a unique isomorphism

δ :< z′ >⟶^{∼}< z >

such that γ ∘ δ is the identity map of < z′ > onto itself. The order of any σ ∈ G divides exp(G). It follows that the eigenvalues λ_{1}, λ_{2}, …, λ_{n} of ρ(σ) all lie in < z′ >. Hence we can lift them via δ to λ_{1}^{∗}, … λ_{n}^{∗} ∈< z >⊆ ℂ. Notice that det(I − ρ(σ)t) = (1 − λ_{1}t) ⋯ (1 − λ_{n}t). So define det^{∗} by

det^{∗}(I − ρ(σ)t) = (1 − λ_{1}^{∗}t) ⋯ (1 − λ_{n}^{∗}t). (4.1.3)

Then we have

P (R^{G}, t) = \frac{1}{|G|} ∑_{σ∈G} \frac{1}{det^{∗}(I − ρ(σ)t)}. (4.1.4)

The Cohen-Macaulay Property. Another classical result of invariant theory is that the invariant algebra is Cohen-Macaulay, a property defined by the theorem below. We begin with some standard definitions and results of commutative algebra.

Let B = B_{0} ⊕ B_{1} ⊕ ⋯ be a graded K-algebra that is finitely generated over K, and hence noetherian. Suppose that B has Krull dimension n, i.e., n is the maximal number of elements of B which are algebraically independent over K. A set {θ_{1}, …, θ_{n}} of n homogeneous elements of (not necessarily the same) positive degrees is called an homogeneous system of parameters (h.s.o.p.) if B is finitely generated as a module over its subring K[θ_{1}, …, θ_{n}]. This implies that θ_{1}, …, θ_{n} are algebraically independent. An h.s.o.p. for B always exists by the Noether Normalization Lemma. B is said to be Cohen-Macaulay if it satisfies the two equivalent conditions of the theorem below.

Theorem 4.1.5. Let θ_{1}, …, θ_{n} be an h.s.o.p. for the graded K-algebra B. Then the following two conditions are equivalent.

(a) B is a finitely generated free module over K[θ_{1}, …, θ_{n}]. In other words, there exist homogeneous η_{1}, …, η_{m} ∈ B such that

B = ⨁_{i=1}^{m} η_{i}K[θ_{1}, …, θ_{n}], (4.1.6)

where η_{i}K[θ_{1}, …, θ_{n}] ≃ K[θ_{1}, …, θ_{n}] for all i = 1, …, n.

(b) For every h.s.o.p ϕ_{1}, …, ϕ_{n} of B, the ring B is a finitely generated free K[ϕ_{1}, …, ϕ_{n}]-module

If condition (a) holds, then the homogeneous elements η_{1}, …, η_{m} satisfy (4.1.6) if and only if their images form a K-vector space basis of the quotient algebra B/I, where I is the ideal of B generated by θ_{1}, …, θ_{n}.

Proof. See [18, Theorem 2.3.1].

If K is a field of characteristic zero or of prime characteristic not dividing |G|, then we have the following theorem of Hochster and Eagon.

Theorem 4.1.7. [10, Proposition 13] If G is a finite group and K is a field of characteristic not dividing |G|, then the invariant algebra K[x_{1}, …, x_{n}]^{G} is Cohen-Macaulay.

These results imply that R^{G} contains invariants θ_{1}, …, θ_{n}, called primary invariants, and η_{1}, …, η_{m}, called secondary invariants, such that

R^{G} = ⨁_{i=1}^{m} η_{i}K[θ_{1}, …, θ_{n}],

where η_{i}K[θ_{1}, …, θ_{n}] ≃ K[θ_{1}, …, θ_{n}] for all i = 1, …, n. In this case, the Poincaré Series is

P (R^{G}, t) = \frac{∑_{i=1}^{m} t^{deg η_{i}}}{∏_{j=1}^{n}(1 − t^{deg θ_{j}})}. (4.1.8)

Constructing Invariants. Now we describe techniques for constructing elements of the invariant ring R^{G}.

A basic tool for the construction of invariants is the Reynolds Operator,

ℛ_{G} : K[x_{1}, …, x_{n}] → K[x_{1}, …, x_{n}]^{G}

defined by

ℛ_{G}(f) = \frac{1}{|G|} ∑_{σ∈G} fσ.

By applying the Reynolds Operator to monomials in R, invariants can be constructed. Algorithms to compute generators for the invariant ring of G using the Reynolds Operator and other techniques are discussed by Sturmfels [18] and Kemper [11, 12]. Algorithms have also been implemented in the MAGMA computer algebra system by Kemper. For more information, see [6] and [12].

Invariants can also be constructed via orbit Chern classes, which we describe now, following [15, 3.1]. The group G acts on the vector space R_{d} of all homogeneous polynomials in x_{1}, …, x_{n} of any given degree d. If 𝒪 is an orbit of the action of G on this space, then define the orbit polynomial of 𝒪,

ϕ_{𝒪} (Y) = ∏_{g∈𝒪}(Y + g)

as an element of R[Y] where Y is a new variable. Since 𝒪 is an invariant subset of R_{d}, the polynomial ϕ_{𝒪} (Y) is an element of R^{G}[Y]. If we write

ϕ_{𝒪} (Y) = ∑_{i+j=|𝒪|} c_{i}(𝒪)Y^{j},

then each c_{i}(𝒪) is an invariant, called the ith orbit Chern class of the orbit 𝒪.

4.2. Generators For 𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}]^{G}

Now we return to our group G, and compute generators for 𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}]^{G}. The group G acts on R = 𝔽_{3}[x_{1}, x_{2}, x_{3}, x_{4}] via ρ, as given by Theorem 3.1.5. First, we apply Molien’s Theorem to compute the Poincaré Series. Table 4.2.1 lists, for a complete set of conjugacy class representatives {σ_{i}} of G, the values of ρ(σ_{i}) and det^{∗}(I − ρ(σ_{i})t), as defined in (4.1.3). To compute det^{∗}(I − ρ(σ_{i})t) for each σ_{i} we computed the eigenvalues λ_{1}, λ_{2}, λ_{3}, λ_{4} of ρ(σ_{i}) in an extension field of 𝔽_{3} of order 3^{4}. We also give the values of the Brauer character χ of ρ. This character is defined by

χ(σ_{i}) = λ_{1}^{∗} + λ_{2}^{∗} + λ_{3}^{∗} + λ_{4}^{∗},

where λ_{1}^{∗}, λ_{2}^{∗}, λ_{3}^{∗}, λ_{4}^{∗} are the lifts to ℂ of λ_{1}, λ_{2}, λ_{3}, λ_{4} defined in Remark 4.1.2. We shall need these values of χ later.

Lemma 4.2.1. The Poincaré Series for R^{G} has the form

P (R^{G}, t) = \frac{1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26}}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})} . (4.2.2)

ρ(σ_{i}) Order |G|/|C_{G}(σ_{i})| det^{∗}(I − ρ(σ_{i})t) χ(σ_{i})
I  1  1  (1 − t)^{4}  4
−I  2  1  (1 + t)^{4}  −4
β_{2}  2  10  (1 − t^{2})^{2}  0
α_{1}  4  10  (1 + t^{2})^{2}  0
α_{2}  4  10  (1 + t^{2})^{2}  0
β_{1}ω  4  20  (1 + t^{2})(1 − t)^{2}  2
α_{2}ω  4  20  (1 + t^{2})(1 + t)^{2} −2
ω  4  40  (1 + t^{2})^{2}  0
τ  5  32  (1 − t^{5})/(1 − t) −1
τ^{2}  5  32  (1 − t^{5})/(1 − t) −1
α_{2}β_{2}ω  8  40  1 + t^{4}  0
α_{1}ω  8  40  1 + t^{4}  0
α_{2}τ  10  32  (1 + t^{5})/(1 + t)  1
α_{2}β_{2}τ^{2} 10  32  (1 + t^{5})/(1 + t)  1

Table 4.2.1

Proof. By Molien’s Theorem, using Table 4.2.1,

P (R^{G}, t) = \frac{1}{|G|} ∑_{σ∈G} \frac{1}{det^{∗}(I − ρ(σ)t)}

= \frac{1}{320} [\frac{1}{(1 − t)^{4}} + \frac{1}{(1 + t)^{4}} + \frac{10}{(1 − t^{2})^{2}} + \frac{60}{(1 + t^{2})^{2}} + \frac{20}{(1 + t^{2})(1 − t)^{2}} +

\frac{20}{(1 + t^{2})(1 + t)^{2}} + \frac{64(1 − t)}{(1 − t^{5})} + \frac{80}{(1 + t^{4})} + \frac{64(1 + t)}{(1 + t^{5})}]

= \frac{(1 − t^{2} + t^{8} + t^{12} − t^{18} + t^{20})(1 − t^{2})}{(1 − t^{2})^{4}(1 + t^{2})^{2}(1 + t^{4})(1 − t^{10})}

= \frac{1 − t^{2} + t^{8} + t^{12} − t^{18} + t^{20}}{(1 − t^{2})(1 − t^{4})(1 − t^{8})(1 − t^{10})}.

To get this in the form (4.1.8) with all positive coefficients in the numerator, we multiply numerator and denominator by (1 + t^{2})(1 + t^{4}) to get

P (R^{G}, t) = \frac{1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26}}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})} .

The form (4.2.2) of the Poincaré Series in Lemma 4.2.1 suggests that there are four primary invariants p_{1}, …, p_{4} of degrees 4,8,8 and 10, respectively, and a further eight secondary invariants q_{1}, …, q_{8} of degrees 0,10,12,12,14,14,16 and 26, respectively, such that R^{G} is a free module

R^{G} = ⨁_{i=1}^{8} q_{i} 𝔽_{3}[p_{1}, p_{2}, p_{3}, p_{4}]

over the subalgebra generated by p_{1}, …, p_{4}.

Note that it does not necessarily follow from Lemma 4.2.1 that such invariants exist. But if we can find such invariants of the appropriate degrees then they will generate the entire invariant ring R^{G}. We now show that this is indeed the case by constructing primary and secondary invariants.

Symplectic Invariants. The fact that G is a subgroup of SP_{4}(3) allows us to construct some basic invariants. G preserves the symplectic form on E defined by (3.1.3). In terms of the basis {e_{1}, e_{2}, e_{3}, e_{4}} for E, this form can be written as

⟨∑_{i=1}^{4} α_{i}e_{i}, ∑_{j=1}^{4} β_{j}e_{j}⟩ = 𝔉 (∑ α_{i}e_{i}, ∑ β_{j}e_{j}) = α_{1}β_{4} + α_{2}β_{3} − α_{3}β_{2} − α_{4}β_{1}, (4.2.3)

for α_{i}, β_{j} ∈ 𝔽_{3}. Let K be an infinite field containing 𝔽_{3}, and set V = K ⊗_{𝔽_{3}} E, so that the symplectic form extends to V. We use the same notation for the extension of the symplectic form. We also extend the duals x_{i} to V, so x_{i}(∑_{j=1}^{4} α_{j}e_{j}) = α_{i} for α_{j} ∈ K. Consider the Frobenius map on V defined by

F (∑ α_{i}e_{i}) = ∑ α_{i}^{3}e_{i}.

Then for any v ∈ E, the quadratic forms < v, F (v) > and < v, F^{2}(v) > are invariant under G, since the action of G commutes with F. If v = ∑ α_{i}e_{i}, then

< v, F (v) > = α_{1}α_{4}^{3} + α_{2}α_{3}^{3} − α_{3}α_{2}^{3} − α_{4}α_{1}^{3}
= x_{1}(v)x_{4}(v)^{3} + x_{2}(v)x_{3}(v)^{3} − x_{3}(v)x_{2}(v)^{3} − x_{4}(v)x_{1}(v)^{3},

since the x_{i} are dual to the e_{i}. Therefore as an element of R,

f_{4} = x_{1}x_{4}^{3} + x_{2}x_{3}^{3} − x_{3}x_{2}^{3} − x_{4}x_{1}^{3}

is an invariant under G. Similarly, the invariance of < v, F^{2}(v) > gives the invariant

f_{10} = x_{1}x_{4}^{9} + x_{2}x_{3}^{9} − x_{3}x_{2}^{9} − x_{4}x_{1}^{9}.

Chern Classes. The orbit 𝒪 of G on R_{2} containing x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} has length 10,

𝒪 = { ± (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2} + x_{2}^{2} + x_{2}x_{3} − x_{3}^{2}),
± (x_{1}x_{2} − x_{1}x_{3} − x_{2}x_{4} − x_{3}x_{4}),
± (x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2}),
± (x_{1}^{2} − x_{2}^{2} − x_{3}^{2} + x_{4}^{2}),
± (x_{1}x_{3} − x_{2}x_{4})}.

The orbit chern classes give invariants

f_{8,1} = c_{4}(𝒪) of degree 8, and
g_{12,1} = c_{6}(𝒪) of degree 12.

Explicitly,

f_{8,1} = x_{1}^{7}x_{4} − x_{1}x_{4}^{7} + x_{2}^{7}x_{3} − x_{2}x_{3}^{7}

+ x_{1}^{5}x_{4}^{3} − x_{1}^{3}x_{4}^{5} + x_{2}^{5}x_{3}^{3} − x_{2}^{3}x_{3}^{5}

+ (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})(x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})^{3}

+ (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})(x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})^{3}

+ x_{1}^{3}x_{4}(−x_{2}^{4} + x_{2}x_{3}^{3} + x_{3}^{4}) + x_{1}x_{4}^{3}(−x_{2}^{4} + x_{2}^{3}x_{3} + x_{3}^{4})

+ x_{2}^{3}x_{3}(−x_{1}^{4} + x_{1}x_{4}^{3} + x_{4}^{4}) + x_{2}x_{3}^{3}(−x_{1}^{4} + x_{1}^{3}x_{4} + x_{4}^{4})

+ (x_{1}^{4} − x_{4}^{4})(x_{2}^{4} − x_{3}^{4}),

g_{12,1} = − (x_{1}^{12} + x_{2}^{12} + x_{3}^{12} + x_{4}^{12})

+ (x_{1}^{11}x_{4} − x_{1}x_{4}^{11} + x_{2}^{11}x_{3} − x_{2}x_{3}^{11})

− (x_{1}^{7}x_{4}^{5} − x_{1}^{5}x_{4}^{7} + x_{2}^{7}x_{3}^{5} − x_{2}^{5}x_{3}^{7})

+ (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})

(x_{1}^{10} − x_{4}^{10} − x_{1}^{8}x_{4}^{2} + x_{1}^{2}x_{4}^{8} + x_{1}^{7}x_{4}^{3} + x_{1}^{3}x_{4}^{7} + x_{1}^{6}x_{4}^{4} − x_{1}^{4}x_{4}^{6} − x_{1}^{5}x_{4}^{5})

+ (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})

(x_{2}^{10} − x_{3}^{10} − x_{2}^{8}x_{3}^{2} + x_{2}^{2}x_{3}^{8} + x_{2}^{7}x_{3}^{3} + x_{2}^{3}x_{3}^{7} + x_{2}^{6}x_{3}^{4} − x_{2}^{4}x_{3}^{6} − x_{2}^{5}x_{3}^{5})

− x_{2}x_{3}(x_{2}^{2} − x_{3}^{2})(x_{1}^{8} + x_{4}^{8}) − x_{1}x_{4}(x_{1}^{2} − x_{4}^{2})(x_{2}^{8} + x_{3}^{8})

+ (x_{2}^{4} + x_{2}^{3}x_{3} + x_{2}^{2}x_{3}^{2} + x_{2}x_{3}^{3})(−x_{1}^{7}x_{4} + x_{1}^{3}x_{4}^{5})

+ (−x_{2}^{3}x_{3} + x_{2}^{2}x_{3}^{2} − x_{2}x_{3}^{3} + x_{3}^{4})(x_{1}x_{4}^{7} − x_{1}^{5}x_{4}^{3})

+ (x_{1}^{4} + x_{1}^{3}x_{4} + x_{1}^{2}x_{4}^{2} + x_{1}x_{4}^{3})(−x_{2}^{7}x_{3} + x_{2}^{3}x_{3}^{5})

+ (−x_{1}^{3}x_{4} + x_{1}^{2}x_{4}^{2} − x_{1}x_{4}^{3} + x_{4}^{4})(x_{2}x_{3}^{7} − x_{2}^{5}x_{3}^{3})

+ (x_{1}^{6} − x_{1}^{5}x_{4} + x_{1}^{4}x_{4}^{2} − x_{1}^{3}x_{4}^{3} − x_{1}^{2}x_{4}^{4} − x_{1}x_{4}^{5} − x_{4}^{6})(x_{2}^{5}x_{3} + x_{2}x_{3}^{5})

+ (x_{2}^{6} − x_{2}^{5}x_{3} + x_{2}^{4}x_{3}^{2} − x_{2}^{3}x_{3}^{3} − x_{2}^{2}x_{3}^{4} − x_{2}x_{3}^{5} − x_{3}^{6})(x_{1}^{5}x_{4} + x_{1}x_{4}^{5})

+ (x_{1}^{6} − x_{3}^{6})(−x_{2}^{6} + x_{4}^{6}) − (x_{1}x_{2}x_{3}x_{4})^{2}[(x_{1}^{2} − x_{2}^{2})^{2} + (x_{3}^{2} − x_{4}^{2})^{2}]

+ x_{1}^{2}x_{4}^{2}[−x_{2}^{6}x_{1}(x_{1} + x_{4}) + x_{3}^{6}x_{4}(x_{1} − x_{4})]

+ x_{2}^{2}x_{3}^{2}[−x_{1}^{6}x_{2}(x_{2} + x_{3}) + x_{4}^{6}x_{3}(x_{2} − x_{3})]

+ x_{2}^{4}x_{3}^{4}(x_{1}^{2} + x_{4}^{2})^{2} + x_{1}^{4}x_{4}^{4}(x_{2}^{2} + x_{3}^{2})^{2}

− (x_{1}x_{2}x_{3}x_{4})^{2}[x_{1}x_{4}(x_{2}^{2} + x_{2}x_{3} − x_{3}^{2}) + x_{2}x_{3}(x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})]

− x_{1}^{6}x_{4}^{2}(x_{3}^{4} + x_{2}x_{3}^{3}) − x_{4}^{6}x_{1}^{2}(x_{2}^{4} − x_{2}^{3}x_{3})

− x_{2}^{6}x_{3}^{2}(x_{4}^{4} + x_{1}x_{4}^{3}) − x_{3}^{6}x_{2}^{2}(x_{1}^{4} − x_{1}^{3}x_{4}).

Remaining Invariants. With the computational use of MAGMA, we applied the Reynolds Operator to specific homogeneous polynomials of the required degrees to obtain the following invariants.

f_{8,2} = ℛ_{G}(x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2})
g_{10} = ℛ_{G}(x_{1}^{4}x_{2}^{2}x_{3}^{2}x_{4}^{2})
g_{12,2} = ℛ_{G}(x_{1}^{7}x_{4}^{5} − x_{1}^{5}x_{4}^{7} + x_{2}^{7}x_{3}^{5} − x_{2}^{5}x_{3}^{7})
g_{14,1} = ℛ_{G}(x_{1}^{13}x_{4} − x_{1}x_{4}^{13} + x_{2}^{13}x_{3} − x_{2}x_{3}^{13})
g_{14,2} = ℛ_{G}(x_{1}^{8}x_{2}^{2}x_{3}^{2}x_{4}^{2})
g_{16} = ℛ_{G}(x_{1}^{15}x_{4} − x_{1}x_{4}^{15} + x_{2}^{15}x_{3} − x_{2}x_{3}^{15})

Explicitly,

f_{8,2} = − (x_{1}^{8} + x_{2}^{8} + x_{3}^{8} + x_{4}^{8})

+ x_{1}^{7}x_{4} − x_{1}x_{4}^{7} + x_{2}^{7}x_{3} − x_{2}x_{3}^{7}

+ x_{1}^{5}x_{4}^{3} − x_{1}^{3}x_{4}^{5} + x_{2}^{5}x_{3}^{3} − x_{2}^{3}x_{3}^{5}

− (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})(x_{1}^{6} − x_{4}^{6} + x_{1}^{5}x_{4} + x_{1}x_{4}^{5} + x_{1}^{4}x_{4}^{2} − x_{1}^{2}x_{4}^{4} − x_{1}^{3}x_{4}^{3})

− (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})(x_{2}^{6} − x_{3}^{6} + x_{2}^{5}x_{3} + x_{2}x_{3}^{5} + x_{2}^{4}x_{3}^{2} − x_{2}^{2}x_{3}^{4} − x_{2}^{3}x_{3}^{3})

+ x_{1}^{2}x_{4}^{2}(x_{2}^{3}x_{3} − x_{2}x_{3}^{3} − x_{2}^{4} − x_{3}^{4}) + x_{2}^{2}x_{3}^{2}(x_{1}^{3}x_{4} − x_{1}x_{4}^{3} − x_{1}^{4} − x_{4}^{4})

− x_{2}x_{3}(x_{1}^{4}x_{3}^{2} − x_{2}^{2}x_{4}^{4}) − x_{1}x_{4}(x_{2}^{4}x_{4}^{2} − x_{1}^{2}x_{3}^{4})

+ (x_{1}x_{2}x_{3}x_{4})(x_{1}^{2}x_{2}^{2} + x_{3}^{2}x_{4}^{2}) + (x_{1}x_{2}x_{3}x_{4})^{2} − (x_{1}^{4} − x_{2}^{4})(x_{3}^{4} − x_{4}^{4}),

g_{10} = − (x_{1}^{9}x_{4} + 2x_{1}x_{4}^{9} + x_{2}^{9}x_{3} + 2x_{2}x_{3}^{9})

+ (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})

[−x_{1}^{7}x_{4} − x_{4}^{7}x_{1} + x_{1}^{6}(x_{2}x_{3} + x_{4}^{2}) + x_{4}^{6}(x_{2}x_{3} − x_{1}^{2}) + x_{1}^{5}x_{4}^{3} + x_{1}^{3}x_{4}^{5}]

+ (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})

[−x_{2}^{7}x_{3} − x_{3}^{7}x_{2} + x_{2}^{6}(x_{1}x_{4} + x_{3}^{2}) + x_{3}^{6}(x_{1}x_{4} − x_{2}^{2}) + x_{2}^{5}x_{3}^{3} + x_{2}^{3}x_{3}^{5}]

− (x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + x_{4}^{4})(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})^{3}

− (x_{1}^{5}x_{4} + x_{4}^{5}x_{1})(x_{2}^{2} + x_{3}^{2})(x_{2}^{2} − x_{2}x_{3} − x_{3}^{2})

− (x_{2}^{5}x_{3} + x_{3}^{5}x_{2})(x_{1}^{2} + x_{4}^{2})(x_{1}^{2} − x_{1}x_{4} − x_{4}^{2})

+ x_{1}^{4}x_{4}^{2}(x_{2}^{4} − x_{2}^{3}x_{3}) + x_{1}^{2}x_{4}^{4}(x_{3}^{4} + x_{2}x_{3}^{3})

+ x_{2}^{4}x_{3}^{2}(x_{1}^{4} − x_{1}^{3}x_{4}) + x_{2}^{2}x_{3}^{4}(x_{4}^{4} + x_{1}x_{4}^{3})

+ (x_{1}x_{2}x_{3}x_{4})^{2}(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})

− x_{1}^{3}x_{2}^{3}(x_{2}^{4} − x_{2}^{3}x_{3} − x_{2}x_{3}^{3} − x_{3}^{4})

− x_{2}^{3}x_{3}^{3}(x_{1}^{4} − x_{1}^{3}x_{4} − x_{1}x_{4}^{3} − x_{4}^{4}),

g_{12,2} = − (x_{1}^{11}x_{4} − x_{1}x_{4}^{11} + x_{2}^{11}x_{3} − x_{2}x_{3}^{11})

+ (x_{1}^{3}x_{4} − x_{1}x_{4}^{3} + x_{2}^{3}x_{3} − x_{2}x_{3}^{3})^{3}

+ (x_{1}^{7}x_{4}^{5} − x_{1}^{5}x_{4}^{7} + x_{2}^{7}x_{3}^{5} − x_{2}^{5}x_{3}^{7})

+ (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})(x_{1}^{10} − x_{4}^{10} − x_{1}^{9}x_{4} − x_{1}x_{4}^{9})

+ (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})(x_{2}^{10} − x_{3}^{10} − x_{2}^{9}x_{3} − x_{2}x_{3}^{9})

− x_{1}^{2}x_{4}^{2}(x_{1}^{4} − x_{4}^{4})^{2} + x_{2}^{2}x_{3}^{2}(x_{2}^{4} − x_{3}^{4})^{2}

− (x_{1}x_{2}x_{3}x_{4})(x_{1}^{2} − x_{4}^{2})(x_{2}^{2} − x_{3}^{2})(x_{1}^{3}x_{4} − x_{1}x_{4}^{3} + x_{2}^{3}x_{3} − x_{2}x_{3}^{3})

− [(x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})(−x_{2}^{2} − x_{2}x_{3} + x_{3}^{2})]^{3},

g_{14,1} = − (x_{1}^{13}x_{4} − x_{4}^{13}x_{1} + x_{2}^{13}x_{3} − x_{3}^{13}x_{2})

+ (x_{1}^{12}x_{4}^{2} + x_{1}^{2}x_{4}^{12} + x_{2}^{12}x_{3}^{2} + x_{2}^{2}x_{3}^{12})

+ (x_{1}^{11}x_{4}^{3} − x_{1}^{3}x_{4}^{11} + x_{2}^{11}x_{3}^{3} − x_{2}^{3}x_{3}^{11})

− (x_{1}^{10}x_{4}^{4} + x_{1}^{4}x_{4}^{10} + x_{2}^{10}x_{3}^{4} + x_{2}^{4}x_{3}^{10})

− (x_{1}^{9}x_{4}^{5} − x_{1}^{5}x_{4}^{9} + x_{2}^{9}x_{3}^{5} − x_{2}^{5}x_{3}^{9})

+ (−x_{2}^{2} − x_{2}x_{3} + x_{3}^{2})(x_{1}^{12} − x_{4}^{12} − x_{1}^{9}x_{4}^{3} − x_{1}^{3}x_{4}^{9})

+ (−x_{1}^{2} − x_{1}x_{4} + x_{4}^{2})(x_{2}^{12} − x_{3}^{12} − x_{2}^{9}x_{3}^{3} − x_{2}^{3}x_{3}^{9})

+ (x_{2}^{4} − x_{2}^{3}x_{3} − x_{2}x_{3}^{3} − x_{3}^{4})(x_{1}^{10} − x_{4}^{10} − x_{1}^{9}x_{4} − x_{1}x_{4}^{9})

+ (x_{1}^{4} − x_{1}^{3}x_{4} − x_{1}x_{4}^{3} − x_{4}^{4})(x_{2}^{10} − x_{3}^{10} − x_{2}^{9}x_{3} − x_{2}x_{3}^{9}),

g_{14,2} = − (x_{1}^{14} + x_{2}^{14} + x_{3}^{14} + x_{4}^{14})

+ (x_{1}^{13}x_{4} − x_{1}x_{4}^{13} + x_{2}^{13}x_{3} − x_{2}x_{3}^{13})

− (x_{1}^{11}x_{4}^{3} − x_{1}^{3}x_{4}^{11} + x_{2}^{11}x_{3}^{3} − x_{2}^{3}x_{3}^{11})

+ (x_{1}^{9}x_{4}^{5} − x_{1}^{5}x_{4}^{9} + x_{2}^{9}x_{3}^{5} − x_{2}^{5}x_{3}^{9})

− (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})(x_{1}^{12} − x_{4}^{12} − x_{1}^{11}x_{4} − x_{1}x_{4}^{11} − x_{1}^{10}x_{4}^{2} + x_{1}^{2}x_{4}^{10})

− (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})(x_{2}^{12} − x_{3}^{12} − x_{2}^{11}x_{3} − x_{2}x_{3}^{11} − x_{2}^{10}x_{3}^{2} + x_{2}^{2}x_{3}^{10})

− (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})^{3}(x_{1}^{7}x_{4} + x_{4}^{7}x_{1}) − (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})^{3}(x_{2}^{7}x_{3} + x_{3}^{7}x_{2})

+ (x_{2}^{4} − x_{2}^{3}x_{3} − x_{2}x_{3}^{3} − x_{3}^{4})(x_{1}^{7}x_{4}^{3} + x_{4}^{7}x_{1}^{3})

+ (x_{1}^{4} − x_{1}^{3}x_{4} − x_{1}x_{4}^{3} − x_{4}^{4})(x_{2}^{7}x_{3}^{3} + x_{3}^{7}x_{2}^{3})

+ (x_{1}x_{2}x_{3}x_{4})^{3}(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})

+ (x_{1}x_{2}x_{3}x_{4})^{2}[x_{2}^{2}x_{3}^{2}(x_{1}^{2} + x_{4}^{2}) + x_{1}^{2}x_{4}^{2}(x_{2}^{2} + x_{3}^{2})]

− x_{1}^{2}x_{4}^{2}(x_{1}^{10} + x_{4}^{10}) − x_{2}^{2}x_{3}^{2}(x_{2}^{10} + x_{3}^{10})

+ x_{2}^{2}x_{4}^{2}[x_{2}^{8}(x_{1} − x_{4})^{2} + x_{4}^{8}(x_{2} + x_{3})^{2}] + x_{1}^{2}x_{3}^{2}[x_{1}^{8}(x_{2} − x_{3})^{2} + x_{3}^{8}(x_{1} + x_{4})^{2}]

− x_{1}^{9}x_{2}^{2}x_{4}(x_{2} − x_{3})^{2} + x_{4}^{9}x_{3}^{2}x_{1}(x_{2} + x_{3})^{2}

− x_{2}^{9}x_{1}^{2}x_{3}(x_{1} − x_{4})^{2} + x_{3}^{9}x_{4}^{2}x_{2}(x_{1} + x_{4})^{2}

− (x_{1}x_{2}x_{3}x_{4})(x_{1}^{8}x_{3}^{2} + x_{1}^{2}x_{3}^{8} + x_{2}^{8}x_{4}^{2} + x_{2}^{2}x_{4}^{8})

+ x_{1}^{6}x_{2}^{2}[x_{3}^{2}(x_{4}^{2} + x_{2}x_{3})(x_{2} − x_{3})^{2} + x_{4}^{4}(x_{2} + x_{3})^{2}]

+ x_{4}^{6}x_{3}^{2}[x_{2}^{2}(x_{1}^{2} − x_{2}x_{3})(x_{2} + x_{3})^{2} + x_{1}^{4}(x_{2} − x_{3})^{2}]

+ x_{2}^{6}x_{1}^{2}[x_{4}^{2}(x_{3}^{2} + x_{1}x_{4})(x_{1} − x_{4})^{2} + x_{3}^{4}(x_{1} + x_{4})^{2}]

+ x_{3}^{6}x_{4}^{2}[x_{1}^{2}(x_{2}^{2} − x_{1}x_{4})(x_{1} + x_{4})^{2} + x_{2}^{4}(x_{1} − x_{4})^{2}]

− x_{1}^{6}x_{2}^{6}(x_{3}^{2} + x_{4}^{2}) − x_{3}^{6}x_{4}^{6}(x_{1}^{2} + x_{2}^{2})

g_{16} = x_{1}^{15}x_{4} − x_{1}x_{4}^{15} + x_{2}^{15}x_{3} − x_{2}x_{3}^{15}
− (x_{1}^{13}x_{4}^{3} − x_{1}^{3}x_{4}^{13} + x_{2}^{13}x_{3}^{3} − x_{2}^{3}x_{3}^{13})
− (x_{1}^{12}x_{4}^{4} + x_{1}^{4}x_{4}^{12} + x_{2}^{12}x_{3}^{4} + x_{2}^{4}x_{3}^{12})
+ (x_{1}^{10}x_{4}^{6} + x_{1}^{6}x_{4}^{10} + x_{2}^{10}x_{3}^{6} + x_{2}^{6}x_{3}^{10})
− (x_{1}^{9}x_{4}^{7} − x_{1}^{7}x_{4}^{9} + x_{2}^{9}x_{3}^{7} − x_{2}^{7}x_{3}^{9})
+ (x_{2}^{2} + x_{2}x_{3} − x_{3}^{2})^{3}(−x_{1}^{10} + x_{4}^{10} + x_{1}^{9}x_{4} + x_{1}x_{4}^{9})
+ (x_{1}^{2} + x_{1}x_{4} − x_{4}^{2})^{3}(−x_{2}^{10} + x_{3}^{10} + x_{2}^{9}x_{3} + x_{2}x_{3}^{9})
+ (x_{2}^{4} − x_{2}^{3}x_{3} − x_{2}x_{3}^{3} − x_{3}^{4})(x_{1}^{12} − x_{4}^{12} − x_{1}^{9}x_{4}^{3} − x_{1}^{3}x_{4}^{9})
+ (x_{1}^{4} − x_{1}^{3}x_{4} − x_{1}x_{4}^{3} − x_{4}^{4})(x_{2}^{12} − x_{3}^{12} − x_{2}^{9}x_{3}^{3} − x_{2}^{3}x_{3}^{9}).

See Appendix (A.1.2) for the MAGMA computations of these invariants. The relations among the secondary invariants are given in Appendix A.2.

Theorem 4.2.4. The set

𝒫 = {f_{4}, f_{8,1}, f_{8,2}, f_{10}}

is a set of primary invariants of G and R^{G} is a free module over 𝒜 = 𝔽_{3}[𝒫] with

𝒮 = {1, g_{10}, g_{12,1}, g_{12,2}, g_{14,1}, g_{14,2}, g_{16} , g_{10}.g_{16}}

as a basis. This module has Poincaré Series

\frac{1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26}}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})} . (4.2.5)

Proof. Let I be the ideal of R generated by 𝒫. Using MAGMA, we compute a Groebner basis for I, consisting of 259 homogeneous polynomials. Included in this Groebner basis are the four polynomials gb[1],gb[155],gb[250] and gb[259] printed out in Appendix (A.3.1). We call these polynomials p_{1}, p_{2}, p_{3} and p_{4}, respectively. They have the form

p_{1}(x) = x_{1}^{8} + \overline{p}_{1} where \overline{p}_{1} lies in the ideal generated by \{x_{2}, x_{3}, x_{4}\},
p_{2}(x) = x_{2}^{18} + \overline{p}_{2} where \overline{p}_{2} lies in the ideal generated by \{x_{3}, x_{4}\},
p_{3}(x) = x_{3}^{23},
p_{4}(x) = x_{4}^{23}.

We claim that the existence of these elements of I implies that Z(I), the zero set of I over \overline{𝔽_{3}} (the algebraic closure of 𝔽_{3}) consists of the zero point alone. For if a = (a_{1}, a_{2}, a_{3}, a_{4})^{t} ∈ Z(I) where a_{i} ∈ \overline{𝔽_{3}}, then p_{4}(a) = p_{3}(a) = 0 implies that a_{3} = a_{4} = 0. Then \overline{p}_{2}(a) = 0. So a_{2} = 0. Therefore \overline{p}_{1}(a) = 0, so a_{1} = 0, and our claim is proved. Therefore, by Hilbert’s Nullstellensatz, x_{i}^{m_{i}} ∈ I for each i = 1, …, 4 and some integer m_{i} > 0. Therefore R is finitely generated as a module over 𝒜 = 𝔽_{3}[𝒫]. So is its subring R^{G}. Hence 𝒫 is a homogeneous system of parameters for R^{G}. But R^{G} is Cohen-Macaulay (Theorem 4.1.5). So R^{G} is a free module over 𝒜 by Theorem 4.1.7.

For each g ∈ 𝒮, by computing a normal form with respect to the Groebner basis of I, we verify that g ∉ I. The MAGMA verifications are given in Appendix (A.3.2). Furthermore, we verify that all elements of 𝒮 are linearly independent modulo I. Since I is a homogeneous ideal of R, it is sufficient to check the linear independence modulo I of the pair g_{12,1} and g_{12,2}, and the pair g_{14,1} and g_{14,2}. Since we are working over the field 𝔽_{3}, it suffices to show that

g_{12,1} − g_{12,2} ≢ 0 ≢ g_{12,1} + g_{12,2} mod I, and
g_{14,1} − g_{14,2} ≢ 0 ≢ g_{14,1} + g_{14,2} mod I.

This is verified by computing normal forms of these polynomials modulo our Groebner basis for I. See Appendix (A.3.3) for the MAGMA computation results.

Therefore 𝒮 forms a free basis for a module over 𝒜. This free module ⊕_{g∈𝒮}g 𝒜 is contained in R^{G} since each polynomial in 𝒫 ∪ 𝒮 is an invariant under G. Because 𝒮 is a set of free module generators of degrees {1, 10, 12, 12, 14, 14, 16, 26} over 𝒜, it has Poincaré Series (4.2.5), which is the same as the Poincaré Series for R^{G} given by Lemma 4.2.1. So this free module is all of R^{G}.

CHAPTER 5 The Cohomology Algebra H^{∗}(ON, 𝔽_{3})

5.1. Polynomial Tensor Exterior Algebras

In order to describe the entire cohomology algebra H^{∗}(ON, 𝔽_{3}) we need to consider invariants of polynomial tensor exterior algebras. These are described by Smith([15, Chapter 9]) and Benson([5, Chapter 5]). We return to our general situation where G is a finite group with representation

ρ : G → GL_{n}(K)

over a field K. Via this representation, G acts on the n-dimensional K-vector space X with basis {x_{1}, …, x_{n}} and hence on the polynomial algebra R = K[x_{1}, …, x_{n}]. Suppose that G also acts via ρ on the K-vector space Y with basis {y_{1}, …, y_{n}}. We extend this action to the exterior algebra S = Λ(y_{1}, …, y_{n}). Then G acts on the polynomial tensor exterior algebra

R ⊗ S = K[x_{1}, …, x_{n}] ⊗ Λ(y_{1}, …, y_{n}).

The action of G on the tensor product R⊗S is the diagonal action, so that (r⊗s)σ = rσ ⊗ sσ for all r ∈ R, s ∈ S and σ ∈ G.

We are interested in computing the invariants (R ⊗ S)^{G} under this action of G. We identify R and S with their images in R⊗S. Then the algebra R^{G} of polynomial invariants is a commutative subalgebra of (R ⊗ S)^{G}. Recall from Chapter 4.1 that R^{G} contains a set {θ_{1}, …, θ_{n}} of primary invariants such that R^{G} is a finitely generated free module over K[θ_{1}, …, θ_{n}].

The Poincaré Series. We give R ⊗ S the bigrading

(R ⊗ S)_{i,j} = R_{i} ⊗ S_{j} for i, j ∈ ℕ. (5.1.1)

A typical element of degree (i, j) is a unique sum of terms of the form

fy_{k_{1}} ⋯ y_{k_{j}}, 1 ≤ k_{1} < k_{2} < ⋯ < k_{j} ≤ n,

where f is a homogeneous polynomial of degree i in x_{1}, …, x_{n}. This bigrading is preserved by the action of G. So it passes to the invariant subalgebra (R ⊗ S)^{G}. In this situation, the Poincaré Series of the invariant algebra is the double series

P ((R ⊗ S)^{G}, t, s) = ∑_{i,j=0}^{∞}(dim_{K}(R ⊗ S)_{i,j}^{G})t^{i}s^{j}.

This Poincaré Series can be computed with a version of Molien’s Theorem, due to L. Solomon [16].

Theorem 5.1.2. Let ρ : G → GL_{n}(K) be a representation of a finite group G over a field K of characteristic zero. If G acts on the algebra

R ⊗ S = K[x_{1}, …, x_{n}] ⊗ Λ(y_{1}, …, y_{n})

via the representation ρ, then the Poincaré Series of the invariant algebra (R ⊗S)^{G} is

P ((R ⊗ S)^{G}, t, s) = \frac{1}{|G|} ∑_{σ∈G} \frac{det(I + ρ(σ)s)}{det(I − ρ(σ)t)}.

Proof. See [Sm,Theorem 9.2.2].

As in the classical version of Molien’s Theorem, this formula also works if K is a field of characteristic p prime to the order of G. In that case (see Remark 4.1.2) we lifted the eigenvalues λ_{1}, …, λ_{n} of ρ(σ) in K to λ_{1}^{∗}, …, λ_{n}^{∗} ∈ ℂ under a fixed isomorphism δ.

Then we defined

det^{∗}(I − ρ(σt)) = (1 − λ_{1}^{∗}t) ⋯ (1 − λ_{n}^{∗}t),

so that

P ((R ⊗ S)^{G}, t, s) = \frac{1}{|G|} ∑_{σ∈G} \frac{det^{∗}(I + ρ(σ)s)}{det^{∗}(I − ρ(σ)t)}. (5.1.3)

In fact, Molien’s Theorem has a more general form. Assume that K is a splitting field of characteristic zero for G. Let 𝒳 be the set of irreducible K-characters of G. The action of G on R can be decomposed into a direct sum of irreducible representations. So we can write R = ⊕T where each T is a G-invariant subspace of R on which G acts irreducibly. For each χ ∈ 𝒳, let R_{χ}^{G} denote the direct sum of those T ’s which afford the character χ. Each R_{χ}^{G} is called an isotypic component of the action of G on R. Then R = ⊕_{χ∈𝒳} R_{χ}^{G}. Note that the ring of invariants R^{G} is R_{ϵ}^{G}, where ϵ is the trivial character.

Proposition 5.1.4. In the above situation, the isotypic component R_{χ}^{G} is an R^{G}-module for each χ ∈ 𝒳. Furthermore, if 𝒫 = {θ_{1}, …, θ_{n}} is a set of primary invariants of R^{G}, so that R^{G} is a finitely generated free module over K[𝒫], then R_{χ}^{G} is also a finitely generated free module over K[𝒫].

Proof. Fix a χ ∈ 𝒳. Let T ⊆ R_{χ}^{G} be an irreducible KG-subspace of R. For any f ∈ R^{G}, the map T → f T given by multiplication by f is a KG-module homomorphism. By Schur’s Lemma, either f T = 0 or f T ≃ T as KG-modules. In both cases, f T ⊆ R_{χ}^{G}, so that R_{χ}^{G} is an R^{G}-module.

The set 𝒫 of primary invariants is an homogeneous system of parameters for the polynomial ring R. Hence R is a finitely generated free module over the polynomial ring K[𝒫]. Therefore the submodule R_{χ}^{G} is also finitely generated over K[𝒫].

Let I be the ideal in R generated by 𝒫. Since each θ_{i} is invariant under G, the ideal I is a G-invariant subspace of R. Therefore I is the direct sum

I = ⨁_{χ∈𝒳} I_{χ}^{G}

of its isotypic components I_{χ}^{G} = (I ∩ R_{χ}^{G}), for χ ∈ 𝒳. Then we also have a direct sum decomposition

R/I = ⨁_{χ∈𝒳}(R/I)_{χ}^{G},

where

(R/I)_{χ}^{G} ≃ R_{χ}^{G}/I_{χ}^{G} for all χ ∈ 𝒳.

For each χ ∈ 𝒳, take θ_{1,χ}, …, θ_{m_{χ},χ} ∈ R_{χ}^{G} whose images form a K-basis \overline{θ}_{1,χ}, …, \overline{θ}_{m_{χ},χ} for (R/I)_{χ}^{G}. Then the \overline{θ}_{i,χ}, for χ ∈ 𝒳 and i = 1, …, m_{χ}, form a K-basis for R/I. Note that R_{χ}^{G} and I are homogeneous, so I_{χ}^{G} is homogeneous. Therefore θ_{1,χ}, …, θ_{m_{χ},χ} may be chosen to be homogeneous. Then Theorem 4.1.5 implies that the θ_{i,χ} for χ ∈ 𝒳 and i = 1, …, m_{χ}, form a K[𝒫]-basis for the free module R. Since this basis is a disjoint union of its intersections {θ_{1,χ}, …, θ_{m_{χ},χ}} with the summands R_{χ}^{G} in R = ⨁ R_{χ}^{G}, we conclude that each R_{χ}^{G} is a free K[𝒫]-module with θ_{1,χ}, …, θ_{m_{χ},χ} as basis.

Note that each R_{χ}^{G} is graded, with ith homogeneous component (R_{χ}^{G})_{i} = R_{χ}^{G} ∩ R_{i} consisting of all homogeneous polynomials in R_{χ}^{G} of degree i. So the module R_{χ}^{G} has an associated Poincaré series

P (R_{χ}^{G}, t) = ∑_{i=0}^{∞} (dim_{K}(R_{χ}^{G})_{i}) t^{i}.

The more general form of Molien’s Theorem gives a formula for this Poincaré Series. We state this as a formula for \frac{1}{χ(1)}P (R_{χ}^{G}, t). We divide by the degree χ(1) of the character χ so that the coefficient of t^{i} in \frac{1}{χ(1)}P (R_{χ}^{G}, t) is equal to the multiplicity of the character χ in the action of G on the homogeneous component R_{i} of R.

Theorem 5.1.5. Let ρ : G → GL_{n}(K) be a representation of a finite group G over a splitting field K of characteristic zero, and let χ ∈ 𝒳(G). Then the Poincaré Series for R_{χ}^{G} is given by

\frac{1}{χ(1)}P (R_{χ}^{G}, t) = \frac{1}{|G|} ∑_{σ∈G} \frac{χ(σ^{−1})}{det(I − ρ(σ)t)}.

Proof. See [17, Theorem 2.1].

The Cohen-Macaulay Property. Although the invariant ring (R ⊗ S)^{G} is not commutative, it is graded-commutative and it has many of the same properties as the commutative invariant ring R^{G}. In particular, we have the following analogue of Theorem 4.1.7. We say that a graded-commutative graded K-algebra B of Krull dimension n is Cohen-Macaulay if it is a finitely generated free module over K[θ_{1}, …, θ_{n}] for every homogeneous system of parameters {θ_{1}, …, θ_{n}} of B.

Theorem 5.1.6. Let G be a finite group and K a field of characteristic not dividing the order of G. If G acts on the polynomial tensor exterior algebra R⊗S = (K[x_{1}, …, x_{n}] ⊗ Λ(y_{1}, …, y_{n})) via the representation ρ : G → GL_{n}(K), then the invariant subalgebra (R ⊗ S)^{G} is a Cohen-Macaulay graded-commutative ring.

Proof. See [5, Proposition 5.3.1].

Proposition 5.1.7. Let 𝒫 = {θ_{1}, …, θ_{n}} be a set of primary invariants for R^{G}. Then 𝒫 is also a set of primary invariants for (R ⊗ S)^{G}.

Proof. It suffices to show that (R⊗S)^{G} is finitely generated as a module over K[𝒫]. Then it will follow from the Cohen-Macaulay property (Theorem 5.1.6 and Theorem 4.1.5) that (R ⊗ S)^{G} is a finitely generated free module over K[𝒫]. Now R ⊗ S is finitely generated as a module over R, with basis the set of monomials

{y_{i_{1}}y_{i_{2}} … y_{i_{k}} : 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n}.

Since R is finitely generated over K[𝒫], it follows that R ⊗ S is finitely generated over K[𝒫]. The polynomial ring K[𝒫] is noetherian. So the submodule (R ⊗ S)^{G} of R ⊗ S is also finitely generated over this ring, as required.

If θ_{1}, …, θ_{n} are as in Proposition 5.1.7, then there are secondary invariants η_{1}, …, η_{t} forming a basis for (R ⊗ S)^{G} as a free module over K[θ_{1}, …, θ_{n}],

(R ⊗ S)^{G} = ⨁_{i=1}^{t} η_{i}K[θ_{1}, …, θ_{n}].

Constructing Invariants. In order to construct invariants in (R ⊗ S)^{G}, we extend the definition of the Reynolds Operator ℛ_{G} to R ⊗ S. So for g ∈ R ⊗ S, we define

ℛ_{G}(g) = \frac{1}{|G|} ∑_{σ∈G} gσ.

Then ℛ_{G}(g) ∈ (R ⊗ S)^{G}.

We also define the exterior derivative on R ⊗ S as follows. Define

d : R ⊗ S → R ⊗ S

to be the unique derivation of the K-algebra R ⊗ S sending each x_{i} to y_{i} and each y_{i} to zero. Then for f ∈ K[x_{1}, …, x_{n}],

df = ∑_{i=1}^{n}(\frac{∂f}{∂x_{i}})y_{i}. (5.1.8)

And for g = fy_{k_{1}} … y_{k_{j}} ∈ R ⊗ S,

dg = ∑_{i=1}^{n}(\frac{∂f}{∂x_{i}})y_{i} y_{k_{1}} … y_{k_{j}}. (5.1.9)

The terms on the right hand side of (5.1.9) with i = k_{1}, …, k_{j} will be zero.

Since G acts in the same way via ρ on the x_{i} as on the y_{i}, this exterior derivative d commutes with the action of G. So for each invariant f ∈ R^{G}, we have df ∈ (R ⊗ S)^{G}. Or, more generally, we have dg ∈ (R ⊗ S)^{G} for each g ∈ (R ⊗ S)^{G}.

5.2. Generators for H^{∗}(ON, 𝔽_{3})

We now compute generators for and describe the cohomology algebra H^{∗}(ON, 𝔽_{3}). By Theorem 3.1.5, this is the invariant algebra (R⊗S)^{G}, where R = 𝔽_{3}[x_{1}, …, x_{4}], S = Λ(y_{1}, …, y_{4}) and G = N_{ON}(E)/E. This group G acts on R ⊗ S via the representation ρ : G → GL_{4}(3) given in Theorem 3.1.5. For convenience, we use the bigrading (5.1.1) of R ⊗ S. Note that the cohomology degree of each x_{i} is 2, and of each y_{i} is 1. So an element of R ⊗ S of bidegree (i, j) has degree 2i + j in cohomology.

First, we compute the Poincaré Series.

Theorem 5.2.1. The Poincaré Series for (R ⊗ S)^{G} has the form

P ((R ⊗ S)^{G}, t, s) = \frac{N_{0}(t, s) + N_{1}(t, s) + N_{2}(t, s) + N_{3}(t, s) + N_{4}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})},

where

N_{0}(t, s) = 1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26}
N_{1}(t, s) = s(t + t^{3} + t^{5} + 3t^{7} + 3t^{9} + 4t^{11} + 6t^{13} + 4t^{15} + 3t^{17} + 3t^{19} + t^{21} + t^{23} + t^{25})
N_{2}(t, s) = s^{2}(1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26})
+ 2s^{2}(t^{4} + t^{6} + 2t^{8} + 3t^{10} + 3t^{12} + 3t^{14} + 3t^{16} + 2t^{18} + t^{20} + t^{22})
N_{3}(t, s) = s^{3}(t + t^{3} + t^{5} + 3t^{7} + 3t^{9} + 4t^{11} + 6t^{13} + 4t^{15} + 3t^{17} + 3t^{19} + t^{21} + t^{23} + t^{25})
N_{4}(t, s) = s^{4}(1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26}).

Proof. By Molien’s Theorem (5.1.3), (using Table 4.2.1),

P ((R ⊗ S)^{G}, t, s)

= \frac{1}{|G|} ∑_{σ∈G} \frac{det^{∗}(I + ρ(σ)s)}{det^{∗}(I − ρ(σ)t)}

= \frac{1}{320} [\frac{(1 + s)^{4}}{(1 − t)^{4}} + \frac{(1 − s)^{4}}{(1 + t)^{4}} + 10\frac{(1 − s^{2})^{2}}{(1 − t^{2})^{2}} + 60\frac{(1 + s^{2})^{2}}{(1 + t^{2})^{2}} + 20\frac{(1 + s^{2})(1 + s)^{2}}{(1 + t^{2})(1 − t)^{2}} +

20\frac{(1 + s^{2})(1 − s)^{2}}{(1 + t^{2})(1 + t)^{2}} + 64\frac{(1 − t)(1 + s^{5})}{(1 − t^{5})(1 + s)} + 80\frac{(1 + s^{4})}{(1 + t^{4})} + 64\frac{(1 + t)(1 − s^{5})}{(1 + t^{5})(1 − s)}]

= \frac{M(t, s) (1 − t^{2})}{(1 − t^{2})^{4}(1 + t^{2})^{2}(1 + t^{4})(1 − t^{10})},

where

M(t, s) = (1 + s^{2} + s^{4})(1 − t^{2} + t^{8} + t^{12} − t^{18} + t^{20})

+ 2s^{2}(t^{4} + t^{8} + t^{10} + t^{12} + t^{16}) + (s + s^{3})(t + 2t^{7} + t^{9} + t^{11} + 2t^{13} + t^{19}).

Hence,

P ((R ⊗ S)^{G}, t, s) = \frac{M(t, s)}{(1 − t^{2})(1 − t^{4})(1 − t^{8})(1 − t^{10})}

= \frac{N_{0}(t, s) + N_{1}(t, s) + N_{2}(t, s) + N_{3}(t, s) + N_{4}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})},

where the last equation was obtained from the previous one by multiplying numerator and denominator by (1 + t^{2})(1 + t^{4}).

Let 𝒫 be the set of primary invariants for R^{G} given in Theorem 4.2.4. Then by Proposition 5.1.7, this set 𝒫 is also a set of primary invariants for (R ⊗ S)^{G}. So we need to find a set 𝒢 = {η_{1}, …, η_{m}} of secondary invariants which forms a set of free module generators for (R ⊗ S)^{G} as a module over 𝒜 = 𝔽_{3}[𝒫]. The Poincaré Series given in Theorem 5.2.1 suggests that we may find these such that if each η_{k} has degree (i_{k}, j_{k}) for each k, then

∑_{k=1}^{m} s^{i_{k}}t^{j_{k}} = N(t, s) = N_{0}(t, s) + N_{1}(t, s) + N_{2}(t, s) + N_{3}(t, s) + N_{4}(t, s).

For j = 0, …, 4, we shall construct submodules M_{j} of (R ⊗ S)^{G} such that M_{j} has Poincaré Series

P (M_{j}, t, s) = \frac{N_{j}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

Each submodule M_{j} will be contained in R ⊗ S_{j}. In fact, the Poincaré Series for (R⊗S)^{G} given in Theorem 5.2.1 shows that such a module M_{j} will be the invariant subalgebra (R⊗S_{j})^{G}. Then it will follow that (R⊗S)^{G} = M_{0}⊕M_{1}⊕M_{2}⊕M_{3}⊕M_{4}.

The submodule M_{0}. Our first submodule M_{0} is (R ⊗ S_{0})^{G} = R^{G}. We constructed generators for this module in the previous chapter. By Theorem 4.2.4, the Poincaré Series of R^{G} is

\frac{N_{0}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})},

as required.

The submodule M_{1}. This submodule will be generated by secondary invariants of degree 1 in y = (y_{1}, y_{2}, y_{3}, y_{4})^{t}. If f ∈ R^{G} has degree (i, 0), then its exterior derivative df (given by (5.1.8)) is also invariant under G. This gives us an invariant of degree (i − 1, 1). We also get invariants of the various degrees by multiplying the derivatives of primary and secondary invariants of R^{G} by the secondary invariants of R^{G}. We can construct further invariants by using the Reynolds Operator and by direct computation, using MAGMA. This gives us

h_{(1,1)} = x_{1}y_{4} + x_{2}y_{3} − x_{3}y_{2} − x_{4}y_{1}
h_{(5,1)} = ℛ_{G}(x_{1}^{5}y_{4})
h_{(7,1)} = ℛ_{G}(x_{1}^{5}x_{4}^{2}y_{4})
h_{(9,1)} = ℛ_{G}(x_{1}^{5}x_{4}^{4}y_{4})
h_{(11,1),1} = ℛ_{G}(x_{1}^{7}x_{4}^{4}y_{4})
h_{(11,1),2} = ℛ_{G}(x_{1}^{6}x_{2}^{2}x_{3}^{2}x_{4}y_{4})
h_{(13,1)} = ℛ_{G}(x_{1}^{9}x_{2}^{2}x_{3}^{2}y_{4})

To describe these explicitly, note that each of these invariants will be of the form

h_{(i,1)} = z_{1}y_{1} + z_{2}y_{2} + z_{3}y_{3} + z_{4}y_{4},

where {z_{1}, …, z_{4}} ⊆ R_{i}. Consider the actions of the generators α_{1}, α_{2} and β_{1} of G (given in Theorem 3.1.5) on h_{i,1},

h_{(i,1)}α_{1} = (z_{1})α_{1}y_{3} + (z_{2})α_{1}y_{4} − (z_{3})α_{1}y_{1} − (z_{4})α_{1}y_{2}
h_{(i,1)}α_{2} = (z_{1})α_{2}y_{2} − (z_{2})α_{2}y_{1} − (z_{3})α_{2}y_{4} + (z_{4})α_{2}y_{3}
h_{(i,1)}β_{1} = (z_{1})β_{1}y_{4} + (z_{2})β_{1}y_{3} − (z_{3})β_{1}y_{2} − (z_{4})β_{1}y_{1}.

Since h_{(i,1)} is invariant under the action of each of these elements of G, we can see by equating coefficients of the y_{i} that z_{1}α_{1} = z_{3}, z_{1}α_{2} = z_{2} and z_{1}β_{1} = z_{4}. Then we can write

h_{(i,1)} = z_{1}y_{1} + (z_{1}α_{2})y_{2} + (z_{1}α_{1})y_{3} + (z_{1}β_{1})y_{4}
= Φ(z_{1}), say.

Using this notation, the above invariants are

h_{(1,1)} = Φ(−x_{4}),

h_{(5,1)} = Φ(−x_{1}^{4}x_{4} − x_{1}^{3}x_{2}^{2} − x_{1}^{3}x_{2}x_{3} + x_{1}^{3}x_{3}^{2} + x_{1}^{3}x_{4}^{2} + x_{1}^{2}x_{4}^{3} + x_{1}x_{2}^{4} − x_{1}x_{2}^{3}x_{3} − x_{1}x_{2}x_{3}^{3} −x_{1}x_{3}^{4} −x_{1}x_{4}^{4} −x_{2}^{4}x_{4} +x_{2}^{3}x_{3}x_{4} +x_{2}^{2}x_{4}^{3} +x_{2}x_{3}^{3}x_{4} +x_{2}x_{3}x_{4}^{3} +x_{3}^{4}x_{4} −x_{3}^{2}x_{4}^{3} −x_{4}^{5}),

h_{(7,1)} = Φ(x_{1}^{4}x_{4}^{3} + x_{1}^{3}x_{2}^{4} + x_{1}^{3}x_{2}^{3}x_{3} − x_{1}^{3}x_{3}^{4} − x_{1}^{3}x_{4}^{4} − x_{1}^{2}x_{4}^{5} − x_{1}x_{2}^{6} − x_{1}x_{2}^{3}x_{3}^{3} − x_{1}x_{2}^{3}x_{3}x_{4}^{2} + x_{1}x_{2}x_{3}^{3}x_{4}^{2} + x_{1}x_{3}^{6} + x_{1}x_{4}^{6} − x_{2}^{6}x_{4} + x_{2}^{4}x_{3}^{2}x_{4} + x_{2}^{4}x_{4}^{3} + x_{2}^{3}x_{3}^{3}x_{4} + x_{2}^{3}x_{3}x_{4}^{3} + x_{2}^{2}x_{3}^{4}x_{4} + x_{2}^{2}x_{3}^{2}x_{4}^{3} + x_{2}x_{3}^{3}x_{4}^{3} + x_{4}^{7})

h_{(9,1)} = Φ(x_{1}^{8}x_{4} −x_{1}^{7}x_{2}^{2} −x_{1}^{7}x_{2}x_{3} +x_{1}^{7}x_{3}^{2} +x_{1}^{7}x_{4}^{2} +x_{1}^{6}x_{2}^{2}x_{4} +x_{1}^{6}x_{2}x_{3}x_{4} −x_{1}^{6}x_{3}^{2}x_{4} − x_{1}^{5}x_{2}^{4} −x_{1}^{5}x_{2}x_{3}^{3} +x_{1}^{5}x_{3}^{4} +x_{1}^{5}x_{4}^{4} +x_{1}^{4}x_{2}^{4}x_{4} +x_{1}^{4}x_{2}x_{3}^{3}x_{4} −x_{1}^{4}x_{3}^{4}x_{4} −x_{1}^{4}x_{4}^{5} +x_{1}^{3}x_{2}^{6} −x_{1}^{3}x_{2}^{5}x_{3} − x_{1}^{3}x_{2}^{4}x_{3}^{2} + x_{1}^{3}x_{2}^{4}x_{4}^{2} − x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{2} + x_{1}^{3}x_{2}^{2}x_{3}^{4} − x_{1}^{3}x_{2}^{2}x_{4}^{4} − x_{1}^{3}x_{2}x_{3}^{5} − x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{2} − x_{1}^{3}x_{2}x_{3}x_{4}^{4} − x_{1}^{3}x_{3}^{6}−x_{1}^{3}x_{3}^{4}x_{4}^{2}+x_{1}^{3}x_{3}^{2}x_{4}^{4}−x_{1}^{3}x_{4}^{6}+x_{1}^{2}x_{2}^{6}x_{4}+x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}−x_{1}^{2}x_{2}^{3}x_{3}x_{4}^{3}+x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{3}−x_{1}^{2}x_{3}^{6}x_{4}+ x_{1}x_{2}^{8}−x_{1}x_{2}^{7}x_{3}−x_{1}x_{2}^{6}x_{3}^{2}+x_{1}x_{2}^{6}x_{4}^{2}+x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{2}−x_{1}x_{2}^{3}x_{3}x_{4}^{4}+x_{1}x_{2}^{2}x_{3}^{6}−x_{1}x_{2}^{2}x_{4}^{6}−x_{1}x_{2}x_{3}^{7}+ x_{1}x_{2}x_{3}^{3}x_{4}^{4} −x_{1}x_{2}x_{3}x_{4}^{6} −x_{1}x_{3}^{8} −x_{1}x_{3}^{6}x_{4}^{2} +x_{1}x_{3}^{2}x_{4}^{6} −x_{1}x_{4}^{8} −x_{2}^{6}x_{3}^{2}x_{4} −x_{2}^{5}x_{3}^{3}x_{4} +x_{2}^{5}x_{3}x_{4}^{3} + x_{2}^{3}x_{3}^{5}x_{4} −x_{2}^{3}x_{3}^{3}x_{4}^{3} +x_{2}^{3}x_{3}x_{4}^{5} +x_{2}^{2}x_{3}^{4}x_{4}^{3} −x_{2}x_{3}^{7}x_{4} +x_{2}x_{3}^{5}x_{4}^{3} −x_{2}x_{3}^{3}x_{4}^{5} −x_{3}^{8}x_{4} +x_{3}^{6}x_{4}^{3} +x_{4}^{9})

h_{(11,1),1} = Φ(x_{1}^{8}x_{4}^{3} − x_{1}^{7}x_{2}^{4} − x_{1}^{7}x_{2}^{3}x_{3} + x_{1}^{7}x_{3}^{4} + x_{1}^{7}x_{4}^{4} + x_{1}^{6}x_{2}^{3}x_{3}x_{4} + x_{1}^{6}x_{2}^{2}x_{4}^{3} − x_{1}^{6}x_{2}x_{3}^{3}x_{4} +x_{1}^{6}x_{2}x_{3}x_{4}^{3} −x_{1}^{6}x_{3}^{2}x_{4}^{3} −x_{1}^{5}x_{2}^{6} −x_{1}^{5}x_{2}^{3}x_{3}^{3} +x_{1}^{5}x_{3}^{6} +x_{1}^{5}x_{4}^{6} +x_{1}^{4}x_{2}^{4}x_{4}^{3} +x_{1}^{4}x_{2}^{3}x_{3}x_{4}^{3} − x_{1}^{4}x_{3}^{4}x_{4}^{3}−x_{1}^{4}x_{4}^{7}+x_{1}^{3}x_{2}^{8}+x_{1}^{3}x_{2}^{6}x_{3}^{2}−x_{1}^{3}x_{2}^{6}x_{4}^{2}−x_{1}^{3}x_{2}^{5}x_{3}^{3}+2x_{1}^{3}x_{2}^{3}x_{3}^{5}−x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{2}+x_{1}^{3}x_{2}^{3}x_{3}x_{4}^{4}+ 2x_{1}^{3}x_{2}^{2}x_{3}^{6}+x_{1}^{3}x_{2}^{2}x_{4}^{6}−x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{4}+x_{1}^{3}x_{2}x_{3}x_{4}^{6}−x_{1}^{3}x_{3}^{8}+x_{1}^{3}x_{3}^{6}x_{4}^{2}−x_{1}^{3}x_{3}^{2}x_{4}^{6}−x_{1}^{3}x_{4}^{8}+x_{1}^{2}x_{2}^{6}x_{4}^{3}+ x_{1}^{2}x_{2}^{3}x_{3}^{3}x_{4}^{3}−x_{1}^{2}x_{3}^{6}x_{4}^{3}+x_{1}x_{2}^{10}−x_{1}x_{2}^{7}x_{3}^{3}+x_{1}x_{2}^{6}x_{3}^{4}−x_{1}x_{2}^{6}x_{4}^{4}−x_{1}x_{2}^{4}x_{3}^{6}+x_{1}x_{2}^{4}x_{4}^{6}−x_{1}x_{2}^{3}x_{3}^{7}− x_{1}x_{2}^{3}x_{3}^{3}x_{4}^{4} − x_{1}x_{2}^{3}x_{3}x_{4}^{6} − x_{1}x_{2}x_{3}^{3}x_{4}^{6} − x_{1}x_{3}^{10} + x_{1}x_{3}^{6}x_{4}^{4} − x_{1}x_{3}^{4}x_{4}^{6} − x_{1}x_{4}^{10} + x_{2}^{10}x_{4} − x_{2}^{9}x_{3}x_{4} + x_{2}^{7}x_{3}^{3}x_{4} − x_{2}^{7}x_{3}x_{4}^{3} + x_{2}^{6}x_{3}^{4}x_{4} + x_{2}^{3}x_{3}^{7}x_{4} − x_{2}^{3}x_{3}^{5}x_{4}^{3} − x_{2}^{3}x_{3}x_{4}^{7} − x_{2}^{2}x_{3}^{6}x_{4}^{3} − x_{2}^{2}x_{4}^{9} + x_{2}x_{3}^{9}x_{4} + x_{2}x_{3}^{7}x_{4}^{3} + x_{2}x_{3}^{3}x_{4}^{7} − x_{2}x_{3}x_{4}^{9} − x_{3}^{8}x_{4}^{3} + x_{3}^{2}x_{4}^{9} + x_{4}^{11})

h_{(11,1),2} = Φ(−x_{1}^{11} +x_{1}^{10}x_{4} −x_{1}^{9}x_{2}^{2} −x_{1}^{9}x_{2}x_{3} +x_{1}^{9}x_{3}^{2} −x_{1}^{9}x_{4}^{2} +x_{1}^{8}x_{2}^{2}x_{4} +x_{1}^{8}x_{2}x_{3}x_{4} − x_{1}^{8}x_{3}^{2}x_{4} +x_{1}^{7}x_{2}^{2}x_{3}^{2} +x_{1}^{7}x_{2}^{2}x_{4}^{2} +x_{1}^{7}x_{2}x_{3}^{3} +x_{1}^{7}x_{2}x_{3}x_{4}^{2} +x_{1}^{7}x_{3}^{4} −x_{1}^{7}x_{3}^{2}x_{4}^{2} +x_{1}^{7}x_{4}^{4} −x_{1}^{6}x_{2}^{3}x_{3}x_{4} − x_{1}^{6}x_{2}^{2}x_{3}^{2}x_{4}−x_{1}^{6}x_{2}^{2}x_{4}^{3}−x_{1}^{6}x_{2}x_{3}x_{4}^{3}−x_{1}^{6}x_{3}^{4}x_{4}+x_{1}^{6}x_{3}^{2}x_{4}^{3}+x_{1}^{6}x_{4}^{5}+x_{1}^{5}x_{2}^{5}x_{3}−x_{1}^{5}x_{2}^{4}x_{4}^{2}−x_{1}^{5}x_{2}^{3}x_{3}^{3}+ x_{1}^{5}x_{2}^{3}x_{3}x_{4}^{2} + x_{1}^{5}x_{2}^{2}x_{3}^{4} + x_{1}^{5}x_{2}x_{3}^{5} + x_{1}^{5}x_{2}x_{3}^{3}x_{4}^{2} + x_{1}^{5}x_{3}^{6} + x_{1}^{5}x_{3}^{4}x_{4}^{2} + x_{1}^{5}x_{4}^{6} − x_{1}^{4}x_{2}^{5}x_{3}x_{4} + x_{1}^{4}x_{2}^{3}x_{3}^{3}x_{4} − x_{1}^{4}x_{2}^{2}x_{3}^{4}x_{4} + x_{1}^{4}x_{2}^{2}x_{3}^{2}x_{4}^{3} − x_{1}^{4}x_{2}^{2}x_{4}^{5} − x_{1}^{4}x_{2}x_{3}^{5}x_{4} + x_{1}^{4}x_{2}x_{3}^{3}x_{4}^{3} − x_{1}^{4}x_{2}x_{3}x_{4}^{5} − x_{1}^{4}x_{3}^{6}x_{4} +x_{1}^{4}x_{3}^{4}x_{4}^{3} +x_{1}^{4}x_{3}^{2}x_{4}^{5} +x_{1}^{4}x_{4}^{7} −x_{1}^{3}x_{2}^{7}x_{3} +x_{1}^{3}x_{2}^{6}x_{3}^{2} +x_{1}^{3}x_{2}^{6}x_{4}^{2} −x_{1}^{3}x_{2}^{5}x_{3}x_{4}^{2} +x_{1}^{3}x_{2}^{4}x_{3}^{4} + x_{1}^{3}x_{2}^{4}x_{3}^{2}x_{4}^{2} +x_{1}^{3}x_{2}^{4}x_{4}^{4} −x_{1}^{3}x_{2}^{3}x_{3}^{5} +x_{1}^{3}x_{2}^{3}x_{3}^{3}x_{4}^{2} −x_{1}^{3}x_{2}^{2}x_{4}^{6} +x_{1}^{3}x_{2}x_{3}^{7} −x_{1}^{3}x_{2}x_{3}^{5}x_{4}^{2} +x_{1}^{3}x_{2}x_{3}^{3}x_{4}^{4} − x_{1}^{3}x_{2}x_{3}x_{4}^{6}−x_{1}^{3}x_{3}^{8}−x_{1}^{3}x_{3}^{4}x_{4}^{4}+x_{1}^{3}x_{3}^{2}x_{4}^{6}−x_{1}^{3}x_{4}^{8}−x_{1}^{2}x_{2}^{8}x_{4}+x_{1}^{2}x_{2}^{7}x_{3}x_{4}+x_{1}^{2}x_{2}^{6}x_{3}^{2}x_{4}−x_{1}^{2}x_{2}^{6}x_{4}^{3}− x_{1}^{2}x_{2}^{4}x_{3}^{2}x_{4}^{3} + x_{1}^{2}x_{2}^{4}x_{4}^{5} − x_{1}^{2}x_{2}^{2}x_{3}^{6}x_{4} − x_{1}^{2}x_{2}^{2}x_{3}^{4}x_{4}^{3} + x_{1}^{2}x_{2}^{2}x_{4}^{7} + x_{1}^{2}x_{2}x_{3}^{7}x_{4} + x_{1}^{2}x_{2}x_{3}^{3}x_{4}^{5} + x_{1}^{2}x_{2}x_{3}x_{4}^{7} + x_{1}^{2}x_{3}^{8}x_{4} − x_{1}^{2}x_{3}^{6}x_{4}^{3} − x_{1}^{2}x_{3}^{4}x_{4}^{5} − x_{1}^{2}x_{3}^{2}x_{4}^{7} − x_{1}x_{2}^{9}x_{3} + x_{1}x_{2}^{8}x_{3}^{2} − x_{1}x_{2}^{7}x_{3}^{3} − x_{1}x_{2}^{6}x_{3}^{4} +x_{1}x_{2}^{5}x_{3}^{5} −x_{1}x_{2}^{5}x_{3}^{3}x_{4}^{2} +x_{1}x_{2}^{4}x_{3}^{6} +x_{1}x_{2}^{4}x_{3}^{4}x_{4}^{2} −x_{1}x_{2}^{3}x_{3}^{7} +x_{1}x_{2}^{3}x_{3}^{5}x_{4}^{2} −x_{1}x_{2}^{3}x_{3}x_{4}^{6} − x_{1}x_{2}^{2}x_{3}^{8} +x_{1}x_{2}^{2}x_{3}^{6}x_{4}^{2} +x_{1}x_{2}^{2}x_{3}^{2}x_{4}^{6} −x_{1}x_{2}x_{3}^{9} −x_{1}x_{2}x_{3}^{7}x_{4}^{2} −x_{1}x_{2}x_{3}^{3}x_{4}^{6} −x_{1}x_{3}^{8}x_{4}^{2} +x_{1}x_{3}^{4}x_{4}^{6} − x_{1}x_{4}^{10} − x_{2}^{8}x_{3}^{2}x_{4} + x_{2}^{8}x_{4}^{3} + x_{2}^{7}x_{3}^{3}x_{4} + x_{2}^{6}x_{3}^{4}x_{4} + x_{2}^{6}x_{3}^{2}x_{4}^{3} + x_{2}^{6}x_{4}^{5} − x_{2}^{5}x_{3}^{5}x_{4} − x_{2}^{4}x_{3}^{6}x_{4} − x_{2}^{4}x_{3}^{4}x_{4}^{3} + x_{2}^{4}x_{3}^{2}x_{4}^{5} + x_{2}^{3}x_{3}^{7}x_{4} + x_{2}^{3}x_{3}^{5}x_{4}^{3} + x_{2}^{2}x_{3}^{8}x_{4} + x_{2}^{2}x_{3}^{6}x_{4}^{3} + x_{2}^{2}x_{3}^{4}x_{4}^{5} + x_{2}^{2}x_{3}^{2}x_{4}^{7} + x_{2}^{2}x_{4}^{9} − x_{2}x_{3}^{9}x_{4} + x_{2}x_{3}^{7}x_{4}^{3} + x_{2}x_{3}^{3}x_{4}^{7} + x_{2}x_{3}x_{4}^{9} + x_{3}^{6}x_{4}^{5} + x_{3}^{4}x_{4}^{7} − x_{3}^{2}x_{4}^{9} + x_{4}^{11})

h_{(13,1)} = Φ(−x_{1}^{4}x_{4}^{9} −x_{1}^{3}x_{2}^{10} −x_{1}^{3}x_{2}^{9}x_{3} +x_{1}^{3}x_{3}^{10} −x_{1}^{3}x_{4}^{10} +x_{1}^{2}x_{4}^{11} +x_{1}x_{2}^{12} −x_{1}x_{2}^{9}x_{3}^{3} + x_{1}x_{2}^{9}x_{3}x_{4}^{2} −x_{1}x_{2}^{3}x_{3}^{9} −x_{1}x_{2}x_{3}^{9}x_{4}^{2} −x_{1}x_{3}^{12} +x_{1}x_{4}^{12} +x_{2}^{12}x_{4} −x_{2}^{10}x_{3}^{2}x_{4} +x_{2}^{10}x_{4}^{3} −x_{2}^{9}x_{3}^{3}x_{4} + x_{2}^{9}x_{3}x_{4}^{3} + x_{2}^{4}x_{4}^{9} − x_{2}^{3}x_{3}x_{4}^{9} − x_{2}^{2}x_{3}^{10}x_{4} − x_{2}^{2}x_{3}^{2}x_{4}^{9} + x_{2}x_{3}^{3}x_{4}^{9} − x_{3}^{10}x_{4}^{3} + x_{3}^{4}x_{4}^{9} − x_{4}^{13})

Proposition 5.2.2. Let 𝒮_{1} be the set of polynomials listed in Table 5.2.1. This set 𝒮_{1} generates a submodule M_{1} of (R ⊗ S)^{G} which is a free module over 𝒜, and with Poincaré Series

P (M_{1}, t, s)

= \frac{s(t + t^{3} + t^{5} + 3t^{7} + 3t^{9} + 4t^{11} + 6t^{13} + 4t^{15} + 3t^{17} + 3t^{19} + t^{21} + t^{23} + t^{25})}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}

= \frac{N_{2}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

degree

(1, 1)  h_{(1,1)}
(3, 1)  df_{4}
(5, 1)  h_{(5,1)}
(7, 1)  df_{8,1}, df_{8,2}, h_{(7,1)}
(9, 1)  df_{10}, dg_{10}, h_{(9,1)}
(11, 1)  dg_{12,1}, g_{10} h_{(1,1)}, h_{(11,1),1}, h_{(11,1),2}
(13, 1)  dg_{14,1}, dg_{14,2}, g_{10} df_{4}, g_{12,1} h_{(1,1)}, g_{12,2} h_{(1,1)}, h_{(13,1)}
(15, 1)  g_{12,1}df_{4}, g_{12,2} df_{4}, g_{14,2} h_{(1,1)}, dg_{16}
(17, 1)  g_{10} df_{8,1}, g_{10} df_{8,2}, g_{14,2} df_{4}
(19, 1)  g_{12,1} df_{8,1}, g_{12,2} df_{8,1}, g_{10} df_{10}
(21, 1)  g_{14,1} df_{8,2}
(23, 1)  g_{14,1} dg_{10}
(25, 1)  g_{16} dg_{10}

Table 5.2.1. The set 𝒮_{1}

Proof. By construction, each element of 𝒮_{1} is invariant under G. This implies that M_{1} ⊆ (R ⊗ S)^{G}.

Let I_{R⊗S} be the ideal in R ⊗ S generated by 𝒫. In order to show that 𝒮_{1} is a set of free module generators over 𝒜 = K[𝒫], it suffices to show that all elements of 𝒮_{1} are linearly independent over 𝔽_{3} modulo I_{R⊗S}. Since all elements of 𝒫 and 𝒮_{1} are homogeneous, we need to check that for each given degree, the set of elements of that degree is linearly independent module I_{R⊗S}. For a fixed i, let g_{1}, …, g_{l} be elements of 𝒮_{1} of degree (i, 1). Any 𝔽_{3}-linear combination of g_{1}, …, g_{l} will have the form

g = z_{1}y_{1} + z_{2}y_{2} + z_{3}y_{3} + z_{4}y_{4},

for z_{1}, …, z_{4} ∈ R_{i}. Now {y_{1}, y_{2}, y_{3}, y_{4}} is a basis for R ⊗ S_{1} over R. Therefore g ∈ I_{R⊗S} if and only if each z_{i} is in the ideal I_{R} of R generated by 𝒫. So linear independence of g_{1}, …, g_{l} modulo I_{R⊗S} will follow from linear independence modulo I_{R} of the coefficients of y_{1} in g_{1}, …, g_{l}. We can check this linear independence using MAGMA which computes normal forms modulo I_{R} by using a Groebner basis for I_{R}. See Appendix (A.4.2) for the MAGMA computational results.

As a further check, we verified directly the linear independence modulo I = I_{R⊗S} of the sets of elements of 𝒮_{1} of a fixed degree. These calculations were done with Macaulay 2 which is able to compute in a graded-commutative ring and compute a Groebner basis for I_{R⊗S}. For this, we used the following observation.

Lemma 5.2.3. Let g_{1}, g_{2}, g_{3} ∈ R_{i} ⊗S_{1} for some i ≥ 1. If g_{1}g_{2}g_{3} ∉ I, then g_{1}, g_{2} and g_{3} are linearly independent modulo I.

Proof. Suppose that

α_{1}g_{1} + α_{2}g_{2} + α_{3}g_{3} ∈ I for some α_{1}, α_{2}, α_{3} ∈ 𝔽_{3}.

Since each g_{i} is of degree 1 in y_{1}, …, y_{4}, it satisfies g_{i}^{2} = 0. Then multiplying the above equation by g_{2}g_{3} shows that α_{1}g_{1}g_{2}g_{3} ∈ I. Similarly, α_{2}g_{2}g_{1}g_{3} ∈ I and α_{3}g_{3}g_{1}g_{2} ∈ I. Since g_{1}g_{2}g_{3} ∉ I, this implies that α_{1} = α_{2} = α_{3} = 0.

To check linear independence of sets of more than three elements, we noted that any relation must eliminate the leading terms of the polynomials in it. So we could reduce to checks of relations between smaller sets, and then check all possible cases. See Appendix A.5 for the Macaulay 2 results.

The Poincaré Series of M_{1} follows from the degrees of the free module generators over 𝒜.

The submodule M_{2}. To construct M_{2} = (R ⊗ S_{2})^{G}, first note that the element

h_{2} = y_{1}y_{4} + y_{2}y_{3}

of S_{2} is invariant under G. So products of h_{2} with the secondary invariants of R^{G} will give invariants in R ⊗ S_{2}.

We may obtain further invariants in R ⊗ S_{2} by taking products of two elements of the set 𝒮_{1} of generators for M_{1} = (R ⊗ S_{1})^{G}. Also, if h ∈ R ⊗ S_{1} is invariant, then so is dh ∈ R ⊗ S_{2}. So exterior derivatives of elements of 𝒮_{1} also give invariants in R ⊗ S_{2}. Many of these derivatives are zero, since d^{2} = 0, but we are able to obtain the invariants dh_{(7,1)}, dh_{(9,1)}, dh_{(11,1),1} and dh_{(13,1)} in this way.

In addition, we needed one further invariant of degree (4, 2). Using the Reynolds Operator, we computed

h_{(4,2)} = ℛ_{G}(x_{2}x_{3}^{2}x_{4}y_{1}y_{2})
= (x_{1}^{3}x_{2} − x_{1}x_{2}^{3} + x_{1}^{3}x_{3} + x_{1}x_{3}^{3} − x_{2}^{3}x_{4} + x_{2}x_{3}^{2}x_{4} − x_{1}x_{3}x_{4}^{2} − x_{2}x_{4}^{3})y_{1}y_{2}

+ (x_{1}x_{2}^{3} − x_{1}^{3}x_{3} + x_{1}x_{3}^{3} + x_{2}^{3}x_{4} + x_{2}^{2}x_{3}x_{4} + x_{1}x_{2}x_{4}^{2} − x_{2}x_{4}^{3} + x_{3}x_{4}^{3})y_{1}y_{3}

+ (x_{1}^{4} − x_{2}^{4} + x_{2}^{3}x_{3} + x_{2}^{2}x_{3}^{2} − x_{2}x_{3}^{3} − x_{3}^{4} − x_{1}^{3}x_{4} − x_{1}^{2}x_{4}^{2} + x_{1}x_{4}^{3} + x_{4}^{4})(y_{1}y_{4} − y_{2}y_{3})

+ (x_{1}^{3}x_{2} + x_{1}^{3}x_{3} + x_{1}x_{2}x_{3}^{2} − x_{1}x_{3}^{3} − x_{2}^{3}x_{4} + x_{1}^{2}x_{3}x_{4} + x_{3}^{3}x_{4} + x_{2}x_{4}^{3})y_{2}y_{4}

+ (−x_{1}^{3}x_{3} + x_{1}x_{2}^{2}x_{3} − x_{1}x_{3}^{3} − x_{1}^{2}x_{2}x_{4}x_{2}^{3}x_{4} + x_{3}^{3}x_{4} + x_{2}x_{4}^{3} − x_{3}x_{4}^{3})y_{3}y_{4}

Proposition 5.2.4. Let 𝒮_{2} be the set of polynomials in R ⊗ S_{2} listed in Table 5.2.2, and let 𝒮 be the set of secondary invariants of R^{G} given in Theorem 4.2.4.

Then the disjoint union of h_{2}𝒮 and 𝒮_{2} is a basis for a free module M_{2} over 𝒜. This free module has Poincaré Series

P (M_{2}, t, s) = \frac{N_{2}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

degree

(4, 2)  df_{4} h_{(1,1)}, h_{(4,2)}
(6, 2)  h_{(5,1)} h_{(1,1)}, dh_{(7,1)}
(8, 2)  df_{8,1} h_{(1,1)}, df_{8,2} h_{(1,1)}, h_{(7,1)} h_{(1,1)}, dh_{(9,1)}
(10, 2)  df_{10} h_{(1,1)}, dg_{10} h_{(1,1)}, h_{(9,1)} h_{(1,1)}, df_{8,1} df_{4}, df_{8,2} df_{4}, dh_{(11,1),1}
(12, 2)  dg_{12,1} h_{(1,1)}, h_{(11,1),1} h_{(1,1)}, h_{(11,1),2} h_{(1,1)}, df_{10} df_{4}, dg_{10} df_{4}, dh_{(13,1)}
(14, 2)  dg_{14,1} h_{(1,1)}, dg_{14,2} h_{(1,1)}, g_{10} df_{4} h_{(1,1)} h_{(13,1)} h_{(1,1)}, h_{(11,1),2} df_{4}, dg_{10} h_{(5,1)}
(16, 2)  g_{12,1} df_{4} h_{(1,1)}, dg_{12,1} h_{(5,1)}, df_{10} df_{8,1}, df_{10} df_{8,2}, h_{(11,1),2} h_{(5,1)}, dg_{10} h_{(7,1)}
(18, 2)  g_{10} df_{8,1} h_{(1,1)}, dg_{16} df_{4}, dg_{14,2} h_{(5,1)}, h_{(11,1),2} df_{8,1}
(20, 2)  dg_{10} dg_{12,1}, df_{10} dg_{12,2}
(22, 2)  df_{10} dg_{14,1}, dg_{10} dg_{14,1}

Table 5.2.2. The set 𝒮_{2}

Proof. As in the proof of Proposition 5.2.2, it suffices to show that the set of elements of h_{2}𝒮 ∪ 𝒮_{2} of each fixed degree is linearly independent modulo I_{R⊗S}, the ideal in R ⊗ S generated by 𝒫. We checked this using Macaulay 2. See Appendix A.6 for the computation results. Since h_{2}𝒮 ∪ 𝒮_{2} is a set of free module generators, the module M_{2,1} over 𝒜 generated by h_{2}𝒮 has Poincaré Series

P (M_{2,1}, t, s) = \frac{s^{2}(1 + t^{10} + 2t^{12} + 2t^{14} + t^{16} + t^{26})}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})},

and the module M_{2,2} over 𝒜 generated by 𝒮_{2} has Poincaré Series

P (M_{2,2}, t, s) = \frac{2s^{2}(t^{4} + t^{6} + 2t^{8} + 3t^{10} + 3t^{12} + 3t^{14} + 3t^{16} + 2t^{18} + t^{20} + t^{22})}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

Therefore M_{2} has Poincaré Series

P (M_{2}, t, s) = P (M_{2,1}, t, s) + P (M_{2,2}, t, s)

= \frac{N_{2}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

The submodules M_{3} and M_{4}. We can generate M_{3} = (R ⊗ S_{3})^{G} and M_{4} = (R ⊗ S_{4})^{G} from our previously constructed invariants.

Proposition 5.2.5. Let h_{2} = y_{1}y_{4} + y_{2}y_{3} be the G-invariant element of S_{2} as above, and let 𝒮_{1} be the set of invariants from Proposition 5.2.2. Then the set h_{2}𝒮_{1} generates a free module M_{3} over 𝒜 with Poincaré Series

P (M_{3}, t, s) = \frac{N_{3}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

Proof. Multiplication by h_{2} is an isomorphism ϕ of S_{1} onto S_{3}, sending the basis {y_{1}, y_{2}, y_{3}, y_{4}} of S_{1} to the basis {y_{2}y_{3}y_{1}, y_{1}y_{4}y_{2}, y_{1}y_{4}y_{3}, y_{2}y_{3}y_{4}} of S_{3}. So it is an isomorphism of R ⊗ S_{1} onto R ⊗ S_{3}. Since h_{2} commutes with all elements of 𝒜 ⊆ R, this is an isomorphism of 𝒜-modules. Therefore ϕ sends the free 𝒜-module M_{1} generated by 𝒮_{1} to the free module M_{3} over 𝒜 generated by h_{2}𝒮_{1}. Then the Poincaré Series of M_{3} is

P (M_{3}, t, s) = s^{2}P (M_{1}, t, s) = \frac{N_{3}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

Proposition 5.2.6. Let h_{4} = y_{1}y_{2}y_{3}y_{4} ∈ S_{4}, and let 𝒮 be the set of secondary invariants of R^{G} given in Theorem 4.2.4. Then the module M_{4} generated over 𝒜 by h_{4}𝒮 is free, and has Poincaré Series

P (M_{4}, t, s) = \frac{N_{4}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})}.

Proof. The homogeneous component S_{4} of S is a 1-dimensional 𝔽_{3}-vector space generated by h_{4}. Therefore multiplication by h_{4} is an isomorphism ϕ of R onto R⊗S_{3}. Since h_{4} commutes with all elements of 𝒜 ⊆ R, this is an isomorphism of 𝒜-modules. The set 𝒮 generates the free module R^{G} over 𝒜. So its image h_{4}𝒮 under ϕ generates a free module M_{4}. The Poincaré Series of M_{4} is

P (M_{4}, t, s) = s^{4}P (R^{G}, t, s) = \frac{N_{4}(t, s)}{(1 − t^{4})(1 − t^{8})^{2}(1 − t^{10})},

as required.

Now we combine these Propositions to give our main result:

Theorem 5.2.7. The cohomology algebra H^{∗}(ON, 𝔽_{3}) is isomorphic to the algebra of invariants H = (𝔽_{3}[x_{1}, …, x_{4}] ⊗ Λ(y_{1}, …, y_{4}))^{G}. This 𝔽_{3}-algebra of invariants may be described as follows. It contains four algebraically independent elements f_{4}, f_{8,1}, f_{8,2}, f_{10} of degrees (4, 0), (8, 0), (8, 0), (10, 0), respectively, such that H is a free module over 𝒜 = 𝔽_{3}[f_{4}, f_{8,1}, f_{8,2}, f_{10}]. As such, H has a basis consisting of the 128 elements in the disjoint union

𝒮 ∪ 𝒮_{1} ∪ h_{2}𝒮 ∪ 𝒮_{2} ∪ h_{2}𝒮_{1} ∪ h_{4}𝒮

of the sets 𝒮, 𝒮_{1}, h_{2}𝒮, 𝒮_{2}, h_{2}𝒮_{1} and h_{4}𝒮 given in Theorem 4.2.4 and propositions 5.2.2 - 5.2.6.

Bibliography

[1] A. Adem, J. Maginnis and R. J. Milgram, The geometry and cohomology of the Mathieu group M_{12}, J. of Algebra, 139 (1991), 90-133.

[2] A. Adem and R. J. Milgram, The cohomology of the Mathieu group M_{22}, Topology, 34 (1995), 389-410.

[3] A. Adem and R. J. Milgram, The subgroup structure and mod 2 cohomology of O’Nan’s sporadic simple group, J. Algebra, 176 (1995), no. 1, 288-315.

[4] A. Adem and R. J. Milgram, The cohomology of the McLaughlin group and some associated groups, Math. Zeit., 224 (1997), 495-517.

[5] D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge University Press (1993).

[6] W. Bosma and J. J. Cannon, Handbook of Magma Functions, Sydney: School of Mathematics and Statistics, University of Sydney (1995).

[7] J. F. Carlson, J. Maginnis and R. J. Milgram, The cohomology of the sporadic groups J_{2} and J_{3}, J. Algebra, 214 (1999), no. 1, 143-173.

[8] L. Evens, The Cohomology of Groups, Oxford University Press, New York (1991).

[9] D. J. Green, The 3-local cohomology of the Mathieu group M_{24}, Glasgow Math. J., 38 (1996), no. 1, 69-75.

[10] M. Hochster and J. A. Eagon, Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci, Am. J. Math, 93 (1971), 1020-1058.

[11] G. Kemper, Calculating invariant rings of finite groups over arbitrary fields, J. Symbolic Computation, 21 (1996), 351-366.

[12] G. Kemper and A. Steel, Some algorithms in invariant theory of finite groups, Computational Methods for Representations of Groups and Algebras, Progress in Mathematics, Birkhauser, Basel, 1998 (to appear).

[13] R. J. Milgram, The cohomology of the Mathieu group M_{23}, preprint.

[14] M. E. O’Nan, Some evidence for the existence of a new simple group, Proc. London Math. Soc., 32 (1976), 421-479.

[15] L. Smith, Polynomial Invariants of Finite Groups, Research Notes in Mathematics 6, A. K. Peters, Wellesley, MA (1995).

[16] L. Solomon, Invariants of finite reflection groups, Nagoya J. of Math., 22 (1963), 57-64.

[17] R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. of the Amer. Math. Soc. (3), 1 (1979), 475-511.

[18] B. Sturmfels, Algorithms in Invariant Theory, Springer-Verlag (1993).

[19] R. G. Swan, The p-period of a finite group, Ill. J. Math, 4 (1960), 341-346.

[20] M. Tezuka and N. Yagita, On odd prime components of cohomologies of sporadic simple groups and the rings of universal stable elements, J. Algebra, 183 (1996), no. 2, 483-513.

APPENDIX A Computer Output

A.1. MAGMA: Construction of Invariants

> SetEchoInput(true);
> load "in_invars";
Loading "in_invars"
>

(A.1.1). Define the group G and its action on the polynomial ring R

> k:= FiniteField(3);
> G<alpha1,alpha2,beta1,beta2,tau,omega>:= MatrixGroup<4,k|
>                               [0,0,1,0, 0,0,0,1, -1,0,0,0, 0,-1,0,0],
>                               [0,1,0,0, -1,0,0,0, 0,0,0,-1, 0,0,1,0],
>                               [0,0,0,1, 0,0,1,0, 0,-1,0,0, -1,0,0,0],
>                               [0,-1,-1,0, 1,0,0,1, 1,0,0,-1, 0,-1,1,0],
>                               [1,1,-1,1, 1,0,1,0, 0,1,0,-1, -1,1,1,1],
>                               [1,0,-1,-1, -1,-1,0,-1, -1,1,1,0, 0,-1,1,-1] >;
>
> RG:=InvariantRing(G);
> R<x1,x2,x3,x4>:=PolynomialRing(RG);
>

(A.1.2). Here we define the primary and secondary invariants. First, define the symplectic invariants f_{4} and f_{10}.

> // -------- symplectic invariants
> f4:=  x1*x4^3 + x2*x3^3 - x3*x2^3 - x4*x1^3;
> f10:= x1*x4^9 + x2*x3^9 - x3*x2^9 - x4*x1^9;
>

Now define f_{8,1} and g_{12,1} via Chern classes.

> // -------- chern classes
> orbit:= (x1^2+x2^2+x3^2+x4^2)^G;
> Q<y1,y2,y3,y4,X>,phi:=VariableExtension(R,1,false);
> chorbit:={ X-phi(p): p in orbit};
>
> chern:=1;
> for p in chorbit do
>       chern:= chern*p;
> end for;
>
> f81:=Coefficient(chern,X,6) @@ phi;

> g121:=Coefficient(chern,X,4) @@ phi;
>

The remaining primary and secondary invariants are obtained with the Reynolds Operator.

> // -------- Reynolds Operator
> f82:= ReynoldsOperator(x1^2*x2^2*x3^2*x4^2,G);
> g10:= ReynoldsOperator(x1^4*x2^2*x3^2*x4^2,G);
> g122:= ReynoldsOperator(x1^7*x4^5-x1^5*x4^7+ x2^7*x3^5-x2^5*x3^7, G);
> g141:= ReynoldsOperator(x1^13*x4-x1*x4^13+ x2^13*x3-x2*x3^13,G);
> g142:= ReynoldsOperator(x1^8*x2^2*x3^2*x4^2,G);
> g16:= ReynoldsOperator(x1^15*x4-x1*x4^15+ x2^15*x3-x2*x3^15,G);
>
>
> // ---------- define sets of primary and secondary invariants
> prim:=[f4,f81,f82,f10];
> second:=[1,g10,g121,g122,g141,g142,g16,g10*g16];
>
> save "res_invars";
Saving Magma state to "res_invars"

A.2. Relations among Secondary Invariants of R^{G}

Here, we give the relations among the secondary invariants of R^{G}. These are given by computing the products of all pairs of elements in the set {g_{10}, g_{12,1}, g_{12,2}, g_{14,1}, g_{14,2}, g_{16}}. For each such product, the output below gives the set of coefficients in 𝔽_{3}[𝒫] of the product as a linear combination of the secondary invariants 1, g_{10}, g_{12,1}, g_{12,2}, g_{14,1}, g_{14,2}, g_{16}, g_{10}g_{16}. Here t1,t2,t3,t4 denote the primary invariants f_{4}, f_{8,1}, f_{8,2}, f_{10}.

> restore "res_invars";
> HomogeneousModuleTest(prim,second, g10*g10);
true [
    2*t1^5 + t1^3*t2 + 2*t1^3*t3 + t1*t2*t3 + t1*t3^2 + t4^2,
    t4,
    2*t1^2 + t2 + t3,
    t1^2 + 2*t2 + 2*t3,
    0,
    0,
    2*t1,
    0
]
> HomogeneousModuleTest(prim,second, g10*g121);
true [
    2*t1^3*t4 + t1*t3*t4,
    t1^3 + t1*t2 + 2*t1*t3,
    t4,
    2*t4,
    0,
    t1^2 + t3,

0,
0
]
> HomogeneousModuleTest(prim,second, g10*g122);
true [
2*t1*t2*t4 + 2*t1*t3*t4,
2*t1^3,
2*t4,
0,
2*t1^2 + 2*t2 + t3,
t1^2 + 2*t2,
0,
0
]
> HomogeneousModuleTest(prim,second, g10*g141);
true [
t1^6 + t1^4*t2 + 2*t1^2*t2*t3 + 2*t1^2*t3^2 + t2^3 + 2*t2^2*t3 + t2*t3^2,
0,
t1^3,
2*t1*t2,
0,
2*t4,
t1^2 + 2*t2 + 2*t3,
0
]
> HomogeneousModuleTest(prim,second, g10*g142);
true [
2*t1^6 + 2*t1^4*t2 + 2*t1^4*t3 + 2*t1^2*t2^2 + t1^2*t2*t3 + t2^3 + t2*t3^2 +
    2*t3^3,
0,
2*t1^3 + 2*t1*t2,
t1^3 + 2*t1*t2 + 2*t1*t3,
2*t4,
t4,
t1^2 + 2*t2 + 2*t3,
0
]
> HomogeneousModuleTest(prim,second, g10*g16);
true [
0,
0,
0,
0,
0,
0,
0,
1
]
> HomogeneousModuleTest(prim,second, g121*g121);

true [
t1^6 + t1^4*t2 + 2*t1^4*t3 + t1^2*t2*t3 + 2*t1*t4^2 + t2^2*t3 + t2*t3^2 +
    2*t3^3,
0,
2*t1*t2 + t1*t3,
0,
t4,
2*t4,
t1^2 + t3,
0
]
> HomogeneousModuleTest(prim,second, g121*g122);
true [
t1^6 + t1^4*t2 + t1^2*t2*t3 + 2*t1*t4^2 + 2*t2*t3^2,
t1*t4,
t1^3,
t1^3 + t1*t2 + 2*t1*t3,
0,
t4,
2*t1^2 + 2*t3,
0
]
> HomogeneousModuleTest(prim,second, g121*g141);
true [
t1^2*t2*t4 + t1^2*t3*t4 + t2^2*t4 + 2*t2*t3*t4 + t3^2*t4,
t1^4 + t1^2*t2 + t1^2*t3 + 2*t2^2 + t2*t3,
t1*t4,
t1*t4,
t1^3 + t1*t2 + 2*t1*t3,
2*t1^3,
0,
2
]
> HomogeneousModuleTest(prim,second, g121*g142);
true [
t1^2*t2*t4 + 2*t1^2*t3*t4 + t2^2*t4 + t2*t3*t4 + 2*t3^2*t4,
t1^2*t2 + 2*t1^2*t3 + t2^2 + t2*t3 + 2*t3^2,
0,
t1*t4,
0,
2*t1^3 + t1*t2 + 2*t1*t3,
t4,
1
]
> HomogeneousModuleTest(prim,second, g121*g16);
true [
2*t1^7 + t1^5*t2 + t1^3*t3^2 + 2*t1^2*t4^2 + 2*t1*t2^3 + 2*t1*t2^2*t3 +
    2*t1*t2*t3^2 + t2*t4^2,
t2*t4 + 2*t3*t4,

2*t1^4 + 2*t1^2*t2 + 2*t1^2*t3 + t2^2 + 2*t2*t3,
2*t1^4 + t1^2*t2 + t2*t3 + t3^2,
2*t1*t4,
2*t1*t4,
t1^3 + t1*t2 + 2*t1*t3,
0
]
> HomogeneousModuleTest(prim,second, g122*g122);
true [
t1^4*t3 + 2*t1^2*t3^2 + t1*t4^2 + 2*t2^3 + 2*t2^2*t3,
t1*t4,
0,
0,
t4,
0,
2*t1^2 + t2,
0
]
> HomogeneousModuleTest(prim,second, g122*g141);
true [
t1^4*t4 + t1^2*t3*t4 + 2*t2*t3*t4,
2*t1^2*t3 + t2^2,
t1*t4,
2*t1*t4,
t1^3,
0,
t4,
0
]
> HomogeneousModuleTest(prim,second, g122*g142);
true [
2*t1^4*t4 + 2*t1^2*t2*t4 + t1^2*t3*t4 + 2*t2^2*t4 + 2*t2*t3*t4 + 2*t3^2*t4,
2*t1^4 + 2*t1^2*t2 + 2*t2*t3,
t1*t4,
t1*t4,
0,
t1^3,
0,
2
]
> HomogeneousModuleTest(prim,second, g122*g16);
true [
t1^5*t3 + 2*t1^3*t2^2 + 2*t1^3*t2*t3 + t1^3*t3^2 + 2*t1^2*t4^2 + t1*t2^3 +
    2*t1*t2^2*t3 + t3*t4^2,
t1^2*t4 + 2*t2*t4,
t1^2*t3 + 2*t2^2,
t1^4 + t1^2*t2 + t1^2*t3 + 2*t2^2 + t2*t3,
t1*t4,
2*t1*t4,

2*t1^3,
0
]
> HomogeneousModuleTest(prim,second, g141*g141);
true [
2*t1^5*t2 + t1^5*t3 + 2*t1^3*t2^2 + t1^3*t2*t3 + 2*t1^3*t3^2 + 2*t1*t2^3 +
    t3*t4^2,
t1^2*t4 + 2*t2*t4,
2*t1^2*t2 + 2*t1^2*t3,
2*t1^4 + 2*t1^2*t3 + 2*t2^2 + 2*t2*t3,
2*t1*t4,
t1*t4,
2*t1^3,
0
]
> HomogeneousModuleTest(prim,second, g141*g142);
true [
2*t1^7 + 2*t1^5*t3 + 2*t1^3*t2*t3 + 2*t1^2*t4^2 + t1*t2^3 + t1*t2^2*t3 +
    2*t1*t2*t3^2 + t2*t4^2,
2*t1^2*t4 + t2*t4 + 2*t3*t4,
2*t1^4 + t1^2*t2 + 2*t2^2 + 2*t2*t3,
2*t1^4 + 2*t1^2*t2 + 2*t1^2*t3 + 2*t2^2 + t2*t3 + 2*t3^2,
t1*t4,
t1*t4,
t1^3 + t1*t2,
0
]
> HomogeneousModuleTest(prim,second, g141*g16);
true [
2*t1^5*t4 + t1^3*t2*t4 + t1*t2^2*t4 + t1*t3^2*t4,
0,
2*t1^2*t4 + t2*t4,
t3*t4,
2*t1^4 + 2*t1^2*t2 + 2*t2*t3,
t1^2*t3 + 2*t2^2,
0,
0
]
> HomogeneousModuleTest(prim,second, g142*g142);
true [
2*t1^5*t2 + t1^5*t3 + 2*t1^3*t2^2 + 2*t1^3*t2*t3 + 2*t1^3*t3^2 + t1^2*t4^2 +
    2*t1*t2^3 + t1*t2^2*t3 + 2*t1*t2*t3^2 + 2*t1*t3^3 + 2*t2*t4^2 + t3*t4^2,
2*t1^2*t4 + t2*t4 + t3*t4,
t1^2*t2 + 2*t1^2*t3 + 2*t2*t3 + 2*t3^2,
2*t1^4 + 2*t1^2*t2 + 2*t1^2*t3 + 2*t2^2 + t2*t3 + 2*t3^2,
t1*t4,
0,
t1^3 + t1*t3,

]
> HomogeneousModuleTest(prim,second, g142*g16);
true [
    2*t1^5*t4 + 2*t1^3*t3*t4 + 2*t1*t2^2*t4 + 2*t1*t2*t3*t4 + 2*t1*t3^2*t4,
    2*t1^5 + t1^3*t2 + t1^3*t3,
    t1^2*t4 + 2*t2*t4 + t3*t4,
    2*t1^2*t4 + t2*t4,
    t1^4 + t1^2*t3 + 2*t2^2 + t3^2,
    t1^4 + t1^2*t2 + t2*t3,
    0,
    0
]

A.3. MAGMA: Check Generators for R^{G}

> load "checkR";
Loading "checkR"
>
> //  ---------- MAGMA Calculations for R^G ---------------

Restore the definitions of the primary and secondary invariants.

> restore "res_invars";
Restoring Magma state from "res_invars"
>

(A.3.1). Computation of the Groebner Basis.

> I:=ideal<R|prim>;
> gb:= GroebnerBasis(I);
>
> gb[1];
x1^8 + 2*x1^4*x2^4 + x1^4*x2^2*x3^2 + 2*x1^4*x3^4 + x1^2*x2^4*x4^2 +
    2*x1^2*x2^2*x3^2*x4^2 + x1^2*x3^4*x4^2 + 2*x1^2*x4^6 + x1*x2^3*x3*x4^3 +
    2*x1*x2*x3^3*x4^3 + x2^8 + x2^6*x3^2 + 2*x2^4*x4^4 + x2^2*x3^6 +
    x2^2*x3^2*x4^4 + x3^8 + 2*x3^4*x4^4 + x4^8
> gb[155];
x2^18 + x2^10*x4^8 + x2^8*x3^8*x4^2 + x2^8*x3^2*x4^8 + x2^7*x3^5*x4^6 +
    x2^7*x3*x4^10 + 2*x2^6*x3^10*x4^2 + 2*x2^6*x3^6*x4^6 + 2*x2^6*x3^4*x4^8 +
    2*x2^6*x3^2*x4^10 + x2^5*x3^11*x4^2 + 2*x2^5*x3^9*x4^4 + x2^5*x3^7*x4^6 +
    x2^5*x3*x4^12 + x2^4*x3^14 + x2^4*x3^10*x4^4 + x2^4*x3^8*x4^6 +
    x2^4*x3^6*x4^8 + 2*x2^4*x3^4*x4^10 + x2^4*x3^2*x4^12 + 2*x2^4*x4^14 +
    2*x2^3*x3^13*x4^2 + 2*x2^3*x3^9*x4^6 + x2^3*x3^3*x4^12 + x2^3*x3*x4^14 +
    2*x2^2*x3^16 + x2^2*x3^14*x4^2 + 2*x2^2*x3^12*x4^4 + x2^2*x3^8*x4^8 +
    2*x2^2*x3^6*x4^10 + x2^2*x3^4*x4^12 + 2*x2^2*x3^2*x4^14 + x2^2*x4^16 +
    x2*x3^17 + 2*x2*x3^13*x4^4 + 2*x2*x3^11*x4^6 + x2*x3^7*x4^10 +
    2*x2*x3^5*x4^12 + 2*x2*x3^3*x4^14 + x2*x3*x4^16 + 2*x3^14*x4^4 + x3^12*x4^6
    + 2*x3^10*x4^8 + x3^8*x4^10 + x3^6*x4^12 + 2*x3^4*x4^14 + x4^18
> gb[250];
x3^23
> gb[259];

x4^23
>
> save "res_gb";
Saving Magma state to "res_gb"
>

(A.3.2). Check that all secondary invariants are non-zero modulo I.

>
> g10 in I;
false
> g122 in I;
false
> g122 in I;
false
> g141 in I;
false
> g142 in I;
false
> g16 in I;
false
> g10*g16 in I;
false
>

(A.3.3). Check linear independence modulo I of the pair g_{12,1}, g_{12,2} and the pair g_{14,1}, g_{14,2}.

>
> g121-g122 in I,  g121+g122 in I;
false false
> g141-g142 in I,  g141+g142 in I;
false false
>

A.4. MAGMA: Check Generators for (R ⊗ S_{1})^{G}

> load "checkRS1";
Loading "checkRS1"
>
> //  ----------  Calculations for R*S_1 ---------------
>

Restore previous results.

> restore "res_gb";
Restoring Magma state from "res_gb"
>

(A.4.1). Definitions of the new invariants.

Define the function d, the derivative with respect to x_{1}.

> d:= func<p| Derivative(p,x1)>;
>

Now we construct a new polynomial ring RS in z_{1}, …, z_{4} and y_{1}, …, y_{4}. G acts diagonally on this polynomial ring. We map RS to R via ϕ, sending each z_{i} to x_{i}.

>
> M:=GModule(G);
> T:=DirectSum(M,M);
> GT:=MatrixGroup(T);
> RSG:=InvariantRing(GT);
> RS<z1,z2,z3,z4,y1,y2,y3,y4>:=PolynomialRing(RSG);
>
> phi:= hom<RS -> R | x1,x2,x3,x4,0,0,0,0>;
>

Now define the invariants.

> /* ----- define invariants h_(i,1) in RS  */
> h1 := z1*y4 + z2*y3 - z3*y2 - z4*y1;
> h5 := ReynoldsOperator(z1^5*y4,GT);
> h7 := ReynoldsOperator(z1^5*z4^2*y4,GT);
> h9 := ReynoldsOperator(z1^5*z4^4*y4,GT);
> h111 := ReynoldsOperator(z1^7*z4^4*y4,GT);
> h112 := ReynoldsOperator( z1^6*z2^2*z3^2*z4*y4,GT);
> h13 := ReynoldsOperator( z1^9*z2^2*z3^2*y4,GT);
>
>
> /* ----- now define j_i, the coefficent of y1 in h_(i,1), and map to R */
> j1:= phi( Coefficient(h1,y1,1) );
> j5:= phi( Coefficient(h5,y1,1) );
> j7:= phi( Coefficient(h7,y1,1) );
> j9:= phi( Coefficient(h9,y1,1) );
> j111:= phi( Coefficient(h111,y1,1) );
> j112:= phi( Coefficient(h112,y1,1) );
> j13:= phi( Coefficient(h13,y1,1) );
>
> save "res_rs1";
Saving Magma state to "res_rs1"
>

(A.4.2). Check linear independence modulo I of the sets of coefficients of y_{1}. The function HomogeneousModuleTest(P,S,g) checks if g is in the module generated by S over k[P].

>
> /* DEG (1,1)
>    --------- */
> j1 in I;
false
>
> /* DEG (3,1)
>    --------- */
> d(f4) in I;

false
>
> /* DEG (5,1)
>    --------- */
> j5 in I;
false
>
> /* DEG (7,1)
>    --------- */
> k1:= NormalForm(d(f81),I);
> k2:= NormalForm(d(f82),I);
> k3:= NormalForm(j7,I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
>
>
> /* DEG (9,1)
>    --------- */
> k1:= NormalForm(d(f10),I);
> k2:= NormalForm(d(g10),I);
> k3:= NormalForm(j9,I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
>
> /* DEG (11,1)
>    --------- */
> k1:= NormalForm(d(g121),I);
> k2:= NormalForm(g10*j1,I);
> k3:= NormalForm(j111,I);
> k4:= NormalForm(j112,I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2,k3], k4 );
false
>
>
> /* DEG (13,1)

>    --------- */
> k1:= NormalForm(d(g141),I);
> k2:= NormalForm(d(g142),I);
> k3:= NormalForm(g10*d(f4),I);
> k4:= NormalForm(g121*j1,I);
> k5:= NormalForm(g122*j1,I);
> k6:= NormalForm(j13,I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2,k3], k4 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2,k3,k4], k5 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2,k3,k4,k5], k6 );
false
>
> /* DEG (15,1)
>    --------- */
> k1:= NormalForm(g121*d(f4),I);
> k2:= NormalForm(g122*d(f4),I);
> k3:= NormalForm(g142*j1,I);
> k4:= NormalForm(d(g16),I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2,k3], k4 );
false
>
> /* DEG (17,1)
>    --------- */
> k1:= NormalForm(g10*d(f81),I);
> k2:= NormalForm(g10*d(f82),I);
> k3:= NormalForm(g142*d(f4),I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
>
> /* DEG (19,1)
>    --------- */

> k1:= NormalForm(g121*d(f81),I);
> k2:= NormalForm(g122*d(f81),I);
> k3:= NormalForm(g10*d(f10),I);
> HomogeneousModuleTest( [Id(R)], [Id(R)], k1 );
false
> HomogeneousModuleTest( [Id(R)], [k1], k2 );
false
> HomogeneousModuleTest( [Id(R)], [k1,k2], k3 );
false
>
> /* DEG (21,1)
>    --------- */
> g141*d(f82) in I;
false
>
> /* DEG (23,1)
>    --------- */
> g141*d(g10) in I;
false
>
> /* DEG (25,1)
>    --------- */
> g16*d(g10) in I;
false
>

A.5. Macaulay 2: Check Generators for (R ⊗ S_{1})^{G}

Macaulay 2, version 0.8.52
  Copyright 1993-1999, all rights reserved, D. R. Grayson and M. E. Stillman
  Factory 1.2c from Singular, copyright 1993-1997, G.-M. Greuel, R. Stobbe
  Factorization and characteristic sets 0.3.1, copyright 1996, M. Messollen
  GC, copyright 1996, Hans-J. Boehm, Alan J. Demers, Xerox, Silicon Graphics
  GNU libc and libg++, copyright 1996, Free Software Foundation
  GNU MP, copyright 1996, Free Software Foundation
i1 : input "checkS1"
i2 : --------------- Macaulay 2 Computations for R S_1 -------------

     load "input.m2"
i3 :
     ----( this defines the polynomials, and I = ideal generated by primaries)
     --- function f % I computes normal form of f modulo I
     --- note: monomial order used here is reverse lexicographic

     --  DEG 1
     h1 % I
o3 = x4*y1 + x3*y2 - x2*y3 - x1*y4
o3 : R
i4 :
     --  DEG 3
     df4 % I
       3       3       3       3
o4 = x4 y1 + x3 y2 - x2 y3 - x1 y4
o4 : R
i5 :
     --  DEG 5
     (h5 % I) == 0
o5 = false
i6 :
     --  DEG 7
     (df81 % I) * (df82 % I) *(h7 % I) == 0
o6 = false
i7 :
     --  DEG 9
     (df10 % I) * (dg10 % I) *(h9 % I) == 0
o7 = false
i8 :
     --  DEG 11
     k1 =dg121 % I;  k2 = (g10 * h1) % I;  k3 = h111 % I;  k4 = h112 % I;
i12 :
      leadTerm k1, leadTerm k2, leadTerm k3, leadTerm k4

         3  8      10           3  8         3  8
o12 = (x1 x2 y1, x2  x4*y1, -(x1 x2 y1), -(x1 x2 y1))
o12 : Sequence
i13 :
      --- k1,k3,k4 all have same lead term - higher than k2.
      --- so any relation must involve two of k1,k3,k4
      k1*k3*k4 == 0
o13 = false
i14 :
      k2*k1*k3 == 0,  k2*k1*k4 == 0,  k2*k3*k4 == 0
o14 = (false, false, false)
o14 : Sequence
i15 :
      --  DEG 13
       k1 = dg141 % I;  k2 = dg142 % I;  k3 = (g10 * df4) % I;
i18 :  k4 = (g121 * h1) % I; k5 = (g122 * h1) % I; k6 = h13 % I;
i21 :
      leadTerm k1,leadTerm k2,leadTerm k3,leadTerm k4,leadTerm k5,leadTerm k6
         5  8        5  8       10  3        2  2  8            10  3         5  8
o21 = (x1 x3 y1, -(x1 x3 y1), x2  x4 y1, -(x1 x2 x3 x4*y1), -(x2  x4 y1), -(x1 x3 y1))
o21 : Sequence
i22 :
      --- highest lead terms are k1,k2,k6. Check they do not cancel each other
      k1*k2*k6 == 0
o22 = false
i23 :
      --- if these lead terms are cancelled, get lead terms of
      leadTerm(k1+k2), leadTerm(k1+k6), leadTerm(k2-k6)
           3  2  8         2  2  8            3  2  8
o23 = (-(x1 x2 x3 y1), -(x1 x2 x3 x4*y1), -(x1 x2 x3 y1))

o23 : Sequence
i24 : --- so only possible relation involving these is k1+k6-k4
      leadTerm( k1+ k6 -k4)
        5  2  4  2
o24 = x1 x2 x3 x4 y1
o24 : R
i25 :
      ---- only other possible relation is k3+k5
      k3+k5 == 0
o25 = false
i26 :
      --  DEG 15
      k1 = (g121*df4) % I; k2 = (g122*df4) % I; k3 = (g142*h1) % I;
i29 : k4 = dg16;
i30 :
      leadTerm k1, leadTerm k2, leadTerm k3, leadTerm k4
         3  8  4        10  5       3  8  4        9  6
o30 = (x1 x3 x4 y1, -(x2  x4 y1), x1 x3 x4 y1, -(x1 x2 y1))
o30 : Sequence
i31 :
      --- highest lead term is k4. It can't cancel, so no relation involving k4.
      --- check relations between k1,k2,k3
      k1*k2*k3 == 0
o31 = false
i32 :
      --  DEG 17
      ( (g10 * df81) % I) * ( (g10*df82) % I) * ( (g10*h7) % I) == 0
o32 = false
i33 :

      --  DEG 19
      ( (g121*df81) % I) * ((g122* df81) % I) * ((g10*df10) % I) == 0
o33 = false
i34 :
      --  DEG 21
      ( (g141*df82) % I) == 0
o34 = false
i35 :
      --  DEG 23
      (g141* dg10) % I == 0
o35 = false
i36 :
      --  DEG 25
      (g16*dg10) % I == 0
o36 = false
i37 :
i38 : quit
Process M2 finished

A.6. Macaulay 2: Check Generators for (R ⊗ S_{2})^{G}

Macaulay 2, version 0.8.52
  Copyright 1993-1999, all rights reserved, D. R. Grayson and M. E. Stillman
  Factory 1.2c from Singular, copyright 1993-1997, G.-M. Greuel, R. Stobbe
  Factorization and characteristic sets 0.3.1, copyright 1996, M. Messollen
  GC, copyright 1996, Hans-J. Boehm, Alan J. Demers, Xerox, Silicon Graphics
  GNU libc and libg++, copyright 1996, Free Software Foundation
  GNU MP, copyright 1996, Free Software Foundation
i1 : input "checkS2"

i2 : ------ Macaulay 2 Computations for R S_2
     load "input.m2"
i3 :
     ---- Define exterior derivative D
     D = f ->  (( transpose (jacobian ideal(f)) )* ( transpose Y))_0_0
o3 = D
o3 : Function
i4 :
     -----  DEG (4,2)
     h41 = (df4*h1) % I;
i5 : h42 = hh42 % I;
i6 :
     h41 == 0, h42 == 0, (h41+h42) == 0, (h41-h42) == 0
o6 = (false, false, false, false)
o6 : Sequence
i7 :
     -----  DEG (6,2)
     h61 = (h5*h1) % I;
i8 : h62 = D(h7) % I;
i9 :
     h61 == 0, h62 == 0, (h61+h62) == 0, (h61-h62) == 0
o9 = (false, false, false, false)
o9 : Sequence
i10 :
      ----  DEG (8,2)
      h81 = (df81*h1) % I;
i11 : h82 = (df82*h1) % I;

i12 : h83 = (h7*h1) % I;
i13 : h84 = D(h9) % I;
i14 :
      leadTerm h81, leadTerm h82, leadTerm h83, leadTerm h84
         3  2  3         7            3  2  3         7
o14 = (x1 x2 x3 y1*y2, x1 x3*y1*y2, x1 x2 x3 y1*y2, x1 x2*y1*y2)
o14 : Sequence
i15 :
      ---- highest lead term is h84, then h82, then h81 and h82
      ---- so only possible relation is h81-h82
      h81 - h82 == 0
o15 = false
i16 :
      ----  DEG (10,2)
      k1 = (df10*h1) % I;
i17 : k2 = (dg10*h1) % I;
i18 : k3 = (h9*h1) % I;
i19 : k4 = (df82*df4) % I;
i20 : k5 = (df81*df4) % I;
i21 : k6 = D(h111) % I;
i22 :
      k7 = (h2*g10) % I;
i23 :
      leadTerm k1, leadTerm k2, leadTerm k3
           9               5  5            3  2  5
o23 = (-(x3 x4*y1*y2), -(x1 x3 y1*y2), -(x1 x2 x3 y1*y2))
o23 : Sequence
i24 : leadTerm k4, leadTerm k5, leadTerm k6, leadTerm k7
         7  3         3  2  5         5  5         2  8

o24 = (x1 x3 y1*y2, x1 x2 x3 y1*y2, x1 x2 y1*y2, x1 x2 y2*y3)
o24 : Sequence
i25 :
      --- only way to cancel lead terms is k3+k5
      leadTerm (k3+k5)
          4  5
o25 = -(x1 x2 x4*y1*y2)
o25 : R
i26 :
      --- this can't cancel, so no relations
      ----  DEG (12,2)
      k1 = (dg121*h1) % I;
i27 : k2 = (h111*h1) % I;
i28 : k3 = (h112*h1) % I;
i29 : k4 = (df10*df4) % I;
i30 : k5 = (dg10*df4) % I;
i31 : k6 = (D(h13)) % I;
i32 :
      k7 = (h2*g121) % I;
i33 : k8 = (h2*g122) % I;
i34 :
      leadTerm k1, leadTerm k2, leadTerm k3
         3  2  7           3  2  7          5  7
o34 = (x1 x2 x3 y1*y2, -(x1 x2 x3 y1*y2), x1 x3 y1*y2)
o34 : Sequence
i35 : leadTerm k4, leadTerm k5, leadTerm k6
           9  3            5  7          5     6
o35 = (-(x3 x4 y1*y2), -(x1 x3 y1*y2), x1 x2*x3 y1*y2)

o35 : Sequence
i36 : leadTerm k7, leadTerm k8
           4  8          4  8
o36 = (-(x1 x3 y2*y3), x1 x3 y2*y3)
o36 : Sequence
i37 :
      --- check all possibilities of lead terms cancelling
      leadTerm (k7+k8)
          2  2  8
o37 = -(x1 x2 x3 y2*y3)
o37 : R
i38 : leadTerm (k3+k5)
           11
o38 = x1*x3  y1*y2
o38 : R
i39 : leadTerm (k1+k2)
          4     6
o39 = -(x1 x2*x3 x4*y1*y2)
o39 : R
i40 :
      ----  DEG (14,2)
      k1 = (dg141*h1) % I;
i41 : k2 = (dg142*h1) % I;
i42 : k3 = (g10*df4*h1) % I;
i43 : k4 = (h13*h1) % I;
i44 : k5 = (h112*df4) % I;
i45 : k6 = (dg10*h5) % I;

i46 :
      k7 = (h2*g141) % I;
i47 : k8 = (h2*g142) % I;
i48 :
      leadTerm k1, leadTerm k2, leadTerm k3
            11  2         3  11            11  2
o48 = (x1*x2  x4 y1*y2, x1 x3  y1*y2, x1*x2  x4 y1*y2)
o48 : Sequence
i49 : leadTerm k4, leadTerm k5, leadTerm k6
              11  2            3  11             13
o49 = (-(x1*x2  x4 y1*y2), -(x1 x3  y1*y2), x1*x2  y1*y2)
o49 : Sequence
i50 : leadTerm k7, leadTerm k8
           2  12          2  12
o50 = (-(x1 x3  y2*y3), x2 x3  y2*y3)
o50 : Sequence
i51 :
      --- check all possibilities of lead terms cancelling
      leadTerm (k7+k8)
          2  12
o51 = -(x1 x3  y2*y3)
o51 : R
i52 :
      leadTerm (k2 + k5)
           12
o52 = x2*x3  x4*y1*y2
o52 : R
i53 : leadTerm (k1 - k3)
          3  2  7  2

o53 = -(x1 x2 x3 x4 y1*y2)
o53 : R
i54 : leadTerm (k1 + k4)
          3  2  7  2
o54 = -(x1 x2 x3 x4 y1*y2)
o54 : R
i55 : leadTerm (k3 + k4)
           10  3
o55 = x2*x3  x4 y1*y2
o55 : R
i56 :
      ----  DEG (16,2)
      k1  = (g121*df4*h1) % I;
i57 : k2  = (dg121*h5) % I;
i58 : k3  = (df10*df81) % I;
i59 : k4  = (df10*df82) % I;
i60 : k5 = (h112*h5) % I;
i61 : k6  = (dg10*h7) % I;
i62 :
      k7 = (h2*g16) % I;
i63 :
      leadTerm k1, leadTerm k2, leadTerm k3
           3  7  6          5  2  3  6           5  2  3  6
o63 = (-(x1 x2 x4 y1*y2), x1 x2 x3 x4 y1*y2, -(x1 x2 x3 x4 y1*y2))
o63 : Sequence
i64 : leadTerm k4, leadTerm k5, leadTerm k6
              15                  14                14

o64 = (-(x1*x3  y1*y2), -(x1*x2*x3  y1*y2), x1*x2*x3  y1*y2)
o64 : Sequence
i65 :
      leadTerm k7
        16
o65 = x3  y2*y3
o65 : R
i66 :
      --- check possibilities of lead terms cancelling
      leadTerm (k2+k3)
        4  5  7
o66 = x1 x2 x4 y1*y2
o66 : R
i67 : --- this can't cancel with others
      leadTerm (k5+k6)
        5  2  3  6
o67 = x1 x2 x3 x4 y1*y2
o67 : R
i68 : --- this could cancel with k2,k3 - check this
      leadTerm (k5+k6-k2)
           9  6
o68 = x1*x3 x4 y1*y2
o68 : R
i69 : leadTerm (k5+k6+k3)
           9  6
o69 = x1*x3 x4 y1*y2
o69 : R
i70 :

      ----  DEG (18,2)
      h181 = (g10*df81*h1) % I;
i71 : h182 = (dg16*df4) % I;
i72 : h183 = (dg142*h5) % I;
i73 : h184 = (h112*df81) % I;
i74 :
      leadTerm h181, leadTerm h182, leadTerm h183, leadTerm h184
         5  3  10           5  3  10          7     10           7     10
o74 = (x1 x2 x4  y1*y2, -(x1 x2 x4  y1*y2), x1 x2*x4  y1*y2, -(x1 x2*x4  y1*y2))
o74 : Sequence
i75 :
      --- check possibilites of lead terms cancelling
      leadTerm(h183+h184)
          3  5  10
o75 = -(x1 x2 x4  y1*y2)
o75 : R
i76 : leadTerm(h181+h182)
        5  2     10
o76 = x1 x2 x3*x4  y1*y2
o76 : R
i77 :
      ----  DEG (20,2)
      h201 = (dg10*dg121) % I;
i78 : h202 = (df10*dg122) % I;
i79 :
      h201 == 0, h202 == 0, (h201+h202) == 0, (h201-h202) == 0
o79 = (false, false, false, false)
o79 : Sequence

i80 :
      ---- DEG (22,2)
      h221 = (df10*dg141) % I;
i81 : h222 = (dg10*dg141) % I;
i82 :
      h221 == 0, h222 == 0, (h221+h222) == 0, (h221-h222) == 0
o82 = (false, false, false, false)
o82 : Sequence
i83 :
i84 : quit
Process M2 finished