A Classification of Polynomial Algebras as Modules over the Steenrod Algebra

Jeanne Duflot Dept. of Mathematics

Colorado State University Ft. Collins, CO 80523

Nick Kuhn^{∗} Dept. of Mathematics University of Virginia Charlottesville, VA 22903

Mark Winstead Dept. of Mathematics University of Virginia Charlottesville, VA 22903

April 6, 1992

1 Introduction

Suppose that A_{2} is the mod-2 Steenrod algebra, and that R = F_{2}[x_{1}, …, x_{n}] is the polynomial ring in n variables over the prime field F_{2}. Campbell and Selick [3] observed that the equations Sq^{1}x_{i} = x_{i−1}^{2} for 2 ≤ i ≤ n, and Sq^{1}x_{1} = x_{n}^{2}, define an action of A_{2} on R that makes R isomorphic as an A_{2}-module to the A_{2}-module defined by the standard action on R. In that paper, they also pose the following question, due to Tom Hunter: does the equation Sq^{1}x_{i} = ∑_{j} m_{ij}x_{j}^{2} (where M = [m_{ij}] is any n-by-n matrix over F_{2}) define an action of A_{2} on R?

In this paper we give an affirmative answer to the above question and classify the actions defined in this way. More precisely, let S(V) be the symmetric algebra generated by a finite dimensional F_{2} vector space V concentrated in degree 1. Then we have

∗Research supported in part by the NSF.

Theorem 1.1 Given M ∈ End(V), the equation Sq^{1}v = (Mv)^{2} extends to an unstable action of A_{2} on S(V) satisfying the Cartan formula.

Let S_{M}(V) denote S(V) with this A_{2}-action.

Theorem 1.2 Given two elements of End(V), M_{1} and M_{2}, S_{M_{1}}(V)is isomorphic to S_{M_{2}}(V) as A_{2}-modules if and only if

dim(ker(M_{1}^{i})) = dim(ker(M_{2}^{i})) for all i ≥ 1.

In particular, if M is invertible, S_{M}(V) is isomorphic to S_{I}(V), i.e., S(V) with the standard A_{2}-action.

Theorem 1.1 is proved in sections 2 and 3 using the Bullett and McDonald approach to the Adem relations introduced in [2]. Theorem 1.2 is proved in section 4. The proof^{1} uses nothing more than basic linear algebra and Galois theory, and avoids the use of Davenport’s normal basis theorem needed in [3]. The last section contains an alternate proof of a critical step of Theorem 1.2, as well as some remarks about the obvious generalizations of our results to odd primes.

2 Higher Derivations and A_{2}-algebras

Suppose F is a commutative ring of characteristic 2, and let R be a Z-graded F -algebra.

It is convenient to introduce the idea of a higher derivation, adapted from commutative algebra (e.g., see [4]).

A higher derivation^{2} of R over F is a sequence

D_{∗} = {D_{0}, D_{1}, …, D_{n}, …}

such that 1. D_{i} is an endomorphism of the graded F -module R of degree i, and

1Special thanks to Thann Ward for showing us that our Proposition 4.5, used in the proof of Theorem 1.2, is really a corollary of Lemma 3.4 of [5].

2Some refer to higher derivations as divided sequences.

2. D_{i}(fg) = ∑_{j=0}^{i} D_{j}(f)D_{i−j}(g), for every f, g ∈ R and for every i ∈ {0, 1, 2, …}.

Suppose that T and U are indeterminates. If D_{∗} is a higher derivation of R over F, then D_{∗} induces a ring homomorphism

D_{∗}(T): R → R[[T]]

defined by

D_{∗}(T)(f) = ∑_{i=0}^{∞} D_{i}(f)T^{i}.

Therefore, D_{∗} also induces a ring homomorphism

D_{∗}(T, U): R[[T]] → R[[T, U]]

defined by

D_{∗}(T, U)(∑_{i=0}^{∞} a_{i}T^{i}) = ∑_{i=0}^{∞} D_{∗}(T)(a_{i})U^{i}
= ∑_{i=0}^{∞}(∑_{j=0}^{∞} D_{j}(a_{i})T^{j})U^{i}
= ∑_{n=0}^{∞}(∑_{k=0}^{n} D_{k}(a_{n−k})T^{k}U^{n−k}).

Mimicking Bullett and MacDonald [2], we say that a higher derivation D_{∗} of R over F satisfies the mod-2 Adem relations on R if and only if

(1) (D_{∗}(s^{2} + st)D_{∗}(t^{2}))(f) = (D_{∗}(t^{2} + st)D_{∗}(s^{2}))(f) for every f ∈ R.

In the equation (1) we interpret D_{∗}(s^{2} +st)D_{∗}(t^{2}) as the ring homomorphism defined by the composite

R ⟶^{D_{∗}(T)} R[[T]] ⟶^{D_{∗}(T,U)} R[[T, U]] → R[[s, t]]

T ↦ s^{2} + st

U ↦ t^{2}

and similarly for the right hand side of the equation (1). (Here, s and t are new indeterminates.)

A higher derivation D_{∗} of R over F is unstable if and only if

D_{i}(f) = 0 for every i > deg f.

Following Bullett and Macdonald [2], we see that the next thing to do is to prove

Theorem 2.1 Let R be a graded F -algebra that is generated by elements of degree 1 as an F -algebra. Suppose that D_{∗} is a higher derivation of R over F such that

(ı) D_{∗} is unstable

(ıı) D_{0}D_{1} = D_{1}D_{0}, and

(ııı) D_{1}D_{1} = 0.

Then D_{∗} satisfies the mod-2 Adem relations on R.

Proof: In view of the fact that R is generated as a ring by F = R_{0} and R_{1}; and the fact that both D_{∗}(s^{2} + st)D_{∗}(t^{2}) and D_{∗}(t^{2} + st)D_{∗}(s^{2}) are ring homomorphisms, we need only verify (1) for f ∈ F and for f ∈ R_{1}.

If f ∈ F, then D_{i}(f) = 0 if i > 0, so both sides of (1) reduce to D_{0}D_{0}(f). If f ∈ R_{1} then D_{∗}(T, U)(D_{∗}(T)(f)) = ∑_{n=0}^{∞}(∑_{k=0}^{n} D_{k}(D_{n−k}(f))T^{k}U^{n−k}). Since D_{k}D_{n−k}(f) = 0 if n − k > 1 or if k > n − k + 1, and D_{1}D_{1} = 0, we see that

D_{∗}(T, U)(D_{∗}(T)(f)) = D_{0}D_{0}(f) + D_{1}D_{0}(f)T + D_{0}D_{1}(f)U + D_{2}D_{1}(f)T^{2}U.

Setting T = s^{2} + st and U = t^{2}, we get

D_{0}D_{0}(f) + D_{1}D_{0}(f)(s^{2} + st) + D_{0}D_{1}(f)t^{2} + D_{2}D_{1}(f)(s^{2} + st)^{2}t^{2},

and setting T = t^{2} + st and U = s^{2}, we get

D_{0}D_{0}(f) + D_{1}D_{0}(f)(t^{2} + st) + D_{0}D_{1}(f)s^{2} + D_{2}D_{1}(f)(t^{2} + st)^{2}s^{2}.

Since (t^{2}+st)^{2}s^{2} = s^{2}t^{4}+2s^{3}t^{3}+s^{4}t^{2} = t^{2}(s^{2}+st)^{2}, and D_{0}D_{1}(f) = D_{1}D_{0}(f), the theorem is proved. □

Let A_{2} denote the usual mod-2 Steenrod algebra.

Corollary 2.2 Suppose that R and D_{∗} are as in Theorem 2.1. In addition, suppose that D_{0} = identity. Then the equation

Sq^{i}(f) ≡ D_{i}(f)

defines an action of A_{2} on R. (This action is F -linear, satisfies the Cartan formula and the unstable A_{2}-module condition

Sq^{i}(f) = 0 if i > deg f,

but does not necessarily satisfy the unstable A_{2}-algebra condition

Sq^{deg f}(f) = f^{2}.)

Proof: The residue calculation given in Bullett and Macdonald ([2], section 3) works fine in R[[s, t]] since the characteristics of F and R are both equal to 2. □

3 Proof of Theorem 1.1 and Other Examples

In this section V is a free F -module of rank n. The symmetric algebra S(V) is graded in the usual way by requiring the elements of V to have degree 1.

A derivation of S(V) (over F) is a linear endomorphism D_{1} of the graded F -module S(V) of degree 1 such that D_{1}(fg) = D_{1}(f)g + fD_{1}(g) for every f, g ∈ S(V).

Lemma 3.1 If ∂ : V = S(V)_{1} → S(V)_{2} is an F -linear map, then there exists a unique derivation D_{1} of S(V) such that D_{1} = ∂ on V, and D_{1} = 0 on F.

Lemma 3.2 Suppose D_{1} is any derivation of S(V) such that D_{1} = 0 on F. Then there exists a unique unstable higher derivation D_{∗} of S(V) over F such that D_{∗} = {id, D_{1}, …}.

The proofs of the above lemmas can be derived from the following formula. Let x_{1}, …, x_{n} be a basis for V. Then

D_{k}(x_{1}^{α_{1}} … x_{n}^{α_{n}}) =

∑_{\begin{matrix}j_{1}+...+j_{n}=k \\ j_{i}≥0 \end{matrix}} ( \begin{matrix}α_{1} \\ j_{1} \end{matrix} ) … ( \begin{matrix}α_{n} \\ j_{n} \end{matrix} ) x_{1}^{α_{1}−j_{1}} … x_{n}^{α_{n}−j_{n}}(∂x_{1})^{j_{1}} … (∂x_{n})^{j_{n}}.

Proof of Theorem 1.1:

Let F = F_{2}. Suppose that M ∈ End_{F} (V). Using Lemmas 3.1 and 3.2, define a higher derivation M_{∗} of S(V) over F by M_{1}(v) = (M(v))^{2} for every v ∈ V. Since F = F_{2}, M_{1} is linear. Since M_{∗} is unstable, to check that M_{1}M_{1} = 0, it suffices to show M_{1}M_{1}(v) = 0 for v ∈ V. Setting w = Mv, we see that

M_{1}M_{1}(v) = M_{1}((Mv)^{2}) = M_{1}(w^{2}) = wM_{1}(w) + M_{1}(w)w = 0

Theorem 1.1 now follows from Corollary 2.2. □

From now on, we refer to this particular example of an A_{2}-module as

S_{M}(V).

Example 3.3 Again, let F = F_{2} and let n = 2 and choose a basis {x_{1}, x_{2}} for V, i.e., S(V) = F_{2}[x_{1}, x_{2}]. Let D_{∗} be the unstable higher derivation of S(V) over F_{2} defined by

D_{1}(x_{1}) = x_{1}^{2} + x_{1}x_{2} and D_{1}(x_{2}) = x_{1}x_{2} + x_{2}^{2}.

Then

D_{1}D_{1}(x_{1}) = D_{1}(x_{1}^{2} + x_{1}x_{2}) = D_{1}(x_{1}x_{2}) = (x_{1}^{2} + x_{1}x_{2})x_{2} + x_{1}(x_{1}x_{2} + x_{2}^{2}) = 0.

Similarly, D_{1}D_{1}(x_{2}) = 0, so that D_{∗} defines an A_{2} action on S(V). Call this module R. This A_{2}-module R cannot be isomorphic (as an A_{2}-module) to any of the examples given in Theorem 1.1. For, we see immediately that D_{1} is zero on R_{2}; however, if M ∈ End_{2}(F) and M = [m_{ij}] as a matrix with respect to our basis {x_{1}, x_{2}}, then

M_{1}(x_{1}x_{2}) = (m_{11}x_{1}^{2} + m_{12}x_{2}^{2})x_{2} + x_{1}(m_{21}x_{1}^{2} + m_{22}x_{2}^{2})
= m_{21}x_{1}^{3} + m_{11}x_{1}^{2}x_{2} + m_{22}x_{1}x_{2}^{2} + m_{12}x_{2}^{3},

and this quantity cannot be zero if M is not zero.

A_{2}-modules similar to those in this section can be constructed over any commutative ring F of characteristic 2.

4 The Classification of the twisted A_{2}-actions on S(V)

Suppose that q = 2^{s} for some s > 0, and let F_{q} be a finite field of order q. Let G = Gal(F_{q}\F_{2}); i.e., G is a cyclic group {1, φ, φ^{2}, …, φ^{s−1}} of order s generated by the Frobenius automorphism φ of F_{q}. From algebra, we know that there exists an element ω ∈ F_{q} such that {ω, φ(ω), …, φ^{s−1}(ω)} is a basis for F_{q} over F_{2}. Thus we have

Lemma 4.1 Corresponding to this choice of basis, there is an isomorphism α : F_{q} → F_{2}[G] of F_{2}[G]-modules given by

α(ξ_{1}ω + … + ξ_{s}φ^{s−1}(ω)) = ξ_{1} + … + ξ_{s}φ^{s−1},

where ξ_{i} ∈ F_{2} for every i.

If N is a graded F_{2}[G]-module, we let N_{triv} denote the same underlying vector space with the trivial G-action. Furthermore, if N is a graded vector space over F_{2}, and N is simultaneously an A_{2}-module and an F_{2}[G]-module such that the actions of A_{2} and G on N commute, then we will say that N is a G-A_{2}-module.

In what follows, let N be a G-A_{2}-module.

The tensor products F_{q}⊗N and F_{2}[G]⊗N (all tensor products are taken over F_{2}) have A_{2}-module structures via the formula

θ(z⊗n) = z⊗θ(n) for θ ∈ A_{2}.

They also have F_{2}[G]-module structures, given by the (standard) diagonal action of G, i.e.,

g(z⊗n) = gz⊗gn.

Note that this action of G commutes with the A_{2}-action on both F_{q}⊗N and F_{2}[G]⊗N, hence both F_{q}⊗N and F_{2}[G]⊗N are G-A_{2}-modules.

Define the map β: F_{2}[G]⊗N_{triv} → F_{2}[G]⊗N by

β(g⊗n) = g⊗gn.

Lemma 4.2 The map β is an isomorphism of F_{2}[G]⊗N_{triv} and F_{2}[G]⊗N as G-A_{2}-modules.

Corollary 4.3 The map

Φ: F_{q}⊗N_{triv} → F_{q}⊗N

given by the composite

F_{q}⊗N_{triv} ⟶^{α⊗1} F_{2}[G]⊗N ⟶^{β} F_{2}[G]⊗N ⟶^{α^{−1}⊗1} F_{q}⊗N

is an isomorphism of G-A_{2}-modules.

Since Φ takes fixed points to fixed points and noting that

N_{triv} ≅ F_{2}⊗N_{triv} ≅ (F_{q}⊗N_{triv})^{G},

we have

Corollary 4.4 The fixed point map is an isomorphism of A_{2}-modules

Φ^{G}: N → (F_{q}⊗N)^{G} ≡ N_{φ}.

Explicitly,

Φ^{G}(n) = ∑_{j=0}^{s−1} φ^{j}(ω)⊗φ^{j}(n). (2)

Now suppose that N is a graded algebra and that G acts on N via algebra maps. Furthermore, suppose that N is an A_{2}-algebra, i.e., an algebra with an A_{2}-module structure satisfying the Cartan formula. For example, N might be H^{∗}(X; F_{2}), where X is a G-space.

By construction, N_{φ} will be a sub-A_{2}-algebra of F_{q} ⊗N. Inspection of formula 2 reveals that the isomorphism of A_{2}-modules, Φ^{G}: N → N_{φ}, is \underline{not} an algebra map. However, if we let ĩ: F_{q}⊗N_{φ} → F_{q}⊗N denote the inclusion i: N_{φ} ↪ F_{q} ⊗N with the scalars in the domain extended to F_{q}, then ĩ is visibly a map of A_{2}-algebras and we have

Proposition 4.5 The A_{2}-algebra map

ĩ: F_{q}⊗N_{φ} → F_{q}⊗N

is an isomorphism.

This is Lemma 3.4 of [5]. The proof is a pleasant and elegant exercise in basic Galois theory, using nothing deeper than the nondegeneracy of the trace form of F_{q} over F_{2}.

Let us investigate what this discussion implies about our modules S_{M}(V). Suppose that M ∈End(V) and A∈Gl(V) satisfy AM = MA and A^{s} = I. If we set q = 2^{s} and G = Gal(F_{q}\F_{2}), then G will act on S_{M}(V) via A_{2}-algebra maps by letting φx = Ax. Applying Corollary 4.4 and Proposition 4.5 to N = S_{M}(V), we conclude

(3) Φ^{G}: S_{M}(V) → S_{M}(V)_{φ} is an isomorphism of A_{2}-modules,

and

(4) F_{q}⊗S_{M}(V)_{φ} is a symmetric algebra.

These statements comprise most of the proof of the next theorem.

Theorem 4.6

S_{M}(V)_{φ} ≅ S_{AM}(V)

as A_{2}-algebras. In particular, S_{M}(V) ≅ S_{AM}(V) as A_{2}-modules.

Proof:

We start by observing that for many graded F_{2}-algebras N, e.g. finitely generated polynomial algebras, it will be true that

F_{q}⊗N′ ≅ F_{q}⊗N as algebras ⇒ N′ ≅ N as algebras.

Thus (4) implies that S_{M}(V)_{φ} is the symmetric algebra generated by its degree 1 part, which, by (3), can be identified with Φ^{G}(V). Therefore to show that S_{M}(V)_{φ} ≅S_{AM}(V) as A_{2}-algebras, it suffices to show that if v ∈V then Sq^{1}(Φ^{G}(v)) = (Φ^{G}(AM(v)))^{2}. We compute:

Sq^{1}(Φ^{G}(v)) = Sq^{1}(∑_{j=0}^{s−1} φ^{j}(ω)⊗A^{j}(v))

= ∑_{j=0}^{s−1} φ^{j}(ω)⊗A^{j}(Sq^{1}(v))

= ∑_{j=0}^{s−1} φ^{j}(ω)⊗A^{j}((Mv)^{2})

= ∑_{j=0}^{s−1} φ^{j}(ω)⊗A^{j}(M)(v^{2})

= (∑_{j=0}^{s−1} φ^{j−1}(ω)⊗A^{j−1}(AM(v)))^{2}
= (Φ^{G}(AM(v)))^{2}. □

With this last theorem, we can now prove Theorem 1.2. Proof of Theorem 1.2:

⇒): We first establish that

(5) ker(M^{j}: V → V) = ker(Sq^{2^{j−1}} … Sq^{2}Sq^{1} :V → S_{M}^{2^{j}}(V)) if j ≥ 1.

To see this, we note that in S_{M}(V), we have

Sq^{deg(f)}(f) = (Mf)^{2} = M(f^{2}).

We can see this by induction on deg(f). It is true for i = 1 by definition; suppose that deg(f) = i > 1. We may suppose that f is a monomial by linearity of both sides of the equation; say f = m_{1}m_{2} where deg(m_{1}) = 1 and deg(m_{2}) = deg(f) − 1. Since the A_{2}-action is unstable and satisfies the Cartan formula,

Sq^{deg(f)}(f) = Sq^{1}(m_{1})Sq^{deg(m_{2})}(m_{2}) = (M(m_{1}))^{2}(M(m_{2}))^{2} = (M(m_{1}m_{2}))^{2}

since M is a ring homomorphism.

It follows that if v ∈V, then in S_{M}(V),

Sq^{2^{j−1}} … Sq^{2}Sq^{1}(v) = (M^{j}(v))^{2^{j}} for all j ≥ 1.

Since S_{M}(V) has no nonzero nilpotent elements, (5) holds.

Now, if S_{M_{1}}(V) is isomorphic to S_{M_{2}}(V) as A_{2}-modules, then (5) implies that

dim(ker(M_{1}^{i})) = dim(ker(M_{2}^{i})) for all i ≥ 1.

⇐): In Theorem 4.6 and the discussion preceding it we have established that S_{M}(V) ≅ S_{AM}(V) as A_{2}-modules if A ∈ Gl(V) commutes with M ∈ End(V). It is also clear S_{M}(V) ≅ S_{CMC^{−1}}(V) as A_{2}-modules for all C ∈ Gl(V). The following exercise in linear algebra completes the proof of Theorem 1.2.

Proposition 4.7 Let ≈ be the equivalence relation on End(V) generated by

a) M ≈ CMC^{−1} if C ∈Gl(V), and

b) M ≈ AM if A∈Gl(V) is such that AM = MA.

Suppose that M_{1} and M_{2} are in End(V). Then M_{1} ≈ M_{2} if and only if dim(ker(M_{1}^{i})) = dim(ker(M_{2}^{i})) for each i ≥ 1.

Proof: ⇒): This is straightforward.

⇐): Regard M_{1} and M_{2} as matrices. Using the primary decomposition theorem, we may conjugate M_{1} and M_{2} into matrices of the form N_{1} ⊕P_{1} and N_{2}⊕P_{2} where N_{1} and N_{2} are nilpotent and P_{1} and P_{2} are invertible. Of course, we may further conjugate N_{1} ⊕P_{1} and N_{2} ⊕P_{2} into matrices of the form

A_{1} = X_{n_{1}} ⊕X_{n_{2}} ⊕…⊕X_{n_{k}} ⊕P_{1}

and

A_{2} = X_{m_{1}} ⊕X_{m_{2}} ⊕…⊕X_{m_{k}} ⊕P_{2};

where X_{j_{i}} is a direct sum of j_{i} blocks of size i × i which are companion matrices for the polynomials t^{i}. (Thus, X_{j_{1}} is an j_{1} × j_{1} matrix of zeroes.) (Note also that we allow j_{i} to be zero.) By inspection,

dim(ker(A_{1}^{i})) = ∑_{j=1}^{i−1} jn_{j} + i(∑_{j=i}^{k} n_{j}),

and

dim(ker(A_{2}^{i})) = ∑_{j=1}^{i−1} jm_{j} + i(∑_{j=i}^{k} m_{j}), if i ≤ k.

Therefore our hypothesis implies that

∑_{j=1}^{i−1} j(n_{j} − m_{j}) + i(∑_{j=i}^{k}(n_{j} − m_{j})) = 0 for 1 ≤ i ≤ k.

But the k × k (integral) matrix of coefficients of this system of equations is visibly invertible, so that n_{i} = m_{i} for each i. Thus, A_{1} = N ⊕ P_{1} and A_{2} = N ⊕P_{2}. Multiplying A_{t} by I⊕P_{t}^{−1} (which commutes with A_{t}) we get equal matrices, so M_{1} ≈ M_{2}. □ □

5 Final Remarks

We begin this section by offering an alternate, more explicit, proof of the fact that, in the situation of Theorem 4.6, the twisted algebra S_{M}(V)_{φ} is,in fact, a symmetric algebra. This proof uses a result about the action of the Frobenius map on algebraic groups defined over finite fields.

Let K be an algebraic closure of F_{q}. The Frobenius map φ of F_{q} is of course defined on K and defines a Frobenius map φ on K⊗V, End_{K}(K⊗V), and S(K ⊗V) = K ⊗S(V) with the property that {z ∈ Z|φ^{s}(z) = z} equals F_{q} ⊗V, End_{F_{q}}(F_{q} ⊗V), or F_{q} ⊗S(V), if Z equals K ⊗V, End_{K}(K ⊗V), or K⊗S(V), respectively. According to a theorem of Lang (see, e.g., [1, V 16.3, p211]), there exists an element B ∈Gl(K⊗V) such that A = B^{−1}φ(B). Since A^{s} = I and φ(A) = A, we see that φ^{s}(B) = B (since I = Aφ(A) … φ^{s−1}(A) = B^{−1}φ^{s}(B)). Thus B ∈ End_{F_{q}}(F_{q} ⊗V) and so can also be considered as an algebra isomorphism of the symmetric algebra F_{q}⊗S_{M}(V).

Proposition 5.1 B induces an isomorphism between the subalgebras S_{M}(V)_{φ} and F_{2}⊗S_{M}(V) of F_{q}⊗S_{M}(V).

Proof: It suffices to show that if y ∈ S_{M}(V)_{φ}, then φ(By) = By where φ is just acting on the scalars of F_{q} ⊗S_{M}(V). To see this, we suppose that y = Φ^{G}x with x∈S_{M}(V). Then

B(Φ^{G}(x)) = ∑_{i=0}^{s−1} φ^{i}(ω) · BA^{i}(x) = ∑_{i=0}^{s−1} φ^{i}(ω) · (φ(B)A^{i−1})(x) = φ(BΦ^{G}(x)).

Finally we note that the odd prime analogues of this proposition and all the arguments in section 4 clearly hold. Thus the odd prime version of Theorem 1.2 will hold, assuming the odd prime version of Theorem 1.1 has been established. For this, most of the arguments in sections 2 and 3 will generalize in a straightforward manner, though property (ııı) of Theorem 2.1 will obviously need modification.

References

[1] Borel,A., Linear Algebraic Groups, Springer-Verlag, 1991.

[2] Bullett, S.R. and Macdonald, I.G., On The Adem Relations, Topology 21 (1982), 329-332.

[3] Campbell, H.E.A. and Selick, P.S., Polynomial Algebras over the Steenrod Algebra, Comment. Math. Helvetici 65 (1990), 171-180.

[4] Matsumura, H., Commutative Ring Theory, Cambridge University Press, 1980.

[5] Ward, H.N., Representations of Symplectic Groups, Journal of Algebra 20 (1972), 182-195.