# Dimension Result for the Polynomial Algebra𝔽2[x1,…,xn] as a Module over the Steenrod Algebra

Authors: Mbakiso Fix Mothebe
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

Forn≥1, let P(n)=𝔽2[x1,…,xn] be the polynomial algebra in n variables xi, of degree one, over the field 𝔽2 of two elements. The mod-2 Steenrod algebra 𝒜 acts on P(n) according to well-known rules. Let 𝒜+P(n) denote the image of the action of the positively graded part of 𝒜. A major problem is that of determining a basis for the quotient vector space Q(n)=P(n)/𝒜+P(n). Both P(n)=⊕d≥0Pd(n) and Q(n) are graded where Pd(n) denotes the set of homogeneous polynomials of degree d. A spike of degree d is a monomial of the form x12λ1-1⋯xn2λn-1 where λi≥0 for each i. In this paper we show that if n≥2 and d≥1 can be expressed in the form d=d(λ)=(n-1)(2λ-1) with λ≥2, then dim(Qd(λ)(n))≥B(n,d(λ))+{∑q=2λ(nq),ifλ<n;2n−(n+1),ifλ≥n}, where B(n,d(λ)) is the number of spikes of degree d(λ).


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## Body

## 1. Introduction

Forn>1 let P(n) be the mod-2 cohomology group of the n-fold product of ℝP∞ with itself. Then P(n) is the polynomial algebra
(1)P(n)=𝔽2[x1,…,xn]
in n variables xi, each of degree 1. P(n) is a module over the mod-2 Steenrod algebra 𝒜 according to well-known rules. A polynomial u is said to be hit if it belongs to the set
(2)𝒜+P(n)={∑i>0‍Sqiui∣ui∈P(n),Sqi∈𝒜}.
The problem of determining 𝒜+P(n) is called the hit problem and has been studied by several authors [1–3]. A closely related problem is that of determining a basis for the quotient vector space
(3)Q(n)=P(n)𝒜+P(n)
which has also been studied by several authors [4–9]. Some of the motivation for studying these problems is mentioned in [6]. It stems from the Peterson conjecture proved in [3] and various other sources [10, 11].The following result is useful for determining𝒜-generators for P(n). Let α(m) denote the number of digits 1 in the binary expansion of m.Theorem 1 (Wood [3]).
Letu∈P(n) be a monomial of degree d. If α(n+d)>n then u is hit.ThusQd(n) is zero unless α(n+d)≤n or, equivalently, unless d can be written in the form d=∑i=1n‍(2λi-1), where λi≥0. Thus Qd(n)≠0 only if Pd(n) contains monomials
(4)v=x12λ1-1⋯xn2λn-1
called spikes.We note that a spike can never appear as a term in a hit polynomial.Q
(
n
) has been explicitly calculated by Peterson [7] for n=1,2, by Kameko [12] for n=3, and independently by Kameko [5] and Sum [8] for n=4.


## 2. Preliminaries

In this section we recall some results in Kameko [12] and Singer [2] on admissible monomials and hit monomials in P(n).Ifb=x1e1⋯xnen is a monomial in P(n), write ei=∑j≥0‍αj(ei)2j for the binary expansion of each exponent ei. The expansions are then assembled into a matrix
(5)β(b)=(αj(ei))
of digits 0 or 1 with αj(ei) in the (i,j)thposition of the matrix.We will associate with a monomialb two sequences
(6)w(b)=(w0(b),w1(b),…,wj(b),…),e(b)=(e1,e2,…,en),
where wj(b)=∑i=1n‍αj(ei) for each j≥0. w(b) is called the weight vector of the monomial b and e(b) is called the exponent vector of the monomial b. Note that wj(b)≤n for all j. The monomial b is said to have length l if wl(b)≠0 and wj(b)=0 for all j>l.Given two sequences(7)p=(u0,u1,…,ul,0,0,⋯),q=(v0,v1,…,vl,0,0,…),
we say that p<q if there is a positive integer k such that ui=vi for all i<k and uk<vk. We are now in a position to define an order relation on monomials.Definition 2.
Leta,b be monomials in P(n). We say that a<b if one of the following holds:(i)
w(a)<w(b),(ii)
w(a)=w(b) and e(a)<e(b). Note that the order relation on the set of sequences is the lexicographical one.Following Kameko [4] we define the following.Definition 3.
A monomialb∈P(n) is said to be inadmissible if there exist monomials b1,b2,…,br∈P(n) with bj<b for each j, 1≤j≤r, such that
(8)b≡(∑j=1r‍bj)mod𝒜+P(n).b is said to be admissible if it is not inadmissible.
Clearly the set of all admissible monomials inP(n) is a minimal set of 𝒜-generators of P(n). Clearly a spike is admissible.We shall require the following result due to Kameko.Definition 4.
Letd be a positive integer. Define a linear mapping h:P2d+n(n)→Pd(n) by
(9)h(b)={cifb=x1x2⋯xnc20otherwise
for any monomial b∈P2d+n(n).
Thenh induces a homomorphism h*:Q2d+n(n)→Qd(n).Letβ(d)=min{m∈Z∣α(m+d)≤m}. In [5, Theorem 4.2] Kameko proved the following.Theorem 5 (Kameko).
Letd be a positive integer. If β(2d+n)=n, then h*:Q2d+n(n)→Qd(n) is an isomorphism.From Wood’s theorem and the above result of Kameko the problem of determining𝒜-generators for P(n) is reduced to the cases β(d)<n.We recall the following result of Singer on hit polynomials inP(n).Definition 6.
A spikev=x12λ1-1⋯xn2λn-1 is called a minimal spike if its weight order is minimal with respect to other spikes of degree d or, equivalently, if λ1≥λ2≥⋯≥λs≥0 and λj-1=λj only if j=s or λj+1=0.
In [11, Theorem 1.2] Singer proved the following.Theorem 7 (Singer).
Letb∈P(n) be a monomial of degree d, where α(n+d)≤n. Let v be the minimal spike of degree d. If w(b)<w(v), then b is hit.


## 3. Main Result

In this section we state our main result, Theorem8, which was obtained in [13]. The proof is deferred until Section 5. Theorem 8 is the basis of a more general result, obtained in [14], which we state at the conclusion of this paper.Letλ≥2 be an integer and let d(λ)=(n-1)(2λ-1). Let B(n,d(λ)) denote the number of spikes of degree d(λ). In [13] it is shown that one has the following.Theorem 8.
Ifn≥2, then
(10)dim(Qd(λ)(n))≥B(n,d(λ))+{∑q=2λ‍(nq)ifλ<n2n-(n+1)ifλ≥n.A general formula for computingB(n,d) for an arbitrary value of d can be found in [15].The result is a consequence of the following lemma.For eachq, 2≤q≤n, and any given λ≥q, let cq be the monomial
(11)cq=x12q-1-1⋯xi(2λ-2q-i)-1⋯xq2λ-2xq+12λ-1⋯xn2λ-1
in Pd(λ)(n) and let
(12)W(n,q)={σ∈Sn∣σ(j)<σ(k)ifj<k≤qandσ(s)<σ(r)ifq<s<r}.Lemma 9.
Letq,λ be integers such that λ≥q and let cq∈Pd(λ)(n). Then for any given σ∈W(n,q) the monomial
(13)cq(σ)=xσ(1)2q-1-1⋯xσ(i)(2λ-2q-i)-1⋯xσ(q)2λ-2xσ(q+1)2λ-1⋯xσ(n)2λ-1
is admissible.The following example illustrates the definitions previously mentioned and typical monomials defined in the lemma.Example 10.
Ifn=3, λ=3, and q=2, then W(3,2)={(1),(23),(123)} and in matrix notation
(14)c2(1)=(100011111)c2(23)=(100111011)c2(123)=(111100011).Note that(15)|W(n,q)|=(nq)
and that if n=q then the only element σ in W(n,q) is the identity (13).


## 4. Proof of Lemma9

We first show that ifq=n, then cn=cn(1) is admissible, that is, that
(16)b=b(n)=cn(1)=x12n-1-1x22λ-2n-2-1⋯xi2λ-2n-i-1⋯xn2λ-2
is admissible. We will suppress mentioning the fact that b(n) is also a function of λ.Suppose thatλ≥n is given and let m∈Pd(λ)(n) be the spike
(17)m=m(n)=x10x22λ-1⋯xj2λ-1⋯xn2λ-1
so that
(18)w(m)=(n-1,n-1,…,n-1︷λ).
Let Dd(λ)(n) denote the subspace of Pd(λ)(n) spanned by monomials a=x1t1⋯xntn such that w(a)=w(m). Further let Dd(λ)(n,b) denote the subspace of Dd(λ)(n) for which ti≠2λ-1 for any i. Let
(19)π:Pd(λ)(n)⟶Dd(λ)(n)
denote the projection of Pd(λ)(n) onto its summand Dd(λ)(n) and let
(20)πb:Dd(λ)(n)⟶Dd(λ)(n,b)
denote the projection of Dd(λ)(n) onto Dd(λ)(n,b). Let Hd(λ)(n,b) be the subspace of Dd(λ)(n,b) spanned by
(21){b1+b2∣b1,b2monomialsinDd(λ)(n,b)}.
We claim that
(22)πbπ(𝒜+P(n)∩Pd(λ)(n))=Hd(λ)(n,b).
Clearly this shall suffice to show that b is admissible (it being the monomial of least order in Dd(λ)(n,b)).Our proof of (22) is by induction on n and λ and splits into two cases depending on whether λ=n or λ>n.Case 1 (i) (λ=n).
In this case our argument is by induction onn, starting with the case n=2. If n=2 then (22) is easily seen to be true since
(23)πb(2)π(𝒜+P(2)∩P3(2))
is spanned by {x12x2+x1x22}.Assume that (22) is true when the number of variables is n-1, where n is fixed and n-1≥2.Letτ=(12⋯n)∈Sn and for each j, 1≤j≤n, let Djd(λ)(n,b) be the vector subspace of Dd(λ)(n,b) generated by monomials
(24)(x1⋯x^j⋯xn)(xτj(1)e1⋯xτj(n-1)en-1xτj(n)2λ-1-1)2,
where x1e1⋯xn-1en-1∈Dd(λ-1)(n-1,b(n-1)) and x1⋯x^j⋯xn is obtained from the monomial x1⋯xj⋯xn by deleting the factor xj. It is easy to see that Dd(λ-1)(n-1,b(n-1)) is isomorphic to Djd(λ)(n,b) for each j. In fact the linear mapping
(25)hj:Dd(λ-1)(n-1,b(n-1))⟶Djd(λ)(n,b)
given on monomials by
(26)hj(x1e1⋯xn-1en-1)=(x1⋯x^j⋯xn)(xτj(1)e1⋯xτj(n-1)en-1xτj(n)2λ-1-1)2
is an isomorphism for each j. It is easy to check that
(27)Dd(λ)(n,b)=⨁j=1nhj(Dd(λ-1)(n-1,b(n-1)))
and further that πb(n)π(𝒜+P(n)∩Pd(λ)(n)) is equal to
(28)πb(n)π(Sq1(⋂Pd(λ)P(n)∩Pd(λ)(n))⋂Pd(λ))⨁⨁j=1nhj(Hd(λ-1)(n-1,b(n-1))).
It is therefore sufficient to show that we can find monomials uj∈Djd(λ)(n) such that
(29)u1+u2,…uj+uj+1,…un-1+un
all belong to πb(n)π(𝒜+P(n)∩Pd(λ)(n)).We chooseuj=hj(b(n-1)) for each j; that is,
(30)uj=(x1⋯x^j⋯xn)(xτj(1)2n-2-1⋯xτj(n-1)2λ-1-2xτj(n)2λ-1-1)2.We first illustrate the argument in the special casen=λ=4. In this case b(3)=x13x25x36,u1=x114x27x311x413,u2=x113x214x37x411,u3=x111x213x314x47, and u4=x17x211x313x414=b(4).Consideru1∈D145(4,b(4)). Put
(31)r1=x114x213x37x411∈D145(4,b(4)).
By the induction hypothesis,
(32)u1+r1∈πb(4)π(𝒜+P(4)∩P45(4)).
But r1+u2∈πb(4)π(Sq1(P(4))∩P45(4)). Put
(33)r2=x111x214x313x47∈D245(4,b(4)).
By the induction hypothesis,
(34)u2+r2∈πb(4)π(𝒜+P(4)∩P45(4)).
But r2+u3∈πb(4)π(Sq1(P(4))∩P45(4)). Put
(35)r3=x17x211x314x413∈D345(4,b(4)).
By the induction hypothesis
(36)u3+r3∈πb(4)π(𝒜+P(4)∩P45(4)).
But r3+u4∈πb(4)π(Sq1(P(4))∩P45(4)), which concludes the argument in this special case.The argument to show thatuj+uj+1∈πb(n)π(𝒜+P(n)∩Pd(λ)(n)) in general proceeds as follows.Consider the element(37)uj=(x1⋯x^j⋯xn)(xτj(1)2n-2-1⋯xτj(n-1)2λ-1-2xτj(n)2λ-1-1)2
in Djd(λ)(n,b). Let rj∈Djd(λ)(n,b) be the monomial
(38)rj=(x1⋯x^j⋯xn)×(xτj+1(1)2n-2-1⋯xτj+1(n-2)2λ-1-4xτj+2(n-1)2λ-1-2xτj(n)2λ-1-1)2.
By the induction hypothesis,
(39)uj+rj∈πb(n)π(𝒜+P(n)∩Pd(λ)(n)).
But rj+uj+1∈πb(n)π(Sq1(P(n))∩Pd(λ)(n)) which shows that uj+uj+1∈πb(n)π(𝒜+P(n)∩Pd(λ)(n)) as required.Case 2 (ii) (λ≥n).
Our argument is by induction on bothn and λ. In the case n=2<λ,
(40)πb(2)π(𝒜+P(2)∩Pd(λ)(2))
is known to be spanned by Hd(λ)(2,b(2)).We have seen that (22) is true if λ=n. Assume therefore that (22) is true when the number of variables is n-1 where n is fixed and n-1≥2 and for λ-1, where λ is fixed and λ-1≥n.Letτ∈Sn be defined as above, let Djd(λ)(n,b) be the vector subspace of Dd(λ)(n,b) defined in the same manner as above, and let
(41)hj:Dd(λ-1)(n-1,b(n-1))⟶Djd(λ)(n,b)
be the mapping given by (26).For eachi, 1≤i≤n, let
(42)fi:Dd(λ-1)(n,b(n))⟶Dd(λ)(n,b(n))
be the linear mapping given on monomials by
(43)fi(x1e1⋯xnen)=x12e1+1⋯xi-12ei-1+1xi2eixi+12ei+1+1⋯xn2en+1.
It is easy to check that
(44)Dd(λ)(n,b(n))=⨁j=1nhj(Dd(λ-1)(n-1,b(n-1)))⨁⨁i=1nfi(Dd(λ-1)(n,b(n)))
and further that πb(n)π(𝒜+P(n)∩Pd(λ)(n)) is equal to
(45)πb(n)π(Sq1(⋂Pd(λ)P(n)∩Pd(λ)(n))⋂Pd(λ))⨁[⨁j=1nhj(Hd(λ-1)(n-1,b(n-1)))]⨁[⨁i=1nfi(Hd(λ-1)(n,b(n)))].
It is therefore sufficient to show that we can find monomials uj∈Djd(λ)(n) and monomials vi∈fi(Dd(λ-1)(n,b(n))) such that
(46)u1+u2,…,uj+uj+1,…,un-1+un,v1+v2,…,vi+vi+1,…,vn-1+vn,
and v1+u1 all belong to πb(n)π(𝒜+P(n)∩Pd(λ)(n)). But this is a direct calculation, similar to the one for the case λ=n, which makes use of induction on n and λ and is omitted.Finally we show that ifq<n and λ≥q, then cq(σ) is admissible. Let d(λ) denote d(λ,q)=(q-1)(2λ-1). We first note that
(47)uq=cq(1)=x12q-1-1x22λ-2q-2-1⋯xi2λ-2q-i-1⋯xq2λ-2
is an admissible monomial in Pd(λ)(q) (by Cases 1 and 2).Now given thatσ∈W(n,q), let
(48)gqσ:P(q)⟶P(n)
be the linear mapping given on monomials by
(49)gqσ(x1e1⋯xqeq)=xσ(1)e1⋯xσ(q)eqxσ(q+1)2λ-1⋯xσ(j)2λ-1⋯xσ(n)2λ-1.Then for allu∈P(q) and θ∈𝒜+ for which θ(u)∈Pd(λ)(q) we have
(50)(gqσ(πuqπ(θ(u))))=πbπ(θ(gqσ(u)))
so that
(51)π(𝒜+P(n)∩Pd(λ)(n))=⨁σ,qgqσ(Hd(λ)(q,uq)).
Since
(52)Dd(λ)(n)=⨁σ,qgqσ(Dd(λ)(q,uq))
and each gqσ preserves the order of monomials, we see that cq(σ)=gqσ(uq) is admissible for each pair σ,q. This completes the proof of the lemma.


## 5. Proof of Theorem8

LetCd(λ)(n) denote the subspace of Qd(λ)(n) with basis the union of B(n,d(λ)) (the spikes of degree d(λ)) and the collection
(53)ℬn(d(λ))={cq(σ)∣2≤q≤n,σ∈W(n,q)}
of monomials defined in Lemma 9. Then
(54)dim(Qd(λ)(n))≥dim(Cd(λ)(n))=B(n,d(λ))+|ℬn(d(λ))|=B(n,d(λ))+∑q=2min{λ,n}|W(n,q)|=B(n,d(λ))+{∑q=2λ(nq)ifλ<n2n-(n+1)ifλ≥n.We note that ifλ≥n and C*(n) is the subspace of C(n) generated by ℬn, then the mapping
(55)f:C*d(λ)(n)⟶C*d(λ+1)(n)
defined on monomials by f(cq(σ))=cq(σ)(xσ(2)⋯xσ(n))2λ is an isomorphism between the vector subspaces C*(n).This completes the proof of Theorem8.Remark 11.
The result of this paper in part fulfills the requirements of the following theorem, obtained in [14], whose proof needed a more detailed but separate treatment of Theorem 8.Theorem 12.
Suppose thatn≥2 and d>0 may be expressed in the form d=∑i=1n-1‍(2λi-1) with λ1>λ2>⋯>λn-2≥λn-1≥1. Then
(56)dim(Qd(n))≥(∑q=1min{λn-1,n}(nq))(dim(Qd′(n-1))),
where d′=∑i=1n-1‍(2λi-λn-1-1).



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