Products of Admissible Monomials in the Polynomial Algebra as a Module over the Steenrod Algebra

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. A major problem in algebraic topology is that of determining AP(n), the image of the action of the positively graded part of A. We are interested in the related problem of determining a basis for the quotient vector space Q(n) = P(n)/AP(n). Both P(n) = ⊕ d≥0 P d(n) and Q(n) are graded, where Pd(n) denotes the set of homogeneous polynomials of degree d. Q(n) has been explicitly calculated for n = 1, 2, 3, 4 but problems remain for n ≥ 5. In this note we show that if u = x1 1 · · · x mk k ∈ Pd(k) and v = x e1 1 · · · x er r ∈ Pd ′ (r) are an admissible monomials, (that is, u and v meet a criterion to be in a certain basis for Q(k) and Q(r) respectively), then for each permutation σ ∈ S k+r for which σ(i) < σ( j), i < j ≤ k and σ(s) < σ(t), k < s < t ≤ k + r, the monomial x1 σ(1) · · · x mk σ(k)x e1 σ(k+1) · · · x er σ(k+r) ∈ Pd+d ′ (k + r) is admissible. As an application we consider a few cases when n = 5.


Introduction
For n ≥ 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 in n variables x i , each of degree 1, over the field F 2 of two elements.The mod-2 Steenrod algebra A is the graded associative algebra generated over F 2 by symbols S q i for i ≥ 0, called Steenrod squares subject to the Adem relations (Adem, 1957) and S q 0 = 1.Let P d (n) denote the homogeneous polynomials of degree d.The action of the Steenrod squares S q i : P d (n) → P d+i (n) is determined by the formula: and the Cartan formula for some u i ∈ P(n) of degree d − i.Let A + P(n) denote the subspace of all hit polynomials.The problem of determining A + P(n) is called the hit problem and has been studied by several authors, (Singer, 1991) and (Wood, 1989).We are interested in the related problem of determining a basis for the quotient vector space which has also been studied by several authors, (Kameko, 1990(Kameko, , 2003)), (Peterson, 1987) and (Sum, 2007).Some of the motivation for studying these problems is mentioned in (Nam, 2004).It stems from the Peterson conjecture proved in (Wood, 1989) and various other sources (Peterson,1989) and (Singer, 1989).
The following result is useful for determining A-generators for P(n).Let α(m) denote the number of digits 1 in the binary expansion of m.
In (Wood, 1989)[Theorem 1], R.M.W. Wood proved that: Theorem 1 (Wood, 1989).Let u ∈ P(n) be a monomial of degree d.If α(n + d) > n, then u is hit.(Peterson,1987) for n = 1, 2, by Kameko in his thesis (Kameko, 1990) for n = 3 and independently by Kameko in (Kameko, 2003) and Sum in (Sum, 2007) for n = 4.In this work we shall, unless otherwise stated, be concerned with a basis for Q(n) consisting of 'admissible monomials', as defined below.Thus when we write u ∈ Q d (n) we mean that u is an admissible monomial of degree d.
We define what it means for a monomial b = x e 1 1 • • • x e n n ∈ P(n) to be admissible.Write e i = ∑ j≥0 α j (e i )2 j for the binary expansion of each exponent e i .The expansions are then assembled into a matrix β(b) = (α j (e i )) of digits 0 or 1 with α j (e i ) in the (i, j)-th position of the matrix.We then associate with b, two sequences, where Given two sequences p = (u 0 , u 1 , . . ., u l , 0, . ..), q = (v 0 , v 1 , . . ., v l , 0, . ..), we say p < q if there is a positive integer k such that u i = v i for all i < k and u k < v k .We are now in a position to define an order relation on monomials.
Definition 1.Let a, b be monomials in P(n).We say that a < b if one of the following holds:

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, (Kameko, 1990) we define: b is said to be admissible if it is not inadmissible.
Clearly the set of all admissible monomials in P(n) form a basis for Q(n).
Our main result is: Theorem 2 is a generalization of the following result of the author and Uys proved in (Mothebe and Uys, 2015).
Theorem 3. Let u ∈ P(n − 1) be a monomial of degree d ′ , where α(d As our main application of Theorem 2 we consider a few cases when n = 5.The relevant result in this case is Theorem 4 stated below.To explain the table that appears in the theorem we note that given any explicit admissible monomial basis for Q(s), 1 ≤ s ≤ n − 1, one may compute GLB(n, d), the dimension of the subspace of Q d (n) generated by all degree d monomials of the form k+r) for all triples (k, r, σ) where k + r = n and σ ∈ S k+r satisfies the hypothesis of the theorem.In general GLB(n, d) ≤ dim(Q d (n)) but there are cases where equality holds.
In (Sum, 2007) Sum gives an explicit admissible monomial basis for Q(4) and in addition recalls the results of Kameko (Kameko, 1990) for Q(3).In this paper we make use of these results to compute GLB(5, d), 1 ≤ d ≤ 30, and compare these values with dim in the given range.The results are given in Table A in Theorem 4. The table is incomplete as dim ) has not yet, in general, been calculated for d ≥ 13.As can be seen from Table A there are cases where GLB(5, d) = dim(Q d (5)).This is demonstrated with the aid of known results for dim(Q d (5)) (cited in Table A).The results also show an improvement from the results obtained in (Mothebe and Uys, 2015) by application of Theorem 3. Theorem 4. Table A gives lower bounds, GLB(5, d), for the dimension of While this approach remains to be explored in general these test results suffice for our purpose in this paper and we hope to make a more general account in subsequent work.We are thus only required to prove Theorem 2. This is the subject of the next section which is also our concluding section.

Proof of Theorem 2
In this section we prove Theorem 2. It shall suffice to show that if u We first note that for any given monomial u given on monomials by h Unless otherwise stated, we shall use product notation uv for h u (v).In this way we see that each monomial u ∈ P d (k) determines a subspace (namely h u (P given on monomials by g Let π u : P d+d ′ (k + r) → h u (P d ′ (r)) denote the projection of P d+d ′ (k + r) onto the summand h u (P d ′ (r)) of P d+d ′ (k + r).Our aim is to show that h u induces an isomorphism from A + P(r) ∩ P d ′ (r) to π u (A + P(k + r) ∩ P d+d ′ (k + r)), given by S q b (z) → uS q b (z).In other words we claim that π u (A + P(k + r) ∩ P d+d ′ (k + r)) is generated by polynomials of the form uS q b (z) where S q b (z) ∈ A + P(r) ∩ P d ′ (r) Under this assumption, suppose that p = u ∑ j u j is a polynomial generated by elements uS q b (z) ∈ π u (A + P(k + r) ∩ P d+d ′ (k + r)).Since h u is order preserving it follows that if uu j l is a term of highest order in p, then u j l is an inadmissible monomial in P d ′ (r).A parallel argument holds for the 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.