Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2015, Article ID 235806, 7 pages http://dx.doi.org/10.1155/2015/235806
Research Article Some Relations between Admissible Monomials for the Polynomial Algebra Mbakiso Fix Mothebe1 and Lafras Uys2 1
Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town, South Africa
2
Correspondence should be addressed to Mbakiso Fix Mothebe;
[email protected] Received 14 April 2015; Accepted 5 July 2015 Academic Editor: Ram N. Mohapatra Copyright Β© 2015 M. F. Mothebe and L. Uys. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let P(π) = F2 [π₯1 , . . . , π₯π ] be the polynomial algebra in π variables π₯π , of degree one, over the field F2 of two elements. The mod-2 Steenrod algebra A acts on P(π) according to well known rules. A major problem in algebraic topology is of determining A+ P(π), 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(π) = P(π)/A+ P(π). Q(π) has been explicitly calculated for π = 1, 2, 3, 4 but problems remain for π β₯ 5. Both P(π) = β¨πβ₯0 Pπ (π) and Q(π) are graded, where Pπ (π) denotes the set of homogeneous polynomials of degree π. In this paper, π
π
σΈ
πβ1 β Pπ (πβ1) is an admissible monomial (i.e., π’ meets a criterion to be in a certain basis for Q(πβ1)), we show that if π’ = π₯1 1 β
β
β
π₯πβ1
π
π
πβ1
then, for any pair of integers (π, π), 1 β€ π β€ π, and π β₯ 0, the monomial βππ (π’) = π₯1 1 β
β
β
π₯πβ1πβ1 π₯π2 admissible. As an application we consider a few cases when π = 5.
1. Introduction
(1)
in π variables π₯π , each of degree 1, over the field F2 of two elements. The mod-2 Steenrod algebra A is the graded associative algebra generated over F2 by symbols Sqπ for π β₯ 0, called Steenrod squares subject to the Adem relations [1] and Sq0 = 1. Let Pπ (π) denote the homogeneous polynomials of degree π. The action of the Steenrod squares Sqπ : Pπ (π) β Pπ+π (π) is determined by the formula π’, π = 0, { { { { 2 Sq (π’) = {π’ , deg (π’) = π, { { { {0, deg (π’) < π, π
σΈ
π
β1)
(π) is
and the Cartan formula
For π β₯ 1 let P(π) be the mod-2 cohomology group of the πfold product of Rπβ with itself. Then P(π) is the polynomial algebra P (π) = F2 [π₯1 , . . . , π₯π ]
π
π₯π+1π β
β
β
π₯πππβ1 β Pπ +(2
π
Sqπ (π’V) = βSqπ (π’) Sqπβπ (V) .
(3)
π=0
A polynomial π’ β Pπ (π) is said to be hit if it is in the image of the action of A on P(π), that is, if π’ = βSqπ (π’π ) , π>0
(4)
for some π’π β P(π) of degree π β π. Let A+ P(π) denote the subspace of all hit polynomials. The problem of determining A+ P(π) is called the hit problem and has been studied by several authors [2β4]. We are interested in the related problem of determining a basis for the quotient vector space
(2) Q (π) =
P (π) A+ P (π)
(5)
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International Journal of Mathematics and Mathematical Sciences
which has also been studied by several authors [5β8]. Some of the motivation for studying these problems is mentioned in [9]. It stems from the Peterson conjecture proved in [4] and various other sources [10, 11]. The following result is useful for determining Agenerators for P(π). Let πΌ(π) denote the number of digits 1 in the binary expansion of π. In [4, Theorem 1], Wood proved the following. Theorem 1 (Wood [4]). Let π’ β P(π) be a monomial of degree π. If πΌ(π + π) > π, then π’ is hit. π
Thus, Q (π) is zero unless πΌ(π + π) β€ π or, equivalently, unless π can be written in the form π = βππ=1 (2π π β 1), where π π β₯ 0. Thus, Qπ (π) =ΜΈ 0 only if Pπ (π) contains monomials π1 ππ V = π₯12 β1 β
β
β
π₯π2 β1 called spikes. For convenience, we will assume that π 1 β₯ π 2 β₯ β
β
β
β₯ π π β₯ 0. We, in addition, will consider a special one when π 1 β₯ π 2 β₯ β
β
β
β₯ π π β₯ 0 and π πβ1 = π π only if π = π or π π+1 = 0. In this case V is called a minimal spike. Q(π) has been explicitly calculated by Peterson [8] for π = 1, 2, by Kameko in his thesis [5] for π = 3, and independently by Kameko [6] and Sum [7] for π = 4. In this work we will, unless otherwise stated, be concerned with a basis for Q(π) consisting of βadmissible monomials,β as defined below. Thus, when we write π’ β Qπ (π) we mean that π’ is an admissible monomial of degree π. π We define what it means for a monomial π = π₯11 β
β
β
π₯πππ β P(π) to be admissible. Write ππ = βπβ₯0 πΌπ (ππ )2π for the binary expansion of each exponent ππ . The expansions are then assembled into a matrix π½(π) = (πΌπ (ππ )) of digits 0 or 1 with πΌπ (ππ ) in the (π, π)th position of the matrix. We then associate with π two sequences π€ (π) = (π€0 (π) , π€1 (π) , . . . , π€π (π) , . . .) , (6) π (π) = (π1 , π2 , . . . , ππ ) , where π€π (π) = βππ=1 πΌπ (ππ ) for each π β₯ 0. π€(π) is called the weight vector of the monomial π and π(π) is called the exponent vector of the monomial π. Given two sequences π = (π’0 , π’1 , . . . , π’π , 0, . . .) and π = (V0 , V1 , . . . , Vπ , 0, . . .), we say π < π if there is a positive integer π such that π’π = Vπ for all π < π and π’π < Vπ . We are now in a position to define an order relation on monomials. Definition 2. Let π, π be monomials in P(π). We say that π < π if one of the following holds: (1) π€(π) < π€(π), (2) π€(π) = π€(π) and π(π) < π(π). Note that the order relation on the set of sequences is the lexicographical one.
Following Kameko [5] we define the following. Definition 3. A monomial π β P(π) is said to be inadmissible if there exist monomials π1 , π2 , . . . , ππ β P(π) with ππ < π for each π, 1 β€ π β€ π, such that π
π β‘ ( β ππ ) mod A+ P (π) .
(7)
π=1
π is said to be admissible if it is not inadmissible. Clearly the set of all admissible monomials in P(π) form a basis for Q(π). ππβ1 π β P(π β 1) be a monoLet π’ = π₯1 1 β
β
β
π₯πβ1 mial of degree πσΈ . Given any pair of integers (π, π), 1 β€ π β€ π, π β₯ 0, we will write βππ (π’) for the monomial π
π
π
π
πβ1 2 β1 π π₯π π₯π+1 β
β
β
π₯πππβ1 π₯1 1 β
β
β
π₯πβ1 result is the following.
σΈ
π
β Pπ +(2
β1)
(π). Our main
Theorem 4. Let π’ β P(π β 1) be a monomial of degree πσΈ , where πΌ(πσΈ + π β 1) β€ π β 1. If π’ is admissible, then, for each pair of integers (π, π), 1 β€ π β€ π, π β₯ 0, βππ (π’) is admissible. Our proof of Theorem 4 is deferred until Section 3. As a corollary to Theorem 4, suppose that π is fixed so that π1 2π πβ2 β1 be the minimal spike of πσΈ is also fixed. Let π₯12 β1 β
β
β
π₯πβ2 degree πσΈ . If π > π 1 , then consider the following. σΈ
Corollary 5. Qπ (π) has a subspace isomorphic to β¨ππ=1 Qπ (πβ 1). As our main application of the theorem we consider a few cases when π = 5. The relevant result in this case is Theorem 6 stated below. To explain Table 1 we recall that, given an integer π such that πΌ(π + π) β€ π, we let πΆ (π, π) = {π β₯ 0 | πΌ (π β (2π β 1) + π β 1) β€ π β 1} .
(8)
Then given any explicit admissible monomial basis for Q(π β 1) one may compute Lπ΅(π, π), the dimension of the subspace of Qπ (π) generated by all monomials of the form βππ (π’) β π
Pπ (π), π β πΆ(π, π) and π’ β Qπβ(2 β1) (π β 1). In general Lπ΅(π, π) β€ dim(Qπ (π)). In [7] Sum gives an explicit admissible monomial basis for Q(4). In this paper we make use of his results to compute Lπ΅(5, π), 1 β€ π β€ 30, and compare these values with dim(Qπ (5)) in the given range. The results are given in Table 1. The table is incomplete as not all values for dim(Qπ (5)) are known in the given range. While in general it is true that Lπ΅(5, π) β€ dim(Qπ (5)), there are cases, even nontrivial, where equality holds. This is demonstrated with the aid of known results for dim(Qπ (5)) (cited in Table 1) but we will show later that the same conclusions can be reached independently. Theorem 6. Table 1 gives lower bounds, Lπ΅(5, π), for the dimension of Qπ (5), 1 β€ π β€ 30.
International Journal of Mathematics and Mathematical Sciences Table 1 π
π 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
dim(Q (5)) 5 10 25 45 46 74 110 174 191 280 315 190 β β 432 β β β β β β β β β β 1024 315 480 β β
Reference
[12]
[13]
[14] [12]
[15]
[16] [12]
Lπ΅(5, π) 5 10 25 45 46 74 110 174 191 275 313 190 246 295 384 414 538 673 778 591 741 780 916 908 974 985 313 480 450 763
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 4. Our work is organized as follows. In Section 2, we recall some results on admissible monomials and hit monomials in P(π). In Section 3, we prove Theorem 4. We conclude with an application of the theorem in Section 4.
2. Preliminaries In this section we recall some results in Kameko [17] and Singer [3] on admissible monomials and hit monomials in P(π). The following theorem has been used to great effect by Kameko and Sum in computing a basis for Q(3) and Q(4), respectively. Theorem 7 (Kameko [5] and Sum [18]). Let π, π be monomials in P(π) such that π€π (π) = 0 for π > π > 0. If π is inadmissible, π
then ππ2 is also inadmissible.
3 Up to permutation of representatives weight order provides a total order relation amongst spikes in a given degree. π1 ππ It is easy to show that a spike V = π₯12 β1 β
β
β
π₯π2 β1 β Pπ (π) is a minimal spike if its weight order is minimal with respect to other spikes of degree π. We say V is a maximal spike if its weight order is maximal with respect to other spikes of degree π. In [17, Theorem 4.2] Kameko proved the following. Theorem 8 (Kameko). Let π be a positive integer and let V be the minimal spike of degree 2π + π. Define a linear mapping, π : Pπ (π) β P2π+π (π), by π
2π1 +1
π (π₯1 1 β
β
β
π₯πππ ) = π₯1
β
β
β
π₯π2ππ +1 .
(9)
If π€0 (V) = π, then f induces an isomorphism πβ : Qπ (π) β Q2π+π (π). From Woodβs theorem and the above result of Kameko the problem of determining A-generators for P(π) is reduced to the cases for which π€0 (V) β€ π β 1 whenever V is a minimal spike of a given degree π. We recall the following result of Singer on hit polynomials in P(π). In [3, Theorem 1.2], Singer proved the following. Theorem 9 (Singer). Let π β P(π) be a monomial of degree π, where πΌ(π + π) β€ π. Let V be the minimal spike of degree π. If π€(π) < π€(V), then π is hit. We will require the following stronger version of Theorem 9. Let π be a monomial of degree π. For π > 0 define ππ (π) to be the integer ππ (π) = βπβ₯π π€π (π)2πβπ . In [2, Theorem 1.2], Silverman proved the following. Theorem 10 (Silverman). Let π β P(π) be a monomial of degree π, where πΌ(π + π) β€ π. Let V be the minimal spike of degree π. If ππ (π) > ππ (V) for some π β₯ 1, then π is hit. Finally we note that for any element Sqπ β A+ and any polynomial π’ β P(π) we have π
π
2π
Sqπ2 (π’2 ) = (Sqπ (π’))
(10)
for a given π β₯ 0.
3. Proof of Theorem 4 In this section we prove Theorem 4. Our main observation in this work is made in terms of pairs of integers π, π, 1 β€ π β€ π, σΈ π π β₯ 0, which determine a monomial βππ (π’) β Pπ +(2 β1) (π) σΈ
once a monomial π’ β Pπ (π β 1) is given. We first note that if ππβ1 ππβ1 2π β1 π π , then βππ (π’) = π₯11 β
β
β
π₯πβ1 π₯π while, for any π’ = π₯11 β
β
β
π₯πβ1 other value of π, βππ (π’) is a permutation of βππ (π’) that replaces the exponent of π₯π by 2π β 1 and those for π₯π+π , π β π β₯ π β₯ 1, by ππ+πβ1 . We may therefore use permutation notation in place of π. Let π (π) = {π β ππ | π (π) < π (π ) if π < π < π} .
(11)
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International Journal of Mathematics and Mathematical Sciences
Then |π(π)| = π and for each integer π, 1 β€ π β€ π, there exists ππβ1 ππβ1 π π1 2π β1 ) = π₯π(1) β
β
β
π₯π(πβ1) π₯π(π) π β π(π) such that βππ (π₯11 β
β
β
π₯πβ1 σΈ
π
π
πβ1 whenever π₯11 β
β
β
π₯πβ1 β Pπ (πβ1). For convenience of presenππβ1 ππβ1 π π tation we will write βππ (π₯11 β
β
β
π₯πβ1 ) in place of βππ (π₯11 β
β
β
π₯πβ1 ).
π
π
πβ1 Proof of Theorem 4. Suppose that π’ = π₯11 β
β
β
π₯πβ1 β P(π β 1) σΈ σΈ is a monomial of degree π , where πΌ(π + π β 1) β€ π β 1. Let π β₯ 0 be an integer. We must show that, for any permutation π β π(π), βππ (π’) is admissible whenever π’ is admissible. Put π = πσΈ + 2π β 1.
We first note that, for each π β π(π), every monomial in βππ (P(π β 1)) is of the form π
π
π
2 β1 πβ1 1 β
β
β
π₯π(πβ1) π₯π(π) . π₯π(1)
(12)
Thus, if Sqπ β A+ and π€ is a monomial in Pπβπ (π) for which σΈ Sqπ (π€) contains a term belonging to βππ (Pπ (π β 1)), then π€ must be of the form π
π
π
2 β1 πβ1 1 β
β
β
π₯π(πβ1) π₯π(π) , π₯π(1) σΈ
π
π
(13)
πβ1 for some π₯11 β
β
β
π₯πβ1 β Pπ βπ (πβ1). This must be the case since for any pair of positive integers π, π there exists no integer π‘ π such that Sqπ (π₯π‘ ) = π₯2 β1 . By the Cartan formula Sqπ (π€) may be written in the form
Sq
π
π1 (π₯π(1)
+ βSq
ππβ1 2π β1 β
β
β
π₯π(πβ1) ) π₯π(π) πβπ‘
π‘β₯1
π1 (π₯π(1)
ππβ1 β
β
β
π₯π(πβ1) ) Sqπ‘
2π β1 (π₯π(π) ).
(14)
σΈ
(15)
σΈ
where P(π,π) (π) is the complement of βππ (Pπ (π β 1)) in Pπ (π) and let ππ denote the projection π
ππ : P (π) σ³¨β
βππ
πσΈ
(P (π β 1))
(16)
π
π
π
2 β1 πβ1 1 a sum of monomials of the form (π₯π(1) β
β
β
π₯π(πβ1) )π₯π(π) plus π
π
π
2 β1 πβ1 1 error terms Sqπβπ‘ (π₯π(1) β
β
β
π₯π(πβ1) )Sqπ‘ (π₯π(π) ) all of order lower π than that of βπ (π’). Since in this event the error terms are of no consequence (in terms of determining whether π’ is admissible or not) we may as well restrict ourselves to that part of the polynomial which consists of terms belonging to ππ (Pπ (π)), that is, to the image of βππ . But the mapping βππ is injective and preserves the order of monomials, so by considering its inverse πππ we see that π’ must also be inadmissible.
Proof of Corollary 5. We must show that if π is fixed and is greater than π 1 , then for any permutation π β π(π) we have σΈ π’ β Qπ (π β 1) if and only if βππ (π’) β Qπ (π), or, by virtue of Theorem 4, that we have π’ admissible whenever βππ (π’) is admissible. But any hit polynomial which has βππ (π’) as a term is generated by polynomial expressions of the form (14) so it suffices to show that an error term in each such polynomial expression, π
π
π
2 β1 πβ1 1 β
β
β
π₯π(πβ1) ) Sqπ‘ (π₯π(π) ), Sqπβπ‘ (π₯π(1)
(20)
is either hit or of lower weight order than that of βππ (π’). But if π1
For a given permutation π β π(π) write Pπ (π) = P(π,π) (π) β¨ βππ (Pπ (π β 1)) ,
σΈ
Now, let π’ β Qπ (π β 1). Then, βππ (π’) is an element of π πσΈ βπ (P (π β 1)). Thus, any hit polynomial that has βππ (π’) as a term is a sum of polynomial expressions of the form (14). Proceeding by contraposition, suppose that βππ (π’) is inadmissible; that is, it is not an element of Qπ (π). Then we can find a polynomial in which modulo image of the action, βππ (π’), is
π πβ2
2 β1 V = π₯12 β1 β
β
β
π₯πβ2 is the minimal spike of degree πσΈ , π > π 1 , and π is a term in a polynomial expression of the form (20), then ππ (π) > ππ (V) for π = π 1 and so by Theorem 10 π is hit. The proof is then completed by noting that if π > π 1 , σΈ then the subspace of Pπ (π) spanned by βπ βππ (Pπ (π β 1)) is σΈ a direct summand β¨π βππ (Pπ (π β 1)) of Pπ (π) isomorphic to σΈ β¨ππ=1 (Pπ (π β 1)) and that the respective subspace of Qπ (π)
with basis, the union over all π β π(π) of the sets {βππ (π’) | π’ β σΈ Qπ (π β 1)} inherits this splitting.
σΈ
of Pπ (π) onto βππ (Pπ (π β 1)) with respect to this splitting. The assignment, π
π
πβ1 ) βππ : Sqπ (π₯11 β
β
β
π₯πβ1 π
π
π
2 β1 πβ1 1 σ³¨σ³¨β Sqπ (π₯π(1) β
β
β
π₯π(πβ1) ) π₯π(π) ,
(17)
gives a bijection between σΈ
A+ P (π β 1) β© Pπ (π β 1) , ππ (A+ P (π) β© Pπ (π)) .
(18)
σΈ
We have already applied Theorem 4 to derive the table of Theorem 6. The table can be extended to all π β₯ 0 but in this work we have limited ourselves to π β€ 30. The purpose of this restriction is to show that, besides the cases, 1 β€ π β€ 9, we also have dim(Qπ (5)) = Lπ΅(5, π), when π = 12 and π = 28. We also compute dim(Qπ (5)) for π = 10 and π = 11. To prove all this we require some preliminary observations. For each π, 1 β€ π β€ π, let π
X (π) = Span {π₯1 1 β
β
β
π₯πππ β P (π) | π1 π2 β
β
β
πr =ΜΈ 0} .
Thus, for each π β π(π), we can think of βππ as having an inverse πππ : ππ (Pπ (π)) σ³¨β Pπ (π β 1) .
4. Application of Theorem 4
(19)
(21)
Then X(π) is an A-submodule of P(π). Let W (π) =
X (π) . A+ X (π)
(22)
International Journal of Mathematics and Mathematical Sciences Then we have a direct sum decomposition π π Q (π) β
β¨ β¨ W (π) . π
( )
(23)
π=1 π=1
Thus, we have the following. For any integer π > 0, π π dim (Qπ (π)) = β ( ) dim (Wπ (π)) . π=1 π
(24)
If we were to compute dim(Qπ (π)) by an inductive procedure on the number of variables, then we would only be required to compute dim(Wπ (π)) in proceeding from the case π β 1 to π. Apart from spikes all subsequent references to monomials will mean those belonging to X(π) and, by virtue of Singerβs Theorem 9, we will assume they are of weight order greater than or equal to that of the minimal spike. We note also that any variation between dim(Qπ (π)) and Lπ΅(π, π) is dependant only on Xπ (π), since, for any permutation π β π(π), whenever π’ β Qπ (π β 1), then choosing π = 0, we have βππ (π’) β Qπ (π). That dim(Qπ (5)) = Lπ΅(5, π), 1 β€ π β€ 9, π = 12 and π = 28 will be consequent from the following results. Let π β₯ 0 and suppose that πΌ(π + π) β€ π. Suppose there π1 2π πβ1 β1 β Pπ (π) for which π€0 (V) = is a spike V = π₯12 β1 β
β
β
π₯πβ1 π π β 1. We will say that a monomial π = π₯1 1 β
β
β
π₯πππ β Xπ (π) of weight order greater than or equal to that of V is associated with V if, for some pair (π, π), we have ππ = 2π π β 1 and is strongly associated with V if π is associated with V and if in addition ππ β€ 2πβ2 β 1 for some π. Note that a monomial π is π βπ π (π’)
associated with a spike V if π = for some π π and some monomial π’ β X(π β 1). There are cases, dependent on π, where every monomial in Xπ (π) is strongly associated with some spike V of degree π. In particular we have the following. Lemma 11. Let π be an integer for which πΌ(π + π) β€ π and let π β Xπ (π) be a monomial. Suppose that for each spike V = π1 2π πβ1 β1 of degree π we have π πβ1 > 0, π 1 β€ π β 1, π₯12 β1 β
β
β
π₯πβ1 π π β π π+1 β€ 1, and π 1 β π πβ1 β€ 2. Then π is strongly associated with at least one of the spikes of degree π. Proof. Suppose that every spike of degree π satisfies the hypothesis of the lemma. We first show that if π = π π₯1 1 β
β
β
π₯πππ β Xπ (π) with π€(π) = π€(V) for some spike V then π is strongly associated with V. To see this represents the spike V in matrix form. Without loss of generality assume that row 1 of the matrix has zero entries so that row π + 1, column π has entry πΌπ (2π π β 1), 1 β€ π β€ π β 1. Note that the columns are indexed by π, 0 β€ π β€ π β 2. The matrix form of a monomial π of the same weight order as V may be obtained from the matrix form of V by moving ones from the π β 1 rows of V vertically upwards or downwards to a different row including row 1. In the worst case scenario we can move at most π β 2
5 ones to row 1 from π β 2 distinct rows of the matrix form of V. Thus, we must be left with at least one row with consecutive ones; that is, for any monomial obtained this way we must have ππ = 2π π β 1 for some π. Since π 1 β€ π β 1, we must have ππ β€ 2πβ2 β 1 for some π. Suppose that π€(π) =ΜΈ π€(V) for any spike V of degree π. It is easy to see that this is possible only if there is a spike V for which π 1 β π πβ1 = 2. We claim that we must have ππ = 2π 1 β1 β 1 for some π. In this case a monomial of higher order than V can only be obtained by applying the splitting 2π +2π = 2π+1 to 2π π where π π = π 1 . The proof is then completed by noting that π 1 β 1 β€ π β 2. We are now in a position to show that dim(Qπ (5)) = Lπ΅(5, π) when 1 β€ π β€ 9, π = 12, and π = 28. We will make use of the following observation. Suppose that π is an integer for which the minimal spike V of degree π satisfies the condition π€π (V) = π β 1. For each π‘, 1 β€ π‘ β€ π, let ππ‘ : Pπ (π) β P2π+πβ1 (π) be the linear mapping given on monomials by π
ππ‘ (π₯11 β
β
β
π₯πππ ) 2π +1
= π₯1 1
2π
+1 2π
2π
β
β
β
π₯π‘β1π‘β1 π₯π‘ π‘ π₯π‘+1π‘+1
+1
β
β
β
π₯π2ππ +1 .
(25)
Then, ignoring monomials of order lower than that of the minimal spike of degree 2π + π β 1, it is easy to see, by virtue of (10), that σΈ
A+ P (π) β© Pπ (π) σΈ
= [Sq1 (P (π)) β© Pπ (π)]
(26)
π
β [β¨ ππ‘ (A+ P (π) β© Pπ (π))] , π‘=1
where πσΈ = 2π + π β 1 and Sq1 (P(π)) is the image of the action of Sq1 on P(π). In other words the image of Sqπ , π > 1, is given by β¨ππ‘=1 ππ‘ (A+ P(π) β© Pπ (π)). When π = 5 then the spikes π₯17 π₯27 π₯37 π₯47 , π₯115 π₯27 π₯33 π₯43 , of degree 28, satisfy the hypothesis of Lemma 11 and each monomial of this degree is of the weight order of one of π π π π π these spikes. Let π = π₯1 1 π₯2 2 π₯3 3 π₯4 4 π₯5 5 β X28 (5). Then ππ = 3, 7, 15 for some π or equivalently we can find π β π(5) and a monomial π’ β X(4) such that π = βππ (π’) where π = 2, 3, or 4. To prove that dim(Q28 (5)) = Lπ΅(5, 28), it is sufficient to show that whenever π is admissible then we can find an admissible monomial π’ β X(4) and a permutation π β π(5) such that βππ (π’) = π (π = 2, 3, or 4). Since π is of the form βππ (π’) for some π’ β X(4) it is sufficient to show that if π is a hit polynomial that has π as a term, then the error terms are either hit or of lower order than π. It is easy to see that if π is of the weight order of the spike π₯17 π₯27 π₯37 π₯47 and Sqπ (π€) is a hit polynomial that has π as a term, then all the error terms are of lower weight order than that of π. Clearly this suffices to show that every admissible monomial of the weight order of the spike π₯17 π₯27 π₯37 π₯47 is of the form βπ3 (π’) for some admissible monomial π’ β X(4).
6
International Journal of Mathematics and Mathematical Sciences
That leaves us with the case where π has weight order that of the minimal spike V = π₯115 π₯27 π₯33 π₯43 . In this case we can find π, 1 β€ π β€ 5, such that ππ is equal to at least one 3, 7, 15. If ππ = 15 for some π, then the hypothesis of Corollary 5 is satisfied if we take πσΈ = 13 and π = 4; hence, if π is admissible then π = βπ4 (π’) for some admissible monomial π’ β X13 (4). We may therefore suppose that ππ =ΜΈ 15 for any π. We show that in all cases each element of A+ P(5) β© P28 (5) that has π as a term has error terms that consist of hit monomials. When π = 5, then 2π + π β 1 = 28 implies π = 12 and the minimal spike when π = 12 is V = π₯17 π₯23 π₯3 π₯4 . Since π€0 (V) = 4 we have A+ P (5) β© P28 (5) = [Sq1 (P (5)) β© P28 (π)]
(27)
5
β [β¨ ππ‘ (A+ P (5) β© P12 (5))] . π‘=1
Since ππ‘ preserves the order of monomials for each π‘ we may, by induction, assume that whenever π β X12 (5) is admissible then we can find an admissible monomial π’ β X(4) and a permutation π β π(5) such that βππ (π’) = π (π = 1, 2, or 3). Thus, we need only to consider the case when π is a term in Sq1 (P(5)). Since π€0 (π) = π€1 (π) = 4, we see that the error part of a hit polynomial that has π as a term consists of monomials which are hit unless if in the matrix form of π we have πΌπ (ππ ) = πΌ1 (ππ ) = 0 for some π. In this case π is a permutation representative of one of π₯17 π₯23 π₯33 π₯43 π₯512 , π₯17 π₯23 π₯33 π₯44 π₯511 , or π₯17 π₯27 π₯33 π₯43 π₯58 . Consider the cases where π is a permutation representative of one of π₯17 π₯23 π₯33 π₯43 π₯512 or π₯17 π₯23 π₯33 π₯44 π₯511 . If we take ππ = 7, then the error terms in a hit polynomial, Sq1 (π€), that has π = βπ3 (π’) as a term are hit monomials. Finally it is easy to check that the case where Sq1 (P(5)) has an element which has a permutation representative of π₯17 π₯27 π₯33 π₯43 π₯58 as a term is when such an element is a sum of hit polynomials in P28 (5). Such action makes no contribution (in terms of adding or deducting) to the basis of Q28 (5). A similar argument applies to monomials of weight order of the spike π₯17 π₯23 π₯3 π₯4 of degree 12 which is also the base of our induction in the case π = 28. We now show that dim(Qπ (5)) = Lπ΅(5, π) when 1 β€ π β€ 9. There is nothing to show in the cases 1 β€ π β€ 4 as dim(Qπ (5)) = Lπ΅(5, π) whenever Xπ (5) = 0. In the case π = 5 the spike V = π₯1 π₯2 π₯3 π₯4 π₯5 is the only element in W5 (5). In the cases π = 6, 7 each monomial in Xπ (5) is associated with one of the respective spikes π₯13 π₯2 π₯3 π₯4 and π₯13 π₯23 π₯3 , π₯13 π₯2 π₯3 π₯4 π₯5 and it is easy to verify that those of the form βππ (π’) for some π’ β W(4) form basis for W6 (5) and W7 (5), respectively. Similarly in the cases π = 8, 9. If π = 10, then one can verify that in addition to the elements obtained from W(4) via the mappings βππ the five monomials π₯1 π₯2 π₯32 π₯42 π₯54 , π₯1 π₯2 π₯32 π₯44 π₯52 , π₯1 π₯22 π₯3 π₯42 π₯54 , π₯1 π₯22 π₯3 π₯44 π₯52 , and π₯1 π₯22 π₯34 π₯4 π₯52 are also admissible so that dim(Qπ (5)) = 280. If π = 11, then it is easy to show that in addition to the monomials generated from W(4) the two
monomials π₯1 π₯2 π₯32 π₯42 π₯55 , π₯1 π₯22 π₯3 π₯42 π₯55 also belong to W11 (5) or are also admissible so that dim(Q11 (5)) = 315. Finally we note that if V β Pπ (π) is a spike that satisfies the hypothesis of Lemma 11 and is one for which π 1 β π πβ1 β€ 1, then V is the maximal spike and π€(V) β₯ π€(π) for every monomial π β Pπ (π). In these cases monomials π β Xπ (π) for which π€(π) = π€(V) are strongly associated with V. Even though there might be other spikes of degree π which do not satisfy the hypothesis of the lemma, one may all the same obtain partial results for dim(Qπ (π)) by applying an inductive procedure on π as above to compute the dimension of the subspace of Qπ (π) generated by monomials in the weight class of V. In [19] one such case is considered. The spike V = 2π β1 2π β1 of degree π(π) = (π β 1)(2π β 1) satisfies the π₯1 β
β
β
π₯πβ1 hypothesis of Lemma 11 and is one for which π 1 β π πβ1 = 0. It is shown that if π β€ π β 1, then the dimension of the subspace of Qπ(π) (π) generated by monomials in the weight class of V is equal to π + βππ=2 ( ππ ) and that if π β₯ π, then the dimension of the respective subspace is 2π β 1.
Disclosure This work was done while the first author was visiting the African Institute for Mathematical Sciences (AIMS).
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments The authors would like to thank AIMS and its director Professor Barry Green for their support and kind hospitality. Finally they would like to thank Sum for making a copy of his document [7] available to them.
References [1] J. Adem, βThe relations on Steenrod powers of cohomology classes,β in Algebraic Geometry and Topology, A Symposium in Honor of S. Lefschetz, pp. 191β238, Princeton University Press, Princeton, NJ, USA, 1957. [2] J. H. Silverman, βHit polynomials and conjugation in the dual Steenrod algebra,β Mathematical Proceedings of the Cambridge Philosophical Society, vol. 123, no. 3, pp. 531β547, 1998. [3] W. M. Singer, βOn the action of Steenrod squares on polynomial algebras,β Proceedings of the American Mathematical Society, vol. 111, no. 2, pp. 577β583, 1991. [4] R. M. Wood, βSteenrod squares of polynomials and the Peterson conjecture,β Mathematical Proceedings of the Cambridge Philosophical Society, vol. 105, no. 2, pp. 307β309, 1989. [5] M. Kameko, Products of projective spaces as Steenrod modules [Ph.D. thesis], John Hopkins University, Baltimore, Md, USA, 1990. [6] M. Kameko, Generators of the Cohomology of BV4 , Toyama University, Toyama, Japan, 2003.
International Journal of Mathematics and Mathematical Sciences [7] N. Sum, The Hit Problem for the Polynomial Algebra of Four Variables, University of Quy Nhon, Quy Nhoβn, Vietnam, 2007. [8] F. P. Peterson, βGenerators of π»* (π
πβ β§ π
πβ ) as a module over the Steenrod algebra,β Abstracts of Papers Presented to the American Mathematical Society, vol. 833, pp. 55β89, 1987. [9] T. N. Nam, βA-gΒ΄enΒ΄erateurs gΒ΄enΒ΄eriques pour lβalg`ebre polynomiale,β Advances in Mathematics, vol. 186, no. 2, pp. 334β362, 2004. [10] F. P. Peterson, βA-generators for certain polynomial algebras,β Mathematical Proceedings of the Cambridge Philosophical Society, vol. 105, no. 02, pp. 311β312, 1989. [11] W. M. Singer, βThe transfer in homological algebra,β Mathematische Zeitschrift, vol. 202, no. 4, pp. 493β523, 1989. [12] N. Sum and D. V. Phuc, On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra [M.S. thesis], University of Quy Nhon, Quy Nhon, Vietnam, 2013. [13] N. K. Tin, βThe admissible monomial basis for the polynomial algebra of five variables in degree eight,β Journal of Mathematical Sciences and Applications, vol. 2, no. 2, pp. 21β24, 2014. [14] M. F. Mothebe, βDimensions of the polynomial algebra F2 [π₯1 ,. . .,π₯π ] as a module over the Steenrod algebra,β JP Journal of Algebra, Number Theory and Applications, vol. 13, no. 2, pp. 161β170, 2009. [15] N. Sum, βOn the Peterson hit problem of five variables and its applications to the fifth Singer Transfer,β East West Journal of Mathematics, vol. 16, no. 1, pp. 47β62, 2014. [16] G. Walker and R. M. W. Wood, βYoung tableaux and the Steenrod algebra,β in Proceedings of the School and Conference in Algebraic Topology, (Hanoi, 2004), vol. 11, pp. 379β397, Geometry and Topology Monographs, 2007. [17] M. Kameko, βGenerators of the cohomology of BV 3 ,β Journal of Mathematics of Kyoto University, vol. 38, no. 3, pp. 587β593, 1998. [18] N. Sum, βThe negative answer to Kamekoβs conjecture on the hit problem,β Advances in Mathematics, vol. 225, no. 5, pp. 2365β 2390, 2010. [19] M. F. Mothebe, βDimension result for the polynomial algebra F2 [π₯1 ,...,π₯π ] as a module over the Steenrod algebra,β International Journal of Mathematics and Mathematical Sciences, vol. 2013, Article ID 150704, 6 pages, 2013.
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