Some Relations between Admissible Monomials for the Polynomial Algebra

Abstract

Let be the polynomial algebra in variables , of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well known rules. A major problem in algebraic topology is of determining , the image of the action of the positively graded part of . We are interested in the related problem of determining a basis for the quotient vector space . has been explicitly calculated for but problems remain for . Both and are graded, where denotes the set of homogeneous polynomials of degree . In this paper, we show that if is an admissible monomial (i.e., meets a criterion to be in a certain basis for ), then, for any pair of integers (), , and , the monomial is admissible. As an application we consider a few cases when .

1. Introduction

For let be the mod-2 cohomology group of the -fold product of with itself. Then is the polynomial algebra in variables , each of degree , over the field of two elements.

The mod-2 Steenrod algebra is the graded associative algebra generated over by symbols for , called Steenrod squares subject to the Adem relations [1] and . Let denote the homogeneous polynomials of degree . The action of the Steenrod squares is determined by the formula and the Cartan formula A polynomial is said to be hit if it is in the image of the action of on , that is, if for some of degree . Let denote the subspace of all hit polynomials. The problem of determining is called the hit problem and has been studied by several authors [2–4]. We are interested in the related problem of determining a basis for the quotient vector space which has also been studied by several authors [5–8]. 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 -generators for . Let denote the number of digits in the binary expansion of .

In [4, Theorem ], Wood proved the following.

Theorem 1 (Wood [4]). Let be a monomial of degree . If , then is hit.

Thus, is zero unless or, equivalently, unless can be written in the form , where . Thus, only if contains monomials called spikes. For convenience, we will assume that We, in addition, will consider a special one when and only if or . In this case is called a minimal spike.

has been explicitly calculated by Peterson [8] for , by Kameko in his thesis [5] for , and independently by Kameko [6] and Sum [7] for . In this work we will, unless otherwise stated, be concerned with a basis for consisting of “admissible monomials,” as defined below. Thus, when we write we mean that is an admissible monomial of degree

We define what it means for a monomial to be admissible. Write for the binary expansion of each exponent . The expansions are then assembled into a matrix of digits or with in the th position of the matrix.

We then associate with two sequences where for each . is called the weight vector of the monomial and is called the exponent vector of the monomial .

Given two sequences and , we say if there is a positive integer such that for all and . We are now in a position to define an order relation on monomials.

Definition 2. Let be monomials in . We say that if one of the following holds: (1),(2) and .

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 is said to be inadmissible if there exist monomials with for each , , such that is said to be admissible if it is not inadmissible.

Clearly the set of all admissible monomials in form a basis for

Let be a monomial of degree Given any pair of integers , , , we will write for the monomial Our main result is the following.

Theorem 4. Let be a monomial of degree , where . If is admissible, then, for each pair of integers , , , is admissible.

Our proof of Theorem 4 is deferred until Section 3.

As a corollary to Theorem 4, suppose that is fixed so that is also fixed. Let be the minimal spike of degree . If , then consider the following.

Corollary 5. has a subspace isomorphic to

As our main application of the theorem we consider a few cases when . The relevant result in this case is Theorem 6 stated below. To explain Table 1 we recall that, given an integer such that , we let Then given any explicit admissible monomial basis for one may compute , the dimension of the subspace of generated by all monomials of the form and In general

Table 1 

In [7] Sum gives an explicit admissible monomial basis for In this paper we make use of his results to compute , , and compare these values with in the given range. The results are given in Table 1. The table is incomplete as not all values for are known in the given range. While in general it is true that , there are cases, even nontrivial, where equality holds. This is demonstrated with the aid of known results for (cited in Table 1) but we will show later that the same conclusions can be reached independently.

Theorem 6. Table 1 gives lower bounds, , 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 4.

Our work is organized as follows. In Section 2, we recall some results on admissible monomials and hit monomials in . 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 .

The following theorem has been used to great effect by Kameko and Sum in computing a basis for and , respectively.

Theorem 7 (Kameko [5] and Sum [18]). Let be monomials in such that for . If is inadmissible, then is also inadmissible.

Up to permutation of representatives weight order provides a total order relation amongst spikes in a given degree.

It is easy to show that a spike is a minimal spike if its weight order is minimal with respect to other spikes of degree . We say is a maximal spike if its weight order is maximal with respect to other spikes of degree . In [17, Theorem ] Kameko proved the following.

Theorem 8 (Kameko). Let be a positive integer and let be the minimal spike of degree . Define a linear mapping, , by If , then f induces an isomorphism .

From Wood’s theorem and the above result of Kameko the problem of determining -generators for is reduced to the cases for which whenever is a minimal spike of a given degree

We recall the following result of Singer on hit polynomials in . In [3, Theorem ], Singer proved the following.

Theorem 9 (Singer). Let be a monomial of degree , where . Let be the minimal spike of degree . If , then is hit.

We will require the following stronger version of Theorem 9. Let be a monomial of degree . For define to be the integer .

In [2, Theorem ], Silverman proved the following.

Theorem 10 (Silverman). Let be a monomial of degree , where . Let be the minimal spike of degree . If for some , then is hit.

Finally we note that for any element and any polynomial we have for a given

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 , , , which determine a monomial once a monomial is given. We first note that if , then while, for any other value of ,   is a permutation of that replaces the exponent of by and those for , , by . We may therefore use permutation notation in place of Let Then and for each integer , , there exists such that whenever For convenience of presentation we will write in place of

Proof of Theorem 4. Suppose that is a monomial of degree , where . Let be an integer. We must show that, for any permutation , is admissible whenever is admissible. Put .

We first note that, for each , every monomial in is of the form Thus, if and is a monomial in for which contains a term belonging to , then must be of the form for some . This must be the case since for any pair of positive integers , there exists no integer such that . By the Cartan formula may be written in the form

For a given permutation write where is the complement of in and let denote the projection of onto with respect to this splitting.

The assignment, gives a bijection between Thus, for each , we can think of as having an inverse

Now, let . Then, is an element of . Thus, any hit polynomial that has as a term is a sum of polynomial expressions of the form (14). Proceeding by contraposition, suppose that is inadmissible; that is, it is not an element of . Then we can find a polynomial in which modulo image of the action, , is a sum of monomials of the form plus error terms all of order lower than that of . Since in this event the error terms are of no consequence (in terms of determining whether is admissible or not) we may as well restrict ourselves to that part of the polynomial which consists of terms belonging to , that is, to the image of . But the mapping is injective and preserves the order of monomials, so by considering its inverse we see that must also be inadmissible.

Proof of Corollary 5. We must show that if is fixed and is greater than , then for any permutation we have if and only if , or, by virtue of Theorem 4, that we have admissible whenever is admissible. But any hit polynomial which has 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, is either hit or of lower weight order than that of But if is the minimal spike of degree , , and is a term in a polynomial expression of the form (20), then for and so by Theorem 10 is hit. The proof is then completed by noting that if , then the subspace of spanned by is a direct summand of isomorphic to and that the respective subspace of with basis, the union over all of the sets inherits this splitting.

4. Application of Theorem 4

We have already applied Theorem 4 to derive the table of Theorem 6. The table can be extended to all but in this work we have limited ourselves to . The purpose of this restriction is to show that, besides the cases, , we also have , when and . We also compute for and

To prove all this we require some preliminary observations. For each , let Then is an -submodule of . Let Then we have a direct sum decomposition Thus, we have the following.

For any integer ,

If we were to compute by an inductive procedure on the number of variables, then we would only be required to compute in proceeding from the case to . Apart from spikes all subsequent references to monomials will mean those belonging to and, by virtue of Singer’s 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 and is dependant only on , since, for any permutation , whenever , then choosing , we have .

That , , and will be consequent from the following results.

Let and suppose that . Suppose there is a spike for which . We will say that a monomial of weight order greater than or equal to that of is associated with if, for some pair , we have and is strongly associated with if is associated with and if in addition for some . Note that a monomial is associated with a spike if for some and some monomial .

There are cases, dependent on , where every monomial in is strongly associated with some spike of degree In particular we have the following.

Lemma 11. Let be an integer for which and let be a monomial. Suppose that for each spike of degree we have , , , and . Then is strongly associated with at least one of the spikes of degree

Proof. Suppose that every spike of degree satisfies the hypothesis of the lemma. We first show that if with for some spike then is strongly associated with To see this represents the spike in matrix form. Without loss of generality assume that row of the matrix has zero entries so that row , column has entry , . Note that the columns are indexed by , . The matrix form of a monomial of the same weight order as may be obtained from the matrix form of by moving ones from the rows of vertically upwards or downwards to a different row including row In the worst case scenario we can move at most ones to row from distinct rows of the matrix form of . Thus, we must be left with at least one row with consecutive ones; that is, for any monomial obtained this way we must have for some Since , we must have for some Suppose that for any spike of degree It is easy to see that this is possible only if there is a spike for which We claim that we must have for some In this case a monomial of higher order than can only be obtained by applying the splitting to where . The proof is then completed by noting that .

We are now in a position to show that when , , and . We will make use of the following observation.

Suppose that is an integer for which the minimal spike of degree satisfies the condition . For each , , let be the linear mapping given on monomials by Then, ignoring monomials of order lower than that of the minimal spike of degree , it is easy to see, by virtue of (10), that where and is the image of the action of on . In other words the image of , , is given by .

When then the spikes , , of degree , satisfy the hypothesis of Lemma 11 and each monomial of this degree is of the weight order of one of these spikes. Let . Then for some or equivalently we can find and a monomial such that where , or 4. To prove that , it is sufficient to show that whenever is admissible then we can find an admissible monomial and a permutation such that (, or ). Since is of the form for some it is sufficient to show that if is a hit polynomial that has as a term, then the error terms are either hit or of lower order than It is easy to see that if is of the weight order of the spike and is a hit polynomial that has as a term, then all the error terms are of lower weight order than that of Clearly this suffices to show that every admissible monomial of the weight order of the spike is of the form for some admissible monomial .

That leaves us with the case where has weight order that of the minimal spike In this case we can find , , such that is equal to at least one If for some , then the hypothesis of Corollary 5 is satisfied if we take and ; hence, if is admissible then for some admissible monomial We may therefore suppose that for any We show that in all cases each element of that has as a term has error terms that consist of hit monomials. When , then implies and the minimal spike when is Since we have Since preserves the order of monomials for each we may, by induction, assume that whenever is admissible then we can find an admissible monomial and a permutation such that (, or ). Thus, we need only to consider the case when is a term in Since , we see that the error part of a hit polynomial that has as a term consists of monomials which are hit unless if in the matrix form of we have for some In this case is a permutation representative of one of , , or Consider the cases where is a permutation representative of one of or If we take , then the error terms in a hit polynomial, , that has as a term are hit monomials. Finally it is easy to check that the case where has an element which has a permutation representative of as a term is when such an element is a sum of hit polynomials in Such action makes no contribution (in terms of adding or deducting) to the basis of A similar argument applies to monomials of weight order of the spike of degree which is also the base of our induction in the case

We now show that when There is nothing to show in the cases as whenever In the case the spike is the only element in In the cases each monomial in is associated with one of the respective spikes and , and it is easy to verify that those of the form for some form basis for and , respectively. Similarly in the cases

If , then one can verify that in addition to the elements obtained from via the mappings the five monomials , , , , and are also admissible so that If , then it is easy to show that in addition to the monomials generated from the two monomials also belong to or are also admissible so that

Finally we note that if is a spike that satisfies the hypothesis of Lemma 11 and is one for which , then is the maximal spike and for every monomial In these cases monomials for which are strongly associated with Even though there might be other spikes of degree which do not satisfy the hypothesis of the lemma, one may all the same obtain partial results for by applying an inductive procedure on as above to compute the dimension of the subspace of generated by monomials in the weight class of

In [19] one such case is considered. The spike of degree satisfies the hypothesis of Lemma 11 and is one for which It is shown that if , then the dimension of the subspace of generated by monomials in the weight class of is equal to and that if , then the dimension of the respective subspace is

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.