Abstract

Let be the polynomial algebra in n variables x_i, 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 that of determining 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 Both and Q(n) are graded, where P^d(n) denotes the set of homogeneous polynomials of degree d. In this note we show that the monomial is the only one among all its permutation representatives that is admissible, (that is, a_n meets a criterion to be in a certain basis for Q(n)). We show further that if with m ≥ n, then there are exactly permutation representatives of the product monomial that are admissible.

2010 MSC : Primary 55S10; Secondary 55S05

1. Introduction

Let be the polynomial algebra in variables , of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on by the formula

and subject to the Cartan formula

for .

A polynomial is in the image of the action of the Steenrod algebra if

for some polynomials . That means belongs to , the subspace of all hit polynomials. The problem of determining is called the hit problem and has been studied by several authors. Our work is motivated by the related problem of finding a basis for the quotient vector space

We define what it means for a monomial to be admissible. Write for the binary expansion of each exponent We then associate with a, two sequences,

where for each is called the weight vector of the monomial and is called the exponent vector of the monomial a.

Given two sequences , 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 1.1 Let a, b be monomials in . We say that if and only if one of the following holds:

1. ,

2. and .

The order on the set of sequences of nonnegative integers is the lexicographical one.

Following Kameko ¹, we define:

Definition 1.2 A monomial is said to be inadmissible if there exists monomials such that

is said to be admissible if it is not inadmissible. The set of all the classes represented by the admissible monomials in is a basis for .

Determination of all admissible monomials therefore provides a general approach in solving the problem of finding a basis for the quotient vector space This approach was used by Kameko ¹ in solving the problem in the case and by Sum ^{4, 5} in the case The problem is unknown in general. Our aim is to show that a certain class of monomials is admissible and to obtain a procedure for counting admissible permutation representatives. Both and are graded by degree and we shall denoted by the subspace of consisting of all the homogeneous polynomials of degree d.

We shall require the following result.

Theorem 1.3 (Kameko ¹; Sum ⁴). Let x, w be monomials in such that for If is inadmissible, then is also inadmissible.

For any we define the homomorphism of algebras by substituting

Then is a homomorphism of -modules. In particular, for , we have

2. Main Results

The following result identifies a class of admissible monomials in

Proposition 2.1 Let and for each integer with, let be a monomial for which If is the monomial of least order in then admissible.

Proof. Our proof of the proposition makes use of the fact that the homomorphism of algebras defined by substituting

is a homomorphism of -modules.

If then is a spike, hence is admissible. Starting with we form a sequence of monomials in respectively as follows. Setting in we obtain the monomial of least order in Setting in we obtain the monomial of least order in Setting in we obtain the monomial of least order in Proceeding in this manner we eventually obtain the monomial of least order

where

Starting with we continue our sequence of monomials of least order by forming as follows. Setting in we obtain the monomial of least order in By setting in we obtain then in and so on we obtain a sequence of monomials of least order Eventually we obtain the monomial of least order

where

Proceeding further in a similar manner we eventually obtain the monomial of least order

With we continue our sequence of monomials of least order by forming by setting in to obtain then in to obtain and so on. Proceeding in this manner we eventually obtain the monomial of least order

which is known to be admissible. Thus all the monomials are not in the image of the action of the Steenrod algebra. Since each is a monomial of least order in it follows than is admissible for each n.

Finally we note if

then no other permutation representative of is admissible. This is the case since for any pair of permutations of we have

In ² we prove that:

Lemma 2.2 (Mothebe ²). If and are admissible monomials, then for each permutation for which and the monomial

is admissible.

As a consequence of Proposition 2.1 and Lemma 2.2 we have:

Theorem 2.3 For each pair of integers the monomials and are admissible. Further if and

is admissible if and only if , and ,

Proof. By the result of Lemma 2.2 if and and then the monomial

is admissible.

Conversely we note that the result of the theorem is true if Proceeding by induction on suppose the result of the theorem is true for some integer and all Let

where and the permutation does not satisfy the condition and , If is admissible, then or is a factor of if or or is a factor of if Then

where By the induction hypothesis a′ is inadmissible so by Theorem 1.3 a is also inadmissible. This completes the proof of the theorem.

In the next section we obtain a procedure for counting the number of distinct monomials that can be obtained from and by means of permutations of their product as outlined in Theorem 2.3.

3. Order Preserving Permutations

Let with be a pair of positive integers and let and let be ordered subsets of N consisting of the first and elements respectively. We shall say that a permutation of the sequence

(1)

is order preserving if for all whenever and for all whenever Let denote the family of all distinct order preserving permutations of the sequence (1). We claim that:

Lemma 3.1

Proof. Let with be a pair of positive integers. We first note that

We know that independent of duplicates there are order preserving permutations of the sequence (1). We therefore need only show that of these permutations there, altogether, are duplicate permutations.

We note further that The statement of the lemma therefore holds for all when since the set of all order preserving permutations of the sequence has only one duplicate so that indeed

If then we have:

since every order preserving permutation of the sequence has duplicates if while it has duplicate if and Thus the formula is true for all when

Proceeding by induction on n, assume and that the formula holds for all pairs of integers m, r whenever and We prove that the formula holds for the pairs of integers n, m with

We first prove the lemma in the case The statement of the lemma becomes

if In this case For purpose of making distinction represent the integers in by and those in by Let

be an order preserving permutation of the sequence

(2)

The number of order preserving permutations of (2) of the form

(3)

is equal to and, by induction, there altogether are

duplicates. Add to these duplicates the duplicates to the sequences of the form we have that the total number of duplicates is

as required.

Finally we prove the formula is true in the cases Consider the set of all order preserving permutations of the sequence

(4)

The set splits into categories of sequences:

(1)

(2)

(3)

(4)

for all

By induction, the total number of duplicates in each such category is

and

Thus the total number of duplicates of the set of all order preserving permutations of the sequences of the form (4) is

as required.

4. Concluding Remarks

It follows from Lemma 3.1 that if and are the monomials given in Theorem 2.3, then the number of permutation representatives of the monomial that are admissible is

The numbers are known to form the Catalan triangle which appears in the OEIS as A009766.

The result of Lemma 3.1 has been shown by Mothebe and Phuc ³ to be closely related to the problem of determining the density of the prime numbers and twin primes in the sequence of natural numbers.

Statement of Competing Interests

The author has no competing interests.

References