International Journal of Mathematics and Mathematical Sciences

Volume 2015, Article ID 235806, 7 pages

http://dx.doi.org/10.1155/2015/235806

## Some Relations between Admissible Monomials for the Polynomial Algebra

^{1}Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana^{2}African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town, South Africa

Received 14 April 2015; Accepted 5 July 2015

Academic Editor: Ram N. Mohapatra

Copyright © 2015 Mbakiso Fix Mothebe and Lafras 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.

#### 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 algebrain 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 formulaand 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 letThen 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