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

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ 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 $\A^+{\P}(n)$, 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 ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ 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 $n \geq 5.$ In this note we show that if $u = x_{1}^{m_1} \cdots x_{k}^{m_{k}} \in {\P}^{d}(k)$ and $v = x_{1}^{e_1} \cdots x_{r}^{e_{r}} \in {\P}^{d'}(r)$ 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 $\sigma \in S_{k+r}$ for which $\sigma(i)<\sigma(j),$ $i<j\leq k$ and $\sigma(s)<\sigma(t),$ $k<s<t\leq k+r,$ the monomial $x_{\sigma(1)}^{m_1} \cdots x_{\sigma(k)}^{m_{k}} x_{\sigma(k+1)}^{e_1} \cdots x_{\sigma(k+r)}^{e_r} \in {\P}^{d+d'}(k+r)$ is admissible. As an application we consider a few cases when $n=5.$