Admissible Monomials and Generating Sets for the Polynomial Algebra as a Module Over the Steenrod Algebra

Abstract

For $n\geq 1,$ let $ \mathbf{P}(n) = \mathbb{F}_2[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i,$ of degree one, over the field $\mathbb{F}_2$ of two elements. The mod2 Steenrod algebra ${\cal A}$ acts on $\mathbf{P}(n)$ according to well known rules. Let ${\cal A}^+\mathbf{P}(n)$ denote the image of the action of the positively graded part of ${\cal A}.$ A major problem is that of determining a basis for the quotient vector space $\mathbf{Q}(n) = \mathbf{P}(n)/{\cal A}^+\mathbf{P}(n).$ Both ${\mathbf{P} }(n) = \oplus_{d\geq0}\mathbf{P}^{d}(n)$ and $\mathbf{Q}(n)$ are graded where $\mathbf{P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d.$ In this paper we show that if $n \geq 2,$ and $d \geq 1$ can be expressed in the form $d = \sum_{i=1}^{n1} (2^{\lambda_i}1)$ with ${\lambda_1}> {\lambda_2} > \ldots >{\lambda_{n2}} \geq {\lambda_{n1}}\geq 1,$ then $${\rm {dim}}(\mathbf{Q}^{d}(n)) \geq \left (\sum_{q=1}^{{\rm min}\{ {\lambda}_{n1},n\}} {{n}\choose {q}}\right ) ({\rm {dim}}(\mathbf{Q}^{d'}(n1)) )$$ where $ d'= \sum_{i=1}^{n1} (2^{\lambda_i \lambda_{n1}}1).$