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

Volume 16, Number 1 (2013), 18 - 27

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

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Abstract

For $n\geq 1,$ let $ {\mathbb 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 mod-2 Steenrod algebra ${\mathcal A}$ acts on ${\mathbb P }(n)$ according to well known rules. Let ${\mathcal A}^+{\mathbb P}(n)$ denote the image of the action of the positively graded part of ${\mathcal A}.$ A major problem is that of determining a basis for the quotient vector space ${\mathbb Q}(n) = {\mathbb P}(n)/{\mathcal A}^+{\mathbb P}(n).$ Both ${\mathbb P }(n) = \oplus_{d\geq0}{\mathbb P}^{d}(n)$ and ${\mathbb Q}(n)$ are graded where ${\mathbb P}^{d}(n)$ 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 $d = \sum_{i=1}^{n-1} (2^{\lambda_i}-1) \; \mbox{with} \; {\lambda_1}> {\lambda_2} > \ldots >{\lambda_{n-2}} \geq {\lambda_{n-1}}\geq 1,$ then $${\rm {dim}}({\mathbb Q}^{d}(n)) \geq \left (\sum_{q=1}^{{\rm min}\{ {\lambda}_{n-1},n\}} {{n}\choose {q}}\right ) ({\rm {dim}}({\mathbb Q}^{d'}(n-1)) )$$ where $ d'= \sum_{i=1}^{n-1} (2^{\lambda_i - \lambda_{n-1}}-1)$.