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

摘要

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 ${P }(n) =igoplus_{d geq 0} {P}^{d}(n)$ 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)