Polynomial Modules Over the Steenrod Algebra and Conjugation in the Milnor Basis

Abstract

Let ${P_s} = {\mathbb {F}_2}[{x_1}, \ldots ,{x_s}]$ be the $\bmod \;2$ cohomology of the s-fold product of $\mathbb {R}{{\text {P}}^\infty }$ with the usual structure as a module over the Steenrod algebra. A monomial in ${P_s}$ is said to be hit if it is in the image of the action $\bar A \otimes {P_s} \to {P_s}$ where $\bar A$ is the augmentation ideal of A. We extend a result of Wood to determine a new family of hit monomials in ${P_s}$. We then use similar methods to obtain a generalization of antiautomorphism formulas of Davis and Gallant.