Hit polynomials and the canonical antiautomorphism of the Steenrod algebra

Silverman, Judith H.

doi:10.1090/S0002-9939-1995-1254854-8

Hit polynomials and the canonical antiautomorphism of the Steenrod algebra
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by Judith H. Silverman

Proc. Amer. Math. Soc. 123 (1995), 627-637

DOI: https://doi.org/10.1090/S0002-9939-1995-1254854-8

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Abstract:

In this paper, we generalize a formula of Davis (Proc. Amer. Math. Soc. 44 (1974), 235-236) for the antiautomorphism of the $\bmod \text {-}2$ Steenrod algebra $\mathcal {A}(2)$, in the process formulating the analogue of the Adem relations for products $Sq(\overbrace {0, \ldots ,0}^{t - 1},a) \cdot Sq(\overbrace {0, \ldots ,0}^{t - 1},b)$. We also state a generalization of a conjecture by the author and Singer (On the action of Steenrod squares on polynomial algebras II, J. Pure Appl. Algebra (to appear)) concerning the $\mathcal {A}(2)$-action on ${\mathbb {F}_2}[{x_1}, \ldots ,{x_s}]$ and use the antiautomorphism formula to prove several cases of the generalized conjecture. We discuss the relationship between the two conjectures and make explicit a sufficient condition for Monks’s work to prove a special case of the original conjecture. Finally, we illustrate in a table the relative strengths of the special cases of the conjectures known to be true.

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