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Hit polynomials and the canonical antiautomorphism of the Steenrod algebra

Author: Judith H. Silverman
Journal: Proc. Amer. Math. Soc. 123 (1995), 627-637
MSC: Primary 55S10; Secondary 20J05, 55R40
MathSciNet review: 1254854
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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$-$ 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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Keywords: Steenrod algebra, antiautomorphism, hit polynomials
Article copyright: © Copyright 1995 American Mathematical Society

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