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In this paper, we generalize a formula of Davis (Proc. Amer. Math. Soc. 44 (1974), 235-236) for the antiautomorphism of the mod-2 Steenrod algebra si (2), in the process formulating the analogue of the Adem relations for i-i i-i products Sq(0, ... , 0, a) ■ Sq(0, ... , 0, 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 j/(2)-action on ¥2[xx > ••• , *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. 1. Statement of results The mod-2 Steenrod algebra sA(2) of cohomology operations is a connected Hopf algebra, and as such admits a unique antiautomorphism x [MÍ158]. In this paper we generalize an antiautomorphism formula of Davis [Dav74] and use the result to study the action of s>A(2) on W2[xx, ... , xs], the mod-2 cohomology algebra of the 5-fold copy of EP°° with itself. The Milnor basis of sA (2) is indexed by the set of sequences of non-negative integers almost all of which are 0. Let A? be the set of such sequences. If S eA? with s¡ = 0 for i > N, the corresponding basis element is denoted Sq(S) = Sq(sx,... ,sN);iXs dimension is \Sq(S)\ = £~ ,(2' l)sj . For / > 1, define it : A7 —► A7 by (sx, ... , sN) ■-» (rx, ... , rtN) with r. = {sj> i**jt, ' \0, t does not divide i. Let Sq, (S) =f Sq(i,(S)). Then \Sqt (S)\ = £(2* l)sj ; in particular, \Sq, (s)\ = (2' l)s. The elements {Sq,(S) : S e S?} form a basis for a Hopf Received by the editors May 19, 1993. 1991 Mathematics Subject Classification. Primary 55S05, 55S10; Secondary 20J05.

Hit polynomials and the canonical antiautomorphism of the Steenrod algebra