Stripping and conjugation in the Steenrod algebra

AbstractLet S(k; f) = Sq(2k−1 f) · Sq(2k−2 f)…Sq(2 f) · Sq(f) in the mod-2 Steenrod algebra A∗, and let χ denote the canonical antiautomorphism of A∗. Given positive integers k, Λ andj with 1 ≤ j ≤ Λ, we prove that χS(k;2Λ − j) = S(Λ − (j − 1); 2j − 1(2k − 1)) · χS(k; 2j−1 − j), generalizing formulae of Davis and the author. Our proof relies on the “stripping” action of the dual Steenrod algebra A∗ or A∗ itself, which we identify as a special case of a general Hopf algebra phenomenon.Given a positive integer f, denote by μ(f) the minimal number of summands in any representation of f in the form ∑(2ik − 1). The antiautomorphism formula above implies that for f = 2Λ − j, 1 ≤ j ≤ Λ + 2, the excess of χS(k; f) satisfies ex(χS(k; f)) = (2k − 1)μ(f) for all k, confirming the conjecture of the author (Silverman, 1993) for such f. We also prove that ex(χS(k; f)) ≤ (2k − 1)μ(f) for all f and k