Stripping and conjugation in the Steenrod algebra

Abstract

Let S( k; f) = Sq(2 k−1 f) · Sq(2 k−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 and j with 1 ≤ j ≤ Λ, we prove that χS( k;2 Λ − j) = S( Λ − ( j − 1); 2 j − 1 (2 k − 1)) · χS( k; 2 j−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 ∑(2 i k − 1). The antiautomorphism formula above implies that for f = 2 Λ − j, 1 ≤ j ≤ Λ + 2, the excess of χS( k; f) satisfies ex( χS( k; f)) = (2 k − 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)) ≤ (2 k − 1) μ( f) for all f and k.