Hit polynomials and conjugation in the dual Steenrod algebra

Abstract

Let [Ascr ]* be the mod-2 Steenrod algebra of cohomology operations and χ its canonical antiautomorphism. For all positive integers f and k, we show that the excess of the element χ[Sq (2k−1f)· Sq (2k−2f)… Sq (2f)·Sq (f)] is (2k−1)μ(f), where μ(f) denotes the minimal number of summands in any representation of f as a sum of numbers of the form 2i−1. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products Sq (a1)…Sq (an) and χ[Sq (a1)…Sq (an)] in terms of the cardinalities of certain sets of matrices.For s[ges ]1, let ℙs=[ ]2= [x1, …, xs] be the mod-2 cohomology of the s-fold product of ℝP∞ with itself, with its usual structure as an [Ascr ]*-module. A polynomial P∈ℙs is hit if it is in the image of the action [Ascr ]*×ℙs→ℙs, where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that if the integers e, f, and k satisfy e<(2k−1)μ(f), then for any polynomials E and F of degrees e and f respectively, the product E·F2k is hit. This generalizes a result of Wood conjectured by Peterson, and proves a conjecture of Singer and Silverman.