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 (2^k−1f)· Sq (2^k−2f)… Sq (2f)·Sq (f)] is (2^k−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 (a_n) and χ[Sq (a₁)…Sq (a_n)] in terms of the cardinalities of certain sets of matrices.

For s[ges ]1, let ℙ_s=[ ]₂= [x₁, …, x_s] 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∈ℙ_s is hit if it is in the image of the action [Ascr ]*×ℙ_s→ℙ_s, where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that if the integers e, f, and k satisfy e<(2^k−1)μ(f), then for any polynomials E and F of degrees e and f respectively, the product E·F^2k is hit. This generalizes a result of Wood conjectured by Peterson, and proves a conjecture of Singer and Silverman.