A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

Abstract

The hit problem for a cohomology module over the Steenrod algebra $\mathcal{A}$ asks for a minimal set of $\mathcal{A}$–generators for the module. In this paper we consider the symmetric algebras over the field ${\mathbb{F}}_p$, for $p$ an arbitrary prime, and treat the equivalent problem of determining the set of ${\mathcal{A}}^{\ast }$–primitive elements in their duals. We produce a method for generating new primitives from known ones via a new action of the Kudo–Araki–May algebra $\mathcal{K}$, and consider the $\mathcal{K}$–module structure of the primitives, which form a sub $\mathcal{K}$–algebra of the dual of the infinite symmetric algebra. Our examples show that the $\mathcal{K}$–action on the primitives is not free. Our new action encompasses, on the finite symmetric algebras, the operators introduced by Kameko for studying the hit problem.