The intersection of the admissible basis and the Milnor basis of the Steenrod algebra
The Steenrod algebra, \(\mathcal{A}_p\), is the collection of those natural transformations of the functor \(X \mapsto H^*(X;\mathbb{F}_p)\) which are `stable', i.e. commute with the suspension operator. This collection naturally forms an \(\mathbb{F}_p\)-algebra with the (non-commutative) product being given by composition and is known to be generated by operations \({\mathcal P}^i\) (or \(Sq^i\) if \(p=2\)) constructed by Steenrod, together with the Bockstein homomorphism \(\beta\). A number of bases have been discovered for \({\mathcal A}_p\), suitable for different purposes, and it is of interest to understand the relations between them. In this article the authors study the relation between two of the oldest bases -- those of the title. The admissible basis consists of those composites \(\beta^{\varepsilon_0} {\mathcal P}^{i_1} \beta^{\varepsilon_1} \cdots {\mathcal P}^{i_n} \beta^{\varepsilon_n}\) such that \(i_j \geq p i_{j+1} + \varepsilon_j\) for all \(j>0\) (\({\mathcal P}^q\) should be read as \(Sq^{2q}\) and \(\beta\) as \(Sq^1\) if \(p=2\)). The Milnor basis arises from Milnor's work showing that the Steenrod algebra has a natural Hopf algebra structure and that its dual is a polynomial algebra if \(p=2\) and a polynomial algebra tensored with an exterior algebra if \(p>2\). One then takes the monomial basis and dualizes it to get the Milnor basis for the Steenrod algebra. The authors give necessary and sufficient conditions for an element of the Milnor basis to be admissible. In the case \(p=2\) their result verifies a conjecture of Monks. For \(p>2\) it should be noted that the authors only deal with Bockstein-free basis elements.
scientific article; zbMATH DE number 1308172