The intersection of the admissible basis and the Milnor basis of the Steenrod algebra
type: Item
instance of: scholarly article
title: The intersection of the admissible basis and the Milnor basis of the Steenrod algebra
zbMATH Open document ID: 0921.55015
DOI: 10.1016/S0022-4049(97)00035-2
author: R. Wood
author: David Carlisle
author: G. Walker
published in: Journal of Pure and Applied Algebra
publication date: 1999-09-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55S10
zbMATH DE Number: 1308172
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: admissible
reviewed by: Martin D. Crossley
MaRDI profile type: Publication
cites work: Q3345437
cites work: The Steenrod algebra and its dual
cites work: Change of basis, monomial relations, and \(P_ t^ s\) bases for the Steenrod algebra
cites work: Stripping and conjugation in the Steenrod algebra
cites work: Hit polynomials and conjugation in the dual Steenrod algebra
cites work: The nilpotence height of $Sq^{2^n}$
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description: scientific article; zbMATH DE number 1308172