The graded algebra of Steenrod $q$th powers

Abstract

The algebra ${\mathsf A}_q$ of Steenrod $q$th powers, where $q = p^e$ is a power of a prime $p$, is isomorphic to a subalgebra ${\mathsf A}'_q$ of the algebra of Steenrod $p$th powers ${\mathsf A}_p$. The filtration of ${\mathsf A}_p$ by powers of its augmentation ideal was studied by J. P. May in his Princeton thesis of 1964. We extend some of May's results to ${\mathsf A}_q$ and obtain a convenient set of defining relations for the graded algebra $E^0({\mathsf A}_q)$. In the case $q=p$, we recover the observation of S. B. Priddy that the subalgebra $E^0({\mathsf A}_p(n-2))$ of $E^0({\mathsf A}_p)$ generated by the elements $P^{p^j}$ for $0 \le j \le n-2$ is isomorphic to the graded algebra associated to the augmentation ideal filtration of the group algebra ${\mathbb F}_p{\mathsf U}(n)$, where ${\mathsf U}(n)$ is the group of upper unitriangular matrices over ${\mathbb F}_p$. The Arnon A basis of ${\mathsf A}_p$ is given by monomials which are minimal in the left lexicographic order of formal monomials in the Steenrod powers. K. G. Monks (for $p=2$) and D. Yu. Emelyanov and Th. Yu. Popelensky (for $p>2$) have found a triangular relation between this basis and the Milnor basis using a certain ordering on the Milnor basis. We introduce a variant of the Arnon A basis which is minimal for the right order, and show that this basis and Arnon's original A basis are also triangularly related to the Milnor basis of ${\mathsf A}_q$ using the right order on the Arnon A basis.