On the module structure over the Steenrod algebra of the Dickson algebra

TL;DR

This paper explicitly determines the module structure of the Dickson algebra over the Steenrod algebra for the case n=2, detailing the action of Steenrod-Milnor operations on its generators.

Contribution

It provides an explicit computation of the Steenrod algebra action on the Dickson algebra for n=2, a case not previously fully described.

Findings

Explicit formulas for Steenrod-Milnor operations on Dickson algebra generators for n=2

Detailed understanding of the module structure over the Steenrod algebra in this case

Foundation for further generalizations to higher n

Abstract

Let $p$ be an odd prime number. We study the problem of determining the module structure over the mod $p$ Steenrod algebra $\mathcal A(p)$ of the Dickson algebra $D_n$ consisting of all modular invariants of general linear group $GL(n,\mathbb F_p)$. Here $\mathbb F_p$ denotes the prime field of $p$ elements. In this paper, we give an explicit answer for $n=2$. More precisely, we explicitly compute the action of the Steenrod-Milnor operations $St^{S,R}$ on the generators of $D_n$ for $n=2$ and for either $S=\emptyset, R=(i)$ or $S=(s), R=(i)$ with $s,i$ arbitrary nonnegative integers.