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

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 $S{t}^{S,R}$ on the generators of $D_n$ for $n=2$ and for either $S=\mathrm{\varnothing },R=(i)$ or $S=(s),R=(i)$ with $s,i$ arbitrary nonnegative integers.