On the action of the Steenrod-Milnor operations on the invariants of the general linear groups

Let $p$ be an odd prime number. Denote by $G{L}_n=GL(n,{\mathbb{F}}_p)$ the general linear group over the prime field ${\mathbb{F}}_p$. Each subgroup of $G{L}_n$ acts on the algebra $P_n=E(x_1,\ldots,x_n)\otimes \mathbb F_p(y_1,\ldots,y_n)$intheusualmanner\mathrm{.}Wegrade$P_n$byassigning$\dim x_i=1$and$\dim y_i=2.$Thisalgebraisamoduleoverthemod$p$ Steenrod algebra ${\mathcal{A}}_p$. The purpose of the paper is to compute the action of the Steenrod-Milnor operations on the generators of $P_2^{G{L}_2}$. More precisely, we explicitly determine the action of $S{t}^{(i,j)}$ on the Dickson invariants $Q_{2,0}$ and $Q_{2,1}$.