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

Abstract

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 $P_n=E(x_1,\dots ,x_n)\otimes {\mathbb{F}}_p(y_1,\dots ,y_n)$ in the usual manner. We grade $P_n$ by assigning $\dim x_i=1$ and $\dim y_i=2.$ This algebra is a module over the mod $p$ 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 $Q_{2,1}$.

7 pages

References in corpus (2)

• On the action of the Steenrod algebra on the modular invariants of special linear group
• On the module structure over the Steenrod algebra of the Dickson algebra