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

TL;DR

This paper computes how Steenrod-Milnor operations act on the invariants of the general linear group over a finite field, specifically on Dickson invariants in a polynomial algebra, advancing understanding in algebraic topology.

Contribution

It explicitly determines the action of Steenrod-Milnor operations on Dickson invariants for the case n=2, providing new detailed calculations in the field.

Findings

Explicit formulas for St^{(i,j)} acting on Q_{2,0}

Explicit formulas for St^{(i,j)} acting on Q_{2,1}

Enhanced understanding of Steenrod operations on invariants

Abstract

Let $p$ be an odd prime number. Denote by $GL_n = GL(n,\mathbb F_p)$ the general linear group over the prime field $\mathbb F_p$. Each subgroup of $GL_n$ acts on the algebra $P_n=E(x_1,\ldots,x_n)\otimes \mathbb F_p(y_1,\ldots,y_n)$ 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 $P_2^{GL_2}$. More precisely, we explicitly determine the action of $St^{(i,j)}$ on the Dickson invariants $Q_{2,0}$ and $Q_{2,1}$.