\({\mathcal A}\)-decomposability of the Dickson algebra (Q1593706)

Let \(P_{k}={\mathbb F}_{2}[x_{1}, \ldots , x_{k}]\) be the polynomial algebra over \({\mathbb F}_{2}\) in \(k\) variables, each of degree \(1\). The general linear group \(GL_{k}= GL(k,{\mathbb F}_{2})\) acts on \(P_{k}\) in the usual way. Dickson has shown that the ring of invariants \(D_{k} = (P_{k})^{GL_{k}}\), is also a polynomial algebra. Let \({\mathcal A}\) be the Steenrod algebra. The usual action of \({\mathcal A}\) on \(P_{k}\) commutes with the action of \(GL_{k}\) so that \(D_{k}\) is also an \({\mathcal A}\)-module. Let \(D_{k}^{+}\) and \({\mathcal A}^{+}\) denote the submodules of positive degree in \(D_{k}\) and \({\mathcal A}\) respectively. The authors show that for all \(k > 2 \;\;\;D_{k}^{+} \subset {\mathcal A}^{+} \cdot P_{k}\). The proof is sketched with details to appear elsewhere.