The hit problem for the Dickson algebra (Q2750933)

Let \(A\) be the mod 2 Steenrod algebra, \(\text{GL}_n= \text{GL}_n (\mathbb{F}_2)\) be the general linear group and \(P_n=\mathbb{F}_2 [t_1,\dots, t_n]\), \(n\geq 1\), be the polynomial algebra on \(\mathbb{F}_2\). The group \(\text{GL}_n\) acts on \(P_n\) and the ring of invariants \(D_n=P_n^{\text{GL}_n}\), called the Dickson algebra, is in particular an unstable \(A\)-module.NEWLINENEWLINENEWLINEThe main result of this paper shows that the natural map \(j_n:\mathbb{F}_2 \otimes_AD_n \to(\mathbb{F}_2 \otimes_A P_n)\) is trivial for \(n>2\). The triviality of the map \(j_n\) is equivalent to:NEWLINENEWLINENEWLINE(i) Every element of positive degree in \(D_n\) is \(A\)-decomposable in \(P_n\) for \(n>2\).NEWLINENEWLINENEWLINE(ii) The only spherical classes in \(Q_0S^0\) are the elements of Hopf invariant one and those of Kervaire invariant one.