\(\mathcal A\)-decomposability of the modular invariants of linear groups. (Q1862731)

Let \({\mathbb F}_2\) denote the field of two elements and let \(P_k\) denote the polynomial algebra \({\mathbb F}_2[x_1,\ldots,x_k]\). Consider \(P_k\) as a graded algebra with \(\deg(x_i)=1\), and as an unstable algebra over the mod-2 Steenrod algebra, \({\mathcal A}\), with the action given by \(Sq^1(x_i)=x_i^2\). There is a natural action of the general linear group, \(GL_k:=GL_k({\mathbb F}_2)\), on \(P_k\) by degree preserving algebra automorphisms. For \(G\) a subgroup of \(GL_k\), denote the subring of \(G\)-invariant polynomials by \(P_k^G\). Let \({\mathcal A}^+\) and \(\left(P_k^G\right)^+\) denote the augmentation ideals of the respective algebras. This paper is concerned with the following problem: For which \(G\) is \[ (P_k^G)^+\subset{\mathcal A}^+P_k? \] It is known that \(GL_1\) and \(GL_2\) do not satisfy this property [see \textit{Nguyên H. V. Hu'ng}, Trans. Am. Math. Soc. 349, No. 10, 3893--3910 (1997; Zbl 0902.55004)] but that \(GL_k\) does satisfy the property for \(k\geq 3\) [see \textit{Nguyên H. V. Hu'ng} and \textit{Tran Ngoc Nam}, Trans. Am. Math. Soc. 353, No. 12, 5029--5040 (2001; Zbl 0979.55011)]. Let \(GL_n\bullet {\mathbf 1}_{k-n}\), for \(n<k\), denote the subgroup of \(GL_k\) consisting of block matrices with elements of \(GL_n\) in the upper left block, the zero matrix in the lower left block, the identity matrix in the lower right block and an arbitrary matrix in the upper right block. This paper provides and outline of the proof that \(GL_3\bullet {\mathbf 1}_{k-3}\) satisfies the property. The details of the proof are to be published elsewhere. Clearly if \(G\) is a subgroup of \(H\) then \(\left(P_k^H\right)^+\subseteq \left(P_k^G\right)^+\). Thus if \(G\) satisfies the property, then \(H\) also satisfies the property. Therefore, as a consequence of the result for \(GL_3\bullet {\mathbf 1}_{k-3}\), the property is also satisfied by \(GL_n\bullet {\mathbf 1}_{k-n}\) for \(n>3\), and for the parabolic subgroups \(GL_{k_1,k_2,\dots,k_m}\) with \(k_1\geq 3\).