The smallest subgroup whose invariants are hit by the Steenrod algebra

Abstract

Let V be a k-dimensional ${\mathbb{F}_2}$-vector space and let W be an n-dimensional vector subspace of V. Denote by GL(n, ${\mathbb{F}_2}$) • 1k-n the subgroup of GL(V) consisting of all isomorphisms ϕ:V → V with ϕ(W) = W and ϕ(v) ≡ v (mod W) for every v ∈ V. We show that GL(3, ${\mathbb{F}_2}$) • 1k-3 is, in some sense, the smallest subgroup of GL(V)≅ GL(k, ${\mathbb{F}_2})$, whose invariants are hit by the Steenrod algebra acting on the polynomial algebra, ${\mathbb{F}_2})\cong{\mathbb{F}_2}[x_{1},\ldots,x_{k}]$. The result is some aspect of an algebraic version of the classical conjecture that the only spherical classes inQ0S0are the elements of Hopf invariant one and those of Kervaire invariant one.