The hit problem for symmetric polynomials over the Steenrod algebra

Abstract

We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [ ]₂[x₁, ;…, x_n] = [oplus ]_{d[ges ]0}P^d(n), viewed as a graded left module over the Steenrod algebra [Ascr ] at the prime 2. The grading is by the homogeneous polynomials P^d(n) of degree d in the n variables x₁, …, x_n of grading 1. The present article investigates the hit problem for the [Ascr ]-submodule of symmetric polynomials B(n) = P(n)^{[sum ]n} , where [sum ]_n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let μ(d) denote the smallest value of k for which d = [sum ]^k_i=1(2^λi−1), where λ_i [ges ] 0.