We are concerned in this paper with polynomials in n variables xi over the field 2 of two elements. The Steenrod squares Sqk are linear operators which act on the variables by the rulesand on the product uv of homogeneous polynomials u and v by the Cartan formula

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We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [ ]2[x1, ;…, xn] = [oplus ]d[ges ]0Pd(n), viewed as a graded left module over the Steenrod algebra [Ascr ] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn 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 ]ki=1(2λi−1), where λi [ges ] 0.

This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.

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