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 paper continues the investigation of the hit problem, started in [5], for the algebra of symmetric polynomials B(n) viewed as a left ${\cal A}$-module graded by degree, where ${\cal A}$ denotes the Steenrod algebra over the field of two elements ${\bb F}_2$. We recall that a homogeneous element f of grading d in a graded left ${\cal A}$-module M is hit if there is a hit equation in the form of a finite sum $f=\sum_{k>0}Sq^{k}(h_k)$, where the homogeneous elements hk in M have grading less than d and the Sqk are the Steenrod squares, which generate ${\cal A}$. We denote by Q = Q(M) = ${\bb F}_2\otimes_{\cal A}$M the quotient of the module M by the hit elements, where ${\bb F}_2$ is here viewed as a right ${\cal A}$-module concentrated in grading 0. Then Q is a graded vector space over ${\bb F}_2$ and a basis for Q lifts to a minimal generating set for M as a module over ${\cal A}$. The hit problem is to find minimal generating sets for M and criteria for elements to be hit. We recall that B(n) = ${\bb F}_{2}\[\sigma_1,\ldots,\sigma_n\]$ is the polynomial subalgebra of P(n) = ${\bb F}_{2}\[x_1,\ldots,x_n\]$ generated by the elementary symmetric functions $\sigma_i$ in the variables xj . In particular, $\sigma_n=x_1\cdots x_n$. The algebras P(n) and B(n) realize respectively the cohomology of the product of n copies of infinite real projective space and the cohomology of the classifying space BO(n) of the orthogonal group O(n) over ${\bb F}_2$, where the usual grading in cohomology corresponds to degree in the polynomial algebra. The ideal M(n) in B(n), generated by $\sigma_n$, can be identified with the cohomology H*(MO(n), ${\bb F}_2$) of the Thom space MO(n) in positive dimensions. It is also convenient to introduce the notation L(n) for the polynomials in P(n) divisible by $\sigma_n$. Topologically, L(n) corresponds in positive degrees to the cohomology of the n-fold smash product of infinite real projective space. From the topological point of view, ${\cal A}$ is the algebra of universal stable operations in ordinary cohomology with ${\bb F}_2$ coefficients and this explains the action of ${\cal A}$ on P(n), L(n), B(n) and M(n). However, the whole subject may be treated in a purely algebraic fashion [5, 11].

Let P(n) = [x1, . . ., xn] = ⊕d≥0Pd(n) be the polynomial algebra viewed as a graded left module over the Steenrod algebra at the prime 2. The grading is by the degree of the homogeneous polynomials Pd(n) of degree d in the n variables x1, . . ., xn. The algebra P(n) realizes the cohomology of the product of n copies of infinite real projective space. We recall that a homogeneous element f of grading d in a graded left -module M is hit if there is a finite sum f = ΣiSqi(hi), called a hit equation, where the pre-images hi ∈ M have grading strictly less than d and the Sqi, called the Steenrod squares, generate . One of the important parts of the hit problem is to check whether a given polynomial in M is hit or not. In this article we study this problem in the 3-variable case.

We find criteria for symmetrized monomials to be non-hit in the A 2-algebra of symmetric polynomials in three variables, where A 2 is the mod 2 Steenrod algebra.

Abstract. We extend some results involved the action of the Steenrod operations on monomials and get some corollaries on the hit problem. Then, by multiplying some special matrices, we obtain an efficient tool to compute the action of these operations. PreliminariesIn 1947, Steenrod [21] introduced the Steenrod squares Sq k in terms of cocycles in simplicial cochain complex by modifying the Alexander-Ĉech-Whitney formula for the cup product construction. Serre [16] showed that they generate all stable operations in cohomology over F 2 under composition. For an overview on algebraic topology we cite [4]. Cartan [2] discovered a formula for working out a Steenrod square on a product of cohomology classes f , g.Adem[1] and Serre [16] established a faithful representation of A by its action on the cohomology of a test space consisting of infinite real projective spaces whose cohomology is the polynomial algebra P(n) = F 2 [x 1 , x 2 , . . . , x n ] = d≥0 P d (n), viewed as a graded module over the Steenrod algebra A at prime 2. The grading is by the homogeneous polynomials P d (n) of degree d in the variables x 1 , x 2 , . . . , x n of grading 1. We cite to [5] and [11] in cohomology operations and to [11] and [22] in the Steenrod algebra.The Steenrod algebra A is defined to be the graded algebra over the field F 2 , generated by the Steenrod squares Sq k , in grading k ≥ 0, subject to the Adem relations [7,24]. From a topological point of view, the Steenrod algebra is the algebra of stable cohomology operations for ordinary cohomology H * over F 2 .For present purpose we only need to know that the Steenrod algebra acts by composition of linear operators on P(n) and the action of the Steenrod squares