Rings of symmetric functions as modules over the Steenrod algebra

Abstract

We write P˝s for the polynomial ring on s letters over the field Z=2, equipped with the standard action of † s , the symmetric group on s letters. This paper deals with the problem of determining a minimal set of generators for the invariant ring .P˝s/ † s as a module over the Steenrod algebra A. That is, we would like to determine the graded vector spaces Z=2˝A .P˝s/ † s . Our main result is stated in terms of a "bigradedSteenrod algebra" H. The generators of this algebra H, like the generators of the classical Steenrod algebra A, satisfy the Adem relations in their usual form. However, the Adem relations for the bigraded Steenrod algebra are interpreted so that Sq 0 is not the unit of the algebra; but rather, an independent generator. Our main work is to assemble the duals of the vector spaces Z=2˝A .P˝s/ † s , for all s 0, into a single bigraded vector space and to show that this bigraded object has the structure of an algebra over H.

A new action of the Kudo–Araki–May algebra on the dual of the symmetric algebras, with applications to the hit problem

“…On the other hand, for the symmetric coinvariants, Singer has recently defined and extended the action of an analogous dual Sq 0 of the Kameko operators to his action of a "bigraded Steenrod algebra" [21], and used it to find primitive elements. We explain at the end of the paper how Singer's action is equivalent to the K-action above at the prime 2.…”

“…For p odd, these are entirely new. While some of these families specialize to zero at the prime 2, those that remain nontrivial coincide with the elements given by Singer [21]. In a paper in preparation [14] we shall treat the hit problem of determining all the primitives in H .BU.2/I F p / at any prime, making use of Proposition 3.1.…”

“…If we take for our grading the length grading in M , then the formulas for our K-action do clearly create a graded cohomology K-module structure on M , and translate to Sq i .a n / D since Sq i .a n / D d 1 i .a n / for all generators. This is the action created by Singer [21] of his "bigraded Steenrod algebra" (note that Sq 0 acts nontrivially, so this is not an action of the Steenrod algebra A).…”

“…Singer [21] developed a method for finding primitives at the prime 2, by defining a 2-primary "bigraded Steenrod algebra" with an action on M . This action is equivalent to the specialization of our action of K to the prime 2, in a sense we shall explain at the end of the paper.…”

The hit problem forH∗(BU(2); 𝔽p)

“…The problem has been studied at the prime p D 2 for polynomial algebras with generators in degree one (cohomology of products of projective spaces), and more recently for algebras of symmetric polynomials in such generators, which are the cohomologies of the classifying spaces BO.l/. The hit problem for various classifying spaces and primes has received considerable attention, and partial results have been obtained in Crossley [1; 2], Janfada and Wood [3; 4], Kameko [5], Pengelley and Williams [6], Peterson [7], Singer [8] and Wood [9]. We refer to [6] for further background.…”