Invariant theory and the lambda algebra

Abstract:

Let $A$ be the Steenrod algebra over the field ${F_2}$. In this paper we construct for any left $A$-module $M$ a chain complex whose homology groups are isomorphic to the groups $\operatorname {Tor}_s^A({F_2},M)$. This chain complex in homological degree $s$ is built from a ring of invariants associated with the action of the linear group $G{L_s}({F_2})$ on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of $G{L_2}({F_2})$. Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.