Computations in generic representation theory: maps from symmetric powers to composite functors

If ${\bold F}_q$ is the finite field of order $q$ and characteristic $p$, let ${\cal F}(q)$ be the category whose objects are functors from finite dimensional ${\bold F}_q$-vector spaces to ${\bold F}_q$-vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in ${\cal F}(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in {\bold F}_q[\Sigma _n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma _n}$,$S^n(V) = V^{\otimes n}/\Sigma _n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$.

Fixing $F$, we discuss the problem of computing $\operatorname{Hom}_{{\cal F}(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $\operatorname{Hom}_{{\cal F}(q)}(S_m, G)$ for all $m$. When $q = p$, we get a complete answer for any functor $F$ chosen from the families listed above.

Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when $F=S^n$, arose in recent work on the homology of iterated loopspaces.