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If Fq is the nite eld of order q and characteristic p ,l et F ( q ) be the category whose objects are functors from nite dimensional Fq{vector spaces to Fq{vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in F(q) include the families Sn ;S n ;n ; Sn ,a ndcTn ,w ith c 2 F q(n), dened by Sn(V )= ( V n ) n , S n ( V )= V n = n , n ( V )= n th exterior power of V , S(V )= S ( V ) = ( p th powers), and cTn(V )= c ( V n). Fixing F , we discuss the problem of computing HomF(q)(Sm ;F G), for all m, given knowledge of HomF(q)(Sm ;G ) for all m .W hen q = p ,w e 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 = Sn, arose in recent work on the homology of iterated loopspaces.

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