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Computations in generic representation theory: maps from symmetric powers to composite functors. (English) Zbl 0940.55019
Summary: If $${\mathbf F}_q$$ is the finite field of order $$q$$ and characteristic $$p$$, let $${\mathcal F}(q)$$ be the category whose objects are functors from finite dimensional $${\mathbf F}_q$$-vector spaces to $${\mathbf F}_q$$-vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in $${\mathcal F}(q)$$ include the families $$S_n$$, $$S^n$$, $$\Lambda^n$$, $$\overline S^n$$, and $$cT^n$$, with $$c\in{\mathbf F}_q[\Sigma_n]$$, defined by $$S_n(V)=(V^{\otimes n})^{\Sigma_n}$$, $$S^n(V)=V^{\otimes n}/\Sigma_n$$, $$\Lambda^n(V)=n$$th exterior power of $$V$$, $$\overline S^*(V) =S^*(V)/(p$$th powers), and $$cT^n(V)=c(V^{\otimes n})$$. Fixing $$F$$, we discuss the problem of computing $$\operatorname{Hom}_{{\mathcal F}(q)}(S_m,F\circ G)$$, for all $$m$$, given knowledge of $$\operatorname{Hom}_{{\mathcal F}(q)}(S_m, G)$$ 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 loop spaces.

##### MSC:
 55S10 Steenrod algebra 20G05 Representation theory for linear algebraic groups 55S12 Dyer-Lashof operations
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