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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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