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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Nicholas J. Kuhn
Journal: Trans. Amer. Math. Soc. 350 (1998), 4221-4233
MSC (1991): Primary 20G05; Secondary 55S10, 55S12
MathSciNet review: 1443197
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Abstract: 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.


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Additional Information

Nicholas J. Kuhn
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email: njk4x@virginia.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02012-1
PII: S 0002-9947(98)02012-1
Received by editor(s): September 11, 1996
Received by editor(s) in revised form: January 3, 1997
Additional Notes: Partially supported by the N.S.F. and the C.N.R.S
Article copyright: © Copyright 1998 American Mathematical Society