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Transactions of the American Mathematical Society

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Computations in generic representation theory: maps from symmetric powers to composite functors
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by Nicholas J. Kuhn PDF
Trans. Amer. Math. Soc. 350 (1998), 4221-4233 Request permission

Abstract:

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 $S_n, S^n, \Lambda ^n, \bar {S}^n$, and $cT^n$, with $c \in \mathbf {F}_q[\Sigma _n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma _n}$,$S^n(V) = V^{\otimes n}/\Sigma _n$, $\Lambda ^n(V) = n^{th} \text { exterior power of } V$, $\bar {S}^*(V) = S^*(V)/(p^{th} \text { powers})$, and $cT^n(V) = c(V^{\otimes n})$. Fixing $F$, we discuss the problem of computing $\operatorname {Hom}_{\mathcal {F}(q)}(S_m, F \circ G)$, 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 loopspaces.
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Additional Information
  • Nicholas J. Kuhn
  • Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
  • Email: njk4x@virginia.edu
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4221-4233
  • MSC (1991): Primary 20G05; Secondary 55S10, 55S12
  • DOI: https://doi.org/10.1090/S0002-9947-98-02012-1
  • MathSciNet review: 1443197