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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • 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:
  • MathSciNet review: 1443197