Taylor towers of symmetric and exterior powers

Volume 201 / 2008

Brenda Johnson, Randy McCarthy Fundamenta Mathematicae 201 (2008), 197-216 MSC: 55P65, 18G99, 55U99. DOI: 10.4064/fm201-3-1


We study the Taylor towers of the $n$th symmetric and exterior power functors, $\mathop{\rm Sp}\nolimits^{n}$ and ${\mit\Lambda} ^{n}$. We obtain a description of the layers of the Taylor towers, $D_{k}\mathop{\rm Sp}\nolimits^{n}$ and $D_{k}{\mit\Lambda} ^{n}$, in terms of the first terms in the Taylor towers of $\mathop{\rm Sp}\nolimits^{t}$ and ${\mit\Lambda} ^{t}$ for $t< n$. The homology of these first terms is related to the stable derived functors (in the sense of Dold and Puppe) of $\mathop{\rm Sp}\nolimits^{t}$ and ${\mit\Lambda} ^{t}$. We use stable derived functor calculations of Dold and Puppe to determine the lowest nontrivial homology groups for $D_{k}\mathop{\rm Sp}\nolimits^{n}$ and $D_{k}{\mit\Lambda} ^{n}$.


  • Brenda JohnsonDepartment of Mathematics
    Union College
    Schenectady, NY 12308, U.S.A.
  • Randy McCarthyDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    1409 W. Green St.
    Urbana, IL 61801, U.S.A.

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