Taylor towers of symmetric and exterior powers
Volume 201 / 2008
Fundamenta Mathematicae 201 (2008), 197-216
MSC: 55P65, 18G99, 55U99.
DOI: 10.4064/fm201-3-1
Abstract
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 $D_{k}{\mit\Lambda} ^{n}$.
