Taylor towers of symmetric and exterior powers

Brenda Johnson¹ ; Randy McCarthy²

¹ Department of Mathematics Union College Schenectady, NY 12308, U.S.A.
² Department of Mathematics University of Illinois at Urbana-Champaign 1409 W. Green St. Urbana, IL 61801, U.S.A.

Fundamenta Mathematicae, Tome 201 (2008) no. 3, pp. 197-216

Résumé

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 $D_{k}{\mit\Lambda} ^{n}$.

Détail

DOI : 10.4064/fm201-3-1

Keywords: study taylor towers nth symmetric exterior power functors mathop nolimits mit lambda obtain description layers taylor towers mathop nolimits mit lambda terms first terms taylor towers mathop nolimits mit lambda homology these first terms related stable derived functors sense dold puppe mathop nolimits mit lambda stable derived functor calculations dold puppe determine lowest nontrivial homology groups mathop nolimits mit lambda