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# Contents

## Definition

Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the field with two elements.

For $V$ an $\mathbb{F}_2$-module, hence an $\mathbb{F}_2$-vector space, and for $n \in \mathbb{N}$, write

$V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod$

for the homotopy quotient of the $n$-fold tensor product of $V$ with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a free resolution $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:

$V^{\otimes n}_{h \Sigma_n} \simeq D_n(V) \coloneqq (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n} \,.$

This is called the $n$th extended power of $V$.

## References

Last revised on June 15, 2017 at 12:23:41. See the history of this page for a list of all contributions to it.