nLab
extended power

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Contents

Definition

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

For VV an 𝔽 2\mathbb{F}_2-module, hence an 𝔽 2\mathbb{F}_2-vector space, and for nn \in \mathbb{N}, write

V hΣ n n𝔽 2Mod V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod

for the homotopy quotient of the nn-fold tensor product of VV with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants D n(V)D_n(V) of the tensor product of V nV^{\otimes n} with a free resolution EΣ n E \Sigma_n^\bullet of 𝔽 2\mathbb{F}_2:

V hΣ n nD n(V)(V nEΣ n) Σ n. V^{\otimes n}_{h \Sigma_n} \simeq D_n(V) \coloneqq (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n} \,.

This is called the nnth extended power of VV.

Application

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.