Lie Groups pp 355-363

# Duality Between Sk and $$\mathrm{GL}(n, \mathbb{C})$$

• Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

## Abstract

Let V be a complex vector space, and let $${\bigotimes }^{k}V = V \otimes \cdots \otimes V$$ be the k-fold tensor of V. (Unadorned ⊗ means $$\otimes _{\mathbb{C}}$$.) We consider this to be a right module over the group ring $$\mathbb{C}[S_{k}]$$, where σ ∈ S k acts by permuting the factors:
$$\displaystyle{ (v_{1} \otimes \cdots \otimes v_{k})\sigma = v_{\sigma (1)} \otimes \cdots \otimes v_{\sigma (k)}. }$$
(34.1)
It may be checked that with this definition
$$\displaystyle{\left ((v_{1} \otimes \cdots \otimes v_{k})\sigma \right )\tau = (v_{1} \otimes \cdots \otimes v_{k})(\sigma \tau ).}$$
If A is $$\mathbb{C}$$-algebra and V is an A-module, then $${\bigotimes }^{k}V$$ has an A-module structure; namely, aA acts diagonally:
$$\displaystyle{a(v_{1} \otimes \cdots \otimes v_{k}) = av_{1} \otimes \cdots \otimes av_{k}.}$$
This action commutes with the action (34.1) of the symmetric group, so it makes $${\bigotimes }^{k}V$$ an $$(A, \mathbb{C}[S_{k}])$$-bimodule. Suppose that $$\rho: S_{k}\longrightarrow \mathrm{GL}(N_{\rho })$$ is a representation. Then N ρ is an S k -module, so by Remark 32.3
$$\displaystyle{ V _{\rho } = \left (\left.{\bigotimes }^{k}V \right ) \otimes _{ \mathbb{C}[S_{k}]}N_{\rho }\right. }$$
(34.2)
is a left A-module.

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