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Lie Groups pp 355-363 | Cite as

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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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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