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kof9595995
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http://planetmath.org/encyclopedia/InducedRepresentation.html
The thing I don't get is the definition of a group element g's action on a vector \sigma v. In the link it defines the action as g(\sigma v):=\tau (hv), where \tau is the unique left coset of G/H containing gg_{\sigma}(i.e. such that gg_{\sigma}=g_{\tau}h for some h belonging to H).
The thing that confuses me is the arbitrariness of h, i.e., if gg_{\sigma}=g_{\tau}h, we can always have gg_{\sigma}=g_{\tau}h_1h_1^{-1}h=(g_{\tau}h_1)(h_1^{-1}h), where h_1 is some arbitrary element of H. Now g_{\tau}h_1 still belongs to coset \tau, but h_1^{-1}h will be a different element in H, say h_1^{-1}h=h' , then according to the previous definition of group action, we have g(\sigma v):=\tau (h'v). So what shall I make of this arbitrariness?
The thing I don't get is the definition of a group element g's action on a vector \sigma v. In the link it defines the action as g(\sigma v):=\tau (hv), where \tau is the unique left coset of G/H containing gg_{\sigma}(i.e. such that gg_{\sigma}=g_{\tau}h for some h belonging to H).
The thing that confuses me is the arbitrariness of h, i.e., if gg_{\sigma}=g_{\tau}h, we can always have gg_{\sigma}=g_{\tau}h_1h_1^{-1}h=(g_{\tau}h_1)(h_1^{-1}h), where h_1 is some arbitrary element of H. Now g_{\tau}h_1 still belongs to coset \tau, but h_1^{-1}h will be a different element in H, say h_1^{-1}h=h' , then according to the previous definition of group action, we have g(\sigma v):=\tau (h'v). So what shall I make of this arbitrariness?
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