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Definition of induced representation

  1. Sep 30, 2011 #1
    http://planetmath.org/encyclopedia/InducedRepresentation.html [Broken]
    The thing I don't get is the definition of a group element [itex]g[/itex]'s action on a vector [itex]\sigma v[/itex]. In the link it defines the action as [itex]g(\sigma v):=\tau (hv)[/itex], where [itex]\tau[/itex] is the unique left coset of G/H containing [itex]gg_{\sigma}[/itex](i.e. such that [itex]gg_{\sigma}=g_{\tau}h[/itex] for some h belonging to H).
    The thing that confuses me is the arbitrariness of h, i.e., if [itex]gg_{\sigma}=g_{\tau}h[/itex], we can always have [itex]gg_{\sigma}=g_{\tau}h_1h_1^{-1}h=(g_{\tau}h_1)(h_1^{-1}h)[/itex], where [itex]h_1[/itex] is some arbitrary element of H. Now [itex]g_{\tau}h_1[/itex] still belongs to coset [itex]\tau[/itex], but [itex]h_1^{-1}h[/itex] will be a different element in H, say [itex]h_1^{-1}h=h'[/itex] , then according to the previous definition of group action, we have [itex]g(\sigma v):=\tau (h'v)[/itex]. So what shall I make of this arbitrariness?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Sep 30, 2011 #2
    Ok I think I figured out my misunderstanding, once a set of representatives of cosets is chosen, you can't change it, so if [itex]g_{\sigma}[/itex] is a representative, then [itex]g_{\sigma}h_1[/itex] can't be on the list of representatives.
     
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