Up until now I have had the belief that finite groups do not supply meaningful examples of Lie groups. However in this paper, Kostant claims that the symmetric group on three elements is an exceptional Lie group (page 66). What is the meaning of this?

  • $\begingroup$ Well, finite group with the discrete topology is a zero dimensional manifold.... $\endgroup$ – Aaron Jan 1 '16 at 8:34
  • $\begingroup$ Yes, this is my point: the Lie group structure doesn't seem to say much, and in particular the Lie algebra vanishes. $\endgroup$ – pre-kidney Jan 1 '16 at 8:37
  • $\begingroup$ To be a little less tongue-in-cheek, I have heard people talk about q-analogues in a way that suggests that the symmetric groups are somehow a degeneration of $GL_n$, and that to the extent that we have a theory of the "field with one element", $GL_n(\mathbb F_1)=S_n$, or something along those lines. I don't know how much of this is a just a collection of analogies and how much can be made rigorous, and I certainly don't know if it is what Kostant was thinking of, but it is at least part of a mathematical fairly tale that I've heard about (though have never really heard directly or in full). $\endgroup$ – Aaron Jan 1 '16 at 8:47

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