Does there exist a finite group representation theory over field with one element $\Bbb F_1$? A naive approach is to consider the orbit of $Hom(G, S_n)$ under conjugacy action of symmetric groups $S_n$, but I don't know any result toward such classifications. For example, what is known between the number of irreducible representations and the number of conjugacy classes?

  • $\begingroup$ Correct me if I'm wrong, but at least with my definition of field, there is no field with just one element. Since a field is required to have a multiplicative and additive identity, and further these are required to be different. ( en.wikipedia.org/wiki/Field_(mathematics) ) $\endgroup$ – Adam Higgins May 24 '18 at 23:31
  • $\begingroup$ I would imagine (though I don't know) that representation theory over $\mathbb{F}_{1}$ would be very interesting. What does $\mathrm{GL}(n,\mathbb{F}_{1})$ look like for instance? $\endgroup$ – Morgan Rodgers May 24 '18 at 23:35
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    $\begingroup$ "Representation theory over $\mathbb{F}_1$" should just mean studying homomorphisms $G \to S_n$, or equivalently studying actions of $G$ on finite sets. $\endgroup$ – Qiaochu Yuan May 25 '18 at 1:12
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    $\begingroup$ @AdamHiggins The "field with one element" is not a field with one element: en.wikipedia.org/wiki/Field_with_one_element $\endgroup$ – Lord Shark the Unknown May 25 '18 at 2:16
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    $\begingroup$ I think the real question is, what precisely do you mean by representations over $\mathbb{F}_1$? $\endgroup$ – Joppy May 25 '18 at 6:20

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