3 added 1 characters in body

Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO question). Paraphrasing someone: we do not not know what is it, but it is not a field :). For this question it is enough to keep in mind J. Tits idea (1957) that Weyl groups should be thought as semisimple groups over F_1. E.g. symmetric group S_n = GL_n(F_1).

Q1 What might be Lie algebras over F_1 ? In particular for GL_n ? What numerology should correspond to gl_n(F_1) ? I.e. are there some numbers related to gl_n(F_q) which have a limit when q->1 (may be renormalized like with GL_n(F_q)) ?

Comments on further questions are also welcome:

Q2 To what extent representation theory of S_n can be thought as limit q->1 of representation theory of GL_n(F_q) ? (There is some paper "Translating the Irreducible Representations of S_n into GL_n(F_q)", but I do not quite understand it).

Q3 What might be "orbit method" to construct representations of S_n = GL_n(F_1) ?

Q4 What might be Langlands correspondence over F_1 ? Should it be related to bijection between irreps of S_n and its conj. classes (keep in mind that GL^L=GL).

Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO question). Paraphrasing someone: we do not not what is it, but it is not a field :). For this question it is enough to keep in mind J. Tits idea (1957) that Weyl groups should be thought as semisimple groups over F_1. E.g. symmetric group S_n = GL_n(F_1).

Q1 What might be Lie algebras over F_1 ? In particular for GL_n ? What numerology should correspond to gl_n(F_1) ? I.e. are there some numbers related to gl_n(F_q) which have a limit to when q->1 (may be renormalized like with GL_n(F_q)) ?

Comments on further questions are also welcome:

Q2 To what extent representation theory of S_n can be thought as limit q->1 of representation theory of GL_n(F_q) ? (There is some paper "Translating the Irreducible Representations of S_n into GL_n(F_q)", but I do not quite understand it).

Q3 What might be "orbit method" to construct representations of S_n = GL_n(F_1) ?

Q3

Q4 What might be Langlands correspondence over F_1 ? Should it be related to bijection between irreps of S_n and its conj. classes (keep in mind that GL^L=GL).

1

Are there F_un Lie algebras ?

Background See WP-article on F_1 = F_{un} = Field with one element. Paraphrasing someone: we do not not what is it, but it is not a field :). For this question it is enough to keep in mind J. Tits idea that Weyl groups should be thought as semisimple groups over F_1. E.g. symmetric group S_n = GL_n(F_1).

Q1 What might be Lie algebras over F_1 ? In particular for GL_n ? What numerology should correspond to gl_n(F_1) ? I.e. are there some numbers related to gl_n(F_q) which have a limit to q->1 (may be renormalized like with GL_n(F_q)) ?

Comments on further questions are also welcome:

Q2 What might be "orbit method" to construct representations of S_n = GL_n(F_1) ?

Q3 What might be Langlands correspondence over F_1 ? Should it be related to bijection between irreps of S_n and its conj. classes (keep in mind that GL^L=GL).