**Background** See [WP-article](http://en.wikipedia.org/wiki/Field_with_one_element) on F\_1 = F\_{un} = Field with one element (and also [this MO](https://mathoverflow.net/questions/2300/what-is-the-field-with-one-element) question). Paraphrasing someone:
we do 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)) ? 




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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](http://www.math.harvard.edu/archive/126_fall_98/papers/dgharris.pdf) "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).



Are there F_un Lie algebras ?  
I will only attempt to answer the first question.


$n$-dimensional vector space over $\mathbf{F}\_1$ is the same as a pointed set with $n+1$ elements. It is natural to call $GL\_n(\mathbf{F}\_1)$ the group of automorphisms of $\mathbf{F}\_1^n$ and $\mathfrak{gl}\_n(\mathbf{F}\_1)$ the monoid of endomorphisms. There are at least two notions of morphisms:


1. Plain morphisms of pointed sets. The monoid of endomorphisms has cardinality $(n+1)^n$.
2. Maps of pointed sets which are injective if you throw away the basepoints (see <http://arxiv.org/abs/1006.0912>). It is not too hard to see that the cardinality of $End(\mathbf{F}\_1)$ is
$$\sum\_{k=0}^n\left(\begin{array}{c}n\\ k\end{array}\right)\frac{n!}{(n-k)!}=\sum\_{k=0}^n\frac{(n!)^2}{k!(n-k)!^2}.$$
Here $k$ is the number of elements that don't go to the basepoint.


Note, that in both cases the group of automorphisms is the same ($S\_n$).


