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Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO 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)) ?

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).

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Lie algebras in characteristic p are already bad enough, and capture so little of the group situation. E.g. a function on the line that's invariant under translation (a group action) must be constant, but a function with derivative zero (the Lie algebra action) is merely required to be a polynomial in x^p. If p=1, this looks like no condition at all...? – Allen Knutson Aug 27 '12 at 13:36
@Allen thank you, it is very valuable comment... may be one should ask about huger algebra including d_x^p/p! ("divided differences" its name if I remember correctly)... – Alexander Chervov Aug 27 '12 at 14:58
up vote 4 down vote accepted

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 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 ($S_n$).

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This is nicer to think of as the monoids of partial transformations and of partial permutations of an n-element set by dropping the base point. They are called the full partial transformation monoid and the symmetric inverse monoid (or rook monoid) respectively. – Benjamin Steinberg Aug 22 '12 at 22:34

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