I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

I understand that this is because GL_{n}(F_{1}) is supposed to be Sigma_{n}, the symmetric group on n letters. Thus K(F_{1}) = K(finite sets) which is the sphere spectrum by the BarrattPriddyQuillenSegal theorem. But I have no idea why GL_{n}(F_{1}) should be Sigma_{n}... 


Yes, taking GL _n(F _1) to be Sigma _n one can make sense both of the Q and the +construction and both yield the same answer as shown by Deitmar in http://arxiv.org/abs/math/0605429. GL _n(F _1)=Sigma _n is suggested by several observations. One is that counting formulas for subspaces of n dimensional vector spaces over F _q turn into counting formulas for subsets of nelement sets, if one sets q=1. So one could say that an ndimensional vector space over F_1 is an nelement set and GL _n(F _1)=Aut(F _1^n)=Sigma _n. See Cohn's very nicely written http://arxiv.org/abs/math/0407093 for this. One gets another hint by looking at the Tits building for GL _n(F _q) (that is a simplicial complex where the group acts). There is a natural limit for q going to one 1  what then remains is the socalled chamber of the building and the symmetry group of that is Sigma _n. Further hints that one should just drop addition (in comparison to the usual notion of module) come from arithmetic geometry, but that is maybe less convincing and a longer story... 


Here is another heuristic, related to what RandalWilliams said above. The sphere spectrum is the unit object in nice categories of spectra. That is, ring spectra are algebras over the sphere spectrum. Now, to every scheme X you can associate a Ktheory ring spectrum K(X), and this is contravariant. So, in the usual theory there is a morphism K(Z)>K(X) for all schemes X. So, finding F_1 also means finding something (its Ktheory spectrum) that maps to the (homotopy) limit of all Ktheory spectra. That this should be the unit object of the category of spectra doesn't seem very surprising. 

