I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of sense intuitively, and actually helps to explain to non-specialists the relations between different things I have done.

However, objects over the (non-existent) field with one element aren't just a metaphor - they are objects that can be defined properly, though that may not have happened *yet*. What is the status for the correspondence between $Alt(n)$ and $SO_n(\mathbb{F}_1)$? Is there really a well-defined homomorphism of some sort, and, if so, are there references where this is worked out?

manyprecise definitions. The problem is that there is no prefered one, and up to know no theory has led to a proof of the Riemann hypothesis. $\endgroup$ – Martin Brandenburg Jan 9 '13 at 18:04