All Questions

This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway. If I understand correctly, for any reductive algebraic group $G$ ...

Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which counts index-n subgroups in terms of homomorphisms to $S_n$. Let me give "field with one element" interpretation ...

The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...

There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element". Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...

It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the ...

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 ...