User:JayBeeEll/Affine symmetric group

Now published at WJS, and incorporated into Wikipedia as Affine symmetric group.

Some things that I don't already have sources for but might be worth adding:
 * $$\widetilde{S}_{n - 1} \hookrightarrow \widetilde{S}_{n}$$
 * from the combinatorial perspective, it's ``affine permutations whose last window entry is n (right??)
 * from the geometric, we can find $$\Lambda_{n - 1} \subset \Lambda_n$$ as those vectors with last coordinate 0, and it's the transformations that stabilize this sublattice (right??)
 * from the algebraic, we send $$s_i \mapsto s_i$$ for $$ i = 1, \ldots, n - 1$$ and $$s_0 \mapsto s_{n - 1}s_n s_{n - 1}$$ (right??)
 * these maps make it a sub-reflection group, but not a parabolic subgroup (in any sense of the word)
 * from the combinatorial perspective, $$\widetilde{S}_{n} \subset \widetilde{S}_{kn}$$ for any integer k, but this inclusion is not as reflection groups. Is there a geometric explanation for this action?  Is it of any use to anyone else for any reason?
 * as a consequence of the previous, we can see that the set $$ \cup_n \widetilde{S}_n$$ of all affine permutations is a group. I know this has appeared in a (unpublished?) paper of Abrams--Cowen-Morton; has it ever appeared anywhere else?