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:

${\displaystyle {\widetilde {S))_{n-1}\hookrightarrow {\widetilde {S))_{n))$ ${\displaystyle {\widetilde {S))_{n-1}\hookrightarrow {\widetilde {S))_{n))$
- from the combinatorial perspective, it's ``affine permutations whose last window entry is n (right??)
- from the geometric, we can find ${\displaystyle \Lambda _{n-1}\subset \Lambda _{n))$ as those vectors with last coordinate 0, and it's the transformations that stabilize this sublattice (right??)
- from the algebraic, we send ${\displaystyle s_{i}\mapsto s_{i))$ for $i=1,\ldots ,n-1$ and ${\displaystyle s_{0}\mapsto s_{n-1}s_{n}s_{n-1))$ (right??)
- these maps make it a sub-reflection group, but not a parabolic subgroup (in any sense of the word)
from the combinatorial perspective, ${\displaystyle {\widetilde {S))_{n}\subset {\widetilde {S))_{kn))$ 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 ${\displaystyle \cup _{n}{\widetilde {S))_{n))$ 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 I know this has appeared in a (unpublished?)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?