I was stuck trying to show that the group of $n$-affine permutations (i.e. bijections of $\mathbb{Z}$ such that $\sum_{i=1}^{n} f(i)-i =0$ and $f(i+n)=f(i)+n$ for a certain $n \in \mathbb{N}$) is a Coxeter group.
More concretely, I wanted to show that it is generated by the “elementary transpositions” that swap $[i]$ and $[i+1]$ as congruence classes modulo $n$ (leaving the rest fixed). I tried by hand to construct a general formula for an arbitrary such permutation, but the notation soon gets very cumbersome and I cannot conclude.
Online/on books I just found this fact stated as “clearly true (by looking at the action on the right of these transpositions on any permutation)”, which I do not really get.