Furthermore, as with every affine Coxeter group, the affine symmetric group acts transitively and freely on the set of alcoves: for each two alcoves, a unique group element takes one alcove to the other. Hence, making an arbitrary choice of alcove A 0 places the group in one-to-one correspondence with the alcoves: the identity element corresponds to A 0 , and every other group element g corresponds to the alcove A = A 0 ⋅ ⋅ g that is the image of A 0 under the action of g.