In (Shi 2002), Shi showed that the affine symmetric group is a generic cover of the family { G ( m , m , n ) : : m ≥ ≥ 1 } , in the following sense: for every positive integer m, there is a surjection π π m from S ~ ~ n to G ( m , m , n ) , and these maps are compatible with the natural surjections G ( m , m , n ) ↠ ↠ G ( p , p , n ) when p ∣ ∣ m that come from raising each entry to the m / p th power. Moreover, these projections respect the reflection group structure, in that the image of every reflection in S ~ ~ n under π π m is a reflection in G ( m , m , n ) ; and similarly when m > 1 the image of the standard Coxeter element s 0 ⋅ ⋅ s 1 ⋯ ⋯ s n − − 1 in S ~ ~ n is a Coxeter element in G ( m , m , n ) .