This point of view allows for a direct translation between the combinatorial and geometric definitions of S ~ ~ n : if one writes [ u ( 1 ) , … … , u ( n ) ] = [ r 1 − − a 1 ⋅ ⋅ n , … … , r n − − a n ⋅ ⋅ n ] where r = [ r 1 , … … , r n ] = π π ( u ) and ( a 1 , a 2 , … … , a n ) ∈ ∈ Λ Λ then the affine permutation u corresponds to the rigid motion of V defined by ( x 1 , … … , x n ) ⋅ ⋅ u = ( x r ( 1 ) + a 1 , … … , x r ( n ) + a n ) .