If u = [ u ( 1 ) , u ( 2 ) , … … , u ( n ) ] is the window notation of an element of this standard copy of S n ⊂ ⊂ S ~ ~ n , its action on the hyperplane V in R n is given by permutation of coordinates: ( x 1 , x 2 , … … , x n ) ⋅ ⋅ u = ( x u ( 1 ) , x u ( 2 ) , … … , x u ( n ) ) . (In this article, the geometric action of permutations and affine permutations is on the right; thus, if u and v are two affine permutations, the action of uv on a point is given by first applying u, then applying v.)