For a fixed element i of { 0 , … … , n − − 1 } , let J = { s 0 , … … , s n − − 1 } ∖ ∖ { s i } be the maximal proper subset of Coxeter generators omitting s i , and let ( S ~ ~ n ) J denote the parabolic subgroup generated by J. Every coset g ⋅ ⋅ ( S ~ ~ n ) J has a unique element of minimum length. The collection of such representatives, denoted ( S ~ ~ n ) J , consists of the following affine permutations: ( S ~ ~ n ) J = { u ∈ ∈ S ~ ~ n : : u ( i − − n + 1 ) < u ( i − − n + 2 ) < ⋯ ⋯ < u ( i − − 1 ) < u ( i ) } .