A reduced word for an element g of a Coxeter group is a tuple ( s i 1 , … … , s i ℓ ℓ ( g ) ) of Coxeter generators of minimum possible length such that g = s i 1 ⋯ ⋯ s i ℓ ℓ ( g ) . The element g is called fully commutative if any reduced word can be transformed into any other by sequentially swapping pairs of factors that commute. For example, in the finite symmetric group S 4 , the element 2143 = ( 12 ) ( 34 ) is fully commutative, since its two reduced words ( s 1 , s 3 ) and ( s 3 , s 1 ) can be connected by swapping commuting factors, but 4132 = ( 142 ) ( 3 ) is not fully commutative because there is no way to reach the reduced word ( s 3 , s 2 , s 3 , s 1 ) starting from the reduced word ( s 2 , s 3 , s 2 , s 1 ) by commutations.