On double cosets with the trivial intersection property and kazhdan-lusztig cells in S_n

For a composition λ of n our aim is to obtain reduced forms for all the elements in the Kazhdan-Lusztig (right) cell containing w_J(λ), the longest element of the standard parabolic subgroup of S_n corresponding to λ. We investigate how far this is possible to achieve by looking at elements of the form w_J(λ)d, where d is a prefix of an element of minimum length in a (W_J(λ), B) double coset with the trivial intersection property, B being a parabolic subgroup of S_n whose type is ‘dual’ to that of W_J(λ).