On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in $S_n$

Abstract

‎For a composition $lambda$ of $n$ our aim is to obtain reduced forms‎ ‎for all the elements in the ‎$w_{J(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $S_n$ corresponding to $lambda$‎. ‎We investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{J(lambda)}d$‎, ‎where $d$ is a prefix of‎ ‎an element of minimum length in a $(W_{J(lambda)},B)$ double coset‎ ‎with the trivial intersection property‎, ‎$B$ being a parabolic subgroup‎ ‎of $S_n$ whose type is `dual' to that of $W_{J(lambda)}$‎.