University of IsfahanInternational Journal of Group Theory2251-76504220150601On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in $S_n$2548979510.22108/ijgt.2015.9795ENThomas P.McDonoughDepartment of Mathematics, Aberystwyth UniversityChristos A.PallikarosDepartment of Mathematics and Statistics, University of CyprusJournal Article20150124For 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 $W_{J(\lambda)}$.https://ijgt.ui.ac.ir/article_9795_baa13d009c9d26331351ea54f83b545d.pdf