On ordered k-paths and rims for certain families of Kazhdan–Lusztig cells of Sn

Abstract

For a composition [Formula: see text] of [Formula: see text] we consider the Kazhdan–Lusztig cell in the symmetric group [Formula: see text] containing the longest element of the standard parabolic subgroup of [Formula: see text] associated to [Formula: see text]. In this paper, we extend some of the ideas and results in [Beiträge zur Algebra und Geometrie, 59(3) (2018) 523–547]. In particular, by introducing the notion of an ordered [Formula: see text]-path, we are able to obtain alternative explicit descriptions for some additional families of cells associated to compositions. This is achieved by first determining the rim of the cell, from which reduced forms for all the elements of the cell are easily obtained.