On ordered $k$-paths and rims for certain families of Kazhdan-Lusztig cells of $S_n$

Abstract

For a composition $\lambda$ of $n$ we consider the Kazhdan-Lusztig cell in the symmetric group $S_n$ containing the longest element of the standard parabolic subgroup of $S_n$ associated to $\lambda$. In this paper we extend some of the ideas and results in [{Beitr{\"a}ge} zur Algebra und Geometrie, \textbf{59} (2018), no.~3, 523--547]. In particular, by introducing the notion of an ordered $k$-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.