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

TL;DR

This paper introduces the concept of ordered $k$-paths to provide explicit descriptions of certain Kazhdan-Lusztig cells in symmetric groups, enhancing understanding of their structure through rim determination and reduced forms.

Contribution

It extends previous work by defining ordered $k$-paths, offering new explicit descriptions of Kazhdan-Lusztig cells related to compositions in symmetric groups.

Findings

Defined ordered $k$-paths for cell description

Determined the rim of specific Kazhdan-Lusztig cells

Provided reduced forms for elements of these cells

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.