On embedding certain Kazhdan--Lusztig cells of $S_n$ into cells of $S_{n+1}$

TL;DR

This paper studies specific Kazhdan-Lusztig cells in symmetric groups, providing explicit descriptions for certain classes and establishing a structured way to relate cells across groups of consecutive sizes.

Contribution

It introduces a method to explicitly describe and relate Kazhdan-Lusztig cells containing involutions in $S_n$ and $S_{n+1}$ using diagram classes and cell induction.

Findings

Explicit descriptions for cells with involutions in certain compositions.

A construction of diagram classes $ ext{E}^{( ext{lambda})}$ linking cells across $S_n$ and $S_{n+1}$.

A framework connecting cell induction and restriction via diagram relations.

Abstract

In this paper, we consider a particular class of Kazhdan-Lusztig cells in the symmetric group $S_n$, the cells containing involutions associated with compositions $\lambda$ of $n$. For certain families of compositions we are able to give an explicit description of the corresponding cells by obtaining reduced forms for all their elements. This is achieved by first finding a particular class of diagrams $\mathcal{E}^{(\lambda)}$ which lead to a subset of the cell from which the remaining elements of the cell are easily obtained. Moreover, we show that for certain cases of related compositions $\lambda$ and $\hat{\lambda}$ of $n$ and $n+1$ respectively, the members of $\mathcal{E}^{(\lambda)}$ and $\mathcal{E}^{(\hat{\lambda})}$ are also related in an analogous way. This allows us to associate certain cells in $S_n$ with cells in $S_{n+1}$ in a well-defined way, which is connected to the induction and restriction of cells.