Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in $\mathbf{a}(2)$-finite Coxeter systems

TL;DR

This paper provides explicit combinatorial descriptions of Kazhdan--Lusztig cells of a specific a-value in certain Coxeter groups, introducing new elements called stubs for parameterization and analyzing cell structures.

Contribution

It introduces the concept of stubs to parameterize cells and characterizes Kazhdan--Lusztig cells of a-value 2 in (2)-finite Coxeter groups using combinatorial methods.

Findings

Explicit descriptions of cells in (2)-finite Coxeter groups.

Introduction of stubs for cell parameterization.

Calculation of cell cardinalities.

Abstract

A Coxeter group is said to be \emph{$\mathbf{a}(2)$-finite} if it has finitely many elements of $\mathbf{a}$-value 2 in the sense of Lusztig. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in an irreducible $\mathbf{a}(2)$-finite Coxeter group. In particular, we introduce elements we call \emph{stubs} to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of all the one-sided and two-sided cells.