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

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.

Final version; to appear in Algebraic Combinatorics

References in corpus (3)

• Lusztig's $a$-function for Coxeter groups with complete graphs
• Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types
• 2-representations of small quotients of Soergel bimodules in infinite types