On the Subregular J-ring of Coxeter Systems

Abstract

Let (W, S) be an arbitrary Coxeter system, and let J be the asymptotic Hecke

algebra associated to (W, S) via Kazhdan-Lusztig polynomials by Lusztig. We study

a subalgebra J_C of J corresponding to the subregular cell C of W . We prove a

factorization theorem that allows us to compute products in J_C without inputs

from Kazhdan-Lusztig theory, then discuss two applications of this result. First, we

describe J_C in terms of the Coxeter diagram of (W, S) in the case (W, S) is simply-

laced, and deduce more connections between the diagram and J_C in some other

cases. Second, we prove that for certain specific Coxeter systems, some subalgebras

of J_C are free fusion rings, thereby connecting the algebras to compact quantum

groups arising in operator algebra theory.