This thesis explores a conjecture of Lusztig about the correspondence of Fourier matrices combinatorially constructed for non-crystallographic Coxeter groups and \ (S\) -matrices of the Drinfeld center of the asymptotic Hecke category associated to a two-sided cell \ (c\) of a Coxeter group \ (W\). We construct these asymptotic Hecke categories using the diagrammatic approach of Elias--Williamson inside the diagrammatic Hecke category of Soergel bimodules. Through explicit calculations in dihedral groups, and some cells in type \ (H₃\), \ (H₄\), and certain infinite Coxeter groups with finite cells, we verify Lusztig's conjecture by showing that the asymptotic Hecke categories are known in the literature as fusion categories with type \ (Aₖ\) fusion rules. For smaller cells these are either Fibonacci categories or \ (Z/2Z\) -graded vector spaces. Our calculations establish an algorithmic framework that, in the future, could be extended to compute the asymptotic Hecke category of the `so-called' middle cell in type \ (H₄\).
Liam Rogel (Thu,) studied this question.