Perfect models for finite Coxeter groups

A model for a finite group is a set of linear characters of subgroups that can be induced to obtain every irreducible character exactly once. A perfect model for a finite Coxeter group is a model in which the relevant subgroups are the quasiparabolic centralizers of perfect involutions. In prior work, we showed that perfect models give rise to interesting examples of W-graphs. Here, we classify which finite Coxeter groups have perfect models. Specifically, we prove that the irreducible finite Coxeter groups with perfect models are those of types A_n, B_n, D_2n+1, H₃, or I₂(n). We also show that up to a natural form of equivalence, outside types A₃, B_n, and H₃, each irreducible finite Coxeter group has at most one perfect model. Along the way, we also prove a technical result about representations of finite Coxeter groups, namely, that induction from standard parabolic subgroups of corank at least two is never multiplicity-free.