Perfect models for finite Coxeter groups

Abstract

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 ${\mathsf{A}}_n$, ${\mathsf{B}}_n$, ${\mathsf{D}}_{2n+1}$, ${\mathsf{H}}_3$, or ${\mathsf{I}}_2(n)$. We also show that up to a natural form of equivalence, outside types ${\mathsf{A}}_3$, ${\mathsf{B}}_n$, and ${\mathsf{H}}_3$, 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.

32 pages

References in corpus (3)

• Automorphisms and Generalized Involution Models of Finite Complex Reflection Groups
• Induced Characters of Type D Weyl Groups and the Littlewood-Richardson Rule
• Gelfand $W$-graphs for classical Weyl groups

Cited by in corpus (1)

• Insertion algorithms for Gelfand $S_n$-graphs