Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products

2025/09/19 by J. Matthew Douglass, Götz Pfeiffer, Douglass, J. Matthew +3
#math.GR

paper · pdf · doi:10.48550/arxiv.2509.15850

Abstract

We revisit the structure of the normalizer N_W(P) of a parabolic subgroup P in a finite Coxeter group W, originally described by Howlett. Building on Howlett's Lemma, which provides canonical complements for reflection subgroups, and inspired by a recent construction of Serre for involution centralizers, we refine this understanding by interpreting N_W(P) as a subdirect product via Goursat's Lemma. Central to our approach is a Galois connection on the lattice of parabolic subgroups, which leads to a new decomposition N_W(P) ≅ (P × Q) \rtimes ((A × B) \rtimes C)\text, where each subgroup reflects a structural feature of the ambient Coxeter system. This perspective yields a more symmetric description of N_W(P), organized around naturally associated reflection subgroups on mutually orthogonal subspaces of the reflection representation of W. Our analysis provides new conceptual clarity and includes a case-by-case classification for all irreducible finite Coxeter groups.

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