Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products

Abstract

We revisit the structure of the normalizer NW(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 NW(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 \begin{align*} N_W(P) \cong (P \times Q) \rtimes ((A \times B) \rtimes C)\text, \end{align*} where each subgroup reflects a structural feature of the ambient Coxeter system. This perspective yields a more symmetric description of NW(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.