Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products
J. Matthew Douglass, Götz Pfeiffer, Gerhard Roehrle
TL;DR
The paper presents a unified, symmetry-enhancing description of the normalizer $N_W(P)$ of a parabolic subgroup $P$ in a finite Coxeter group $W$ by viewing $N_W(P)$ as a subdirect product via Goursat's Lemma and a $W$-equivariant Galois connection on parabolic subgroups. Central to the approach is decomposing the ambient space as $V \cong X \oplus X^{\perp}$ with $X = \mathrm{Fix}_V(P)$ and $X^{\perp} = \mathrm{Fix}_V(P)^{\perp}$, yielding a canonical factorization $N_W(P) \cong (P \times Q) \rtimes D$ and a refinement $N_W(P) \cong (P \times Q) \rtimes ((A \times B) \rtimes C)$ where $Q$ is the orthogonal complement of $P$ and $A,B,C$ arise from Howlett complements and Goursat-induced graphs. The framework provides a uniform, case-by-case classification for irreducible finite Coxeter groups, clarifies the roles of orthogonal closure ${P}^{\dagger\dagger}$ and parabolic closure $\overline{PQ}$, and connects normalizers to involution centralizers via FCA-inspired concepts. The results extend to explicit tables for all irreducible types and reveal a rich interplay between geometric decompositions and algebraic decompositions, with implications for understanding centralizers of involutions and potential extensions to complex reflection groups. The work thus offers a coherent, conceptually symmetric lens on parabolic normalizers and their canonical complements in finite Coxeter groups, organized around orthogonal subspace decompositions and Goursat-type graphs.
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 \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 $N_W(P)$, 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.