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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2509.15850 |
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| _version_ | 1866911547092107264 |
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| author | Douglass, J. Matthew Pfeiffer, Götz Roehrle, Gerhard |
| author_facet | Douglass, J. Matthew Pfeiffer, Götz Roehrle, Gerhard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15850 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products Douglass, J. Matthew Pfeiffer, Götz Roehrle, Gerhard Group Theory 20F55, 06A15 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. |
| title | Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products |
| topic | Group Theory 20F55, 06A15 |
| url | https://arxiv.org/abs/2509.15850 |