Prof. Dr. Gerhard Röhrle

Publications on zbMATH.

Publications on MathSciNet.

Recent preprints on the ArXiv.

Latest preprints:

Parabolic Normalizers in Finite Coxeter Groups as Subdirect Products
with J. Matthew Douglass and Götz Pfeiffer

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) \cong (P \times Q) \rtimes ((A \times B) \rtimes C), 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.
math.GR/2509.15850

On connected subgraph arrangements
with L. Giordani, T. Möller, and P. Mücksch

Abstract:
Recently, Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph G, so called connected subgraph arrangements A_G. In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if A_G is an aspherical connected subgraph arrangement, then A_G is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.
math.CO/2502.18144

The subgroup structure of pseudo-reductive groups
with Michael Bate, Ben Martin, and Damian Sercombe

Abstract:
Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of definition for the geometric unipotent radical of $G$, and let $\pi':G_{k'} \to G'$ be the quotient map. We first characterise those smooth subgroups $H$ of $G$ for which $\pi'(H_{k'})=G'$. We next consider the following questions: given a subgroup $H'$ of $G'$, does there exist a subgroup $H$ of $G$ such that $\pi'(H_{k'})=H'$, and if $H'$ is smooth can we find such a $H$ that is smooth? We find sufficient conditions for a positive answer to these questions. In general there are various obstructions to the existence of such a subgroup $H$, which we illustrate with several examples. Finally, we apply these results to relate the maximal smooth subgroups of $G$ with those of $G'$.
math.GR/2406.11286