Prof. Dr. Gerhard Röhrle

Publications on zbMATH.

Publications on MathSciNet.

Recent preprints on the ArXiv.

Latest preprints:

On overgroups of distinguished unipotent elements in reductive groups and finite groups of Lie type
with M. Bate, S. Böhm, and B. Martin

Abstract:
Suppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short and uniform proof of this result using so-called good A_1 subgroups of G, introduced by Seitz. We also formulate a counterpart of Korhonen's theorem for overgroups of u which are finite groups of Lie type. Moreover, we generalize both results above by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen's theorem for Lie algebras.
math.GR/2407.16379

Free multiderivations of connected subgraph arrangements
with Paul Mücksch and Sven Wiesner

Abstract:
Cuntz and Kühne introduced the class of connected subgraph arrangements A_G, depending on a graph G, and classified all graphs G such that the corresponding arrangement A_G is free. We extend their result to the multiarrangement case and classify all graphs G for which the corresponding arrangement A_G supports some multiplicity m such that the multiarrangement (A_G,m) is free.
math.CO/2406.19866

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

G-complete reducibility and saturation
Michael Bate, Sören Böhm, Alastair Litterick, and Benjamin Martin

Abstract:
Let H < G be connected reductive linear algebraic groups defined over an algebraically closed field of characteristic p> 0. In our first principal theorem we show that if a closed subgroup K of H is H-completely reducible, then it is also G-completely reducible in the sense of Serre, under some restrictions on p, generalising the known case for G = GL(V). Our second main theorem shows that if K is H-completely reducible, then the saturation of K in G is completely reducible in the saturation of H in G (which is again a connected reductive subgroup of G), under suitable restrictions on p, again generalising the known instance for G = GL(V). We also study saturation of finite subgroups of Lie type in G. Here we generalise a result due to Nori from 1987 in case G = GL(V).
math.RT/2401.16927

Inductive Freeness of Ziegler's Canonical Multiderivations for Restrictions of Reflection Arrangements
with Torsten Hoge and Sven Wiesner

Abstract:
Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A'' of A to any hyperplane endowed with the natural multiplicity k is then a free multiarrangement. Recently, in [Hoge-Röhrle2022], an analogue of Ziegler's theorem for the stronger notion of inductive freeness was proved: if A is inductively free, then so is the free multiarrangement (A'',k)$. In [Hoge-Röhrle2018], all reflection arrangements which admit inductively free Ziegler restrictions are classified. The aim of this paper is an extension of this classification to all restrictions of reflection arrangements utilizing the aforementioned fundamental result from [Hoge-Röhrle2022].
math.GR/2210.00436

Invariants and semi-invariants in the cohomology of the complement of a reflection arrangement
with J. Matthew Douglass and Götz Pfeiffer

Abstract:
Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^* denotes rational singular cohomology, in the case when G is generated by reflections in V and A is the set of reflecting hyperplanes determined by G, or a closely related arrangement. Our first main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to providing a conceptual proof of a conjecture due to Felder and Veselov for Coxeter groups, this result extends the latter to all finite complex reflection groups. Moreover, we prove that determinant-like characters of complex reflection groups do not occur in H^*(M(A)). This extends to all finite complex reflection groups a result proved for Weyl groups by Lehrer.
math.RT/2009.12847