# Prof. Dr. Gerhard Röhrle

**Address:**

Ruhr-Universität Bochum

Fakultät für Mathematik

Universitätsstraße 150

44801 Bochum

**Room:**

IB 2/133

**Phone:**

(+49)(0)234 / 32-28304

Bildarchiv des Mathematischen Forschungsinstituts Oberwolfach

## Research Areas

- Algebraic Lie theory
- Algebraic groups
- Representation theory
- Finite Lie type groups
- Hyperplane arrangements
- Reflection groups

## Research projects

- DFG project: On Ziegler Extensions of Multiarrangements (within the DFG Priority Programme Combinatorial Synergies).
- DFG project: On connected subgraph arrangements (within the DFG Priority Programme Combinatorial Synergies).
- DFG project: Serre's notion of Complete Reducibility and Geometric Invariant Theory (within the DFG Priority Programme in Representation Theory).
- DFG project: Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov (within the DFG Priority Programme "Algorithmic and experimental methods in Algebra, Geometry and Number Theory").
- DFG project: Arrangements of complex reflection groups: Geometry and combinatorics (joint with M. Cuntz (Kaiserslautern) within the DFG Priority Programme "Algorithmic and experimental methods in Algebra, Geometry and Number Theory").
- DFG project: Inductive freeness and rank-generating functions for arrangements of ideal type: Two conjectures of Sommers and Tymoczko revisited (within the DFG GEPRIS).
- DFG project: On the Cohomology of complements of complex reflection arrangements (within the DFG GEPRIS).
- DFG project: Inductive freeness of Ziegler's canonical multiplicity (within the DFG GEPRIS).
- DFG project: Overgroups of distinguished unipotent elements in reductive groups (within the DFG GEPRIS).
- DFG project: On hyperfactored and recursively factored arrangements (within the DFG GEPRIS).

## Publications

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