Ruhr-Universität Bochum

Fakultät für Mathematik

IB 2/133

Universitätsstrasse 150

D-44780 Bochum

Tel.: +49 (0)234/32-28304

Fax: +49 (0)234/32-14025

eMail: gerhard.roehrle@rub.de

Sprechstunde:

nach Vereinbarung

Bildarchiv des Mathematischen Forschungsinstituts Oberwolfach

## Arbeitsgebiete

- Algebraische Lie Theorie
- Algebraische Gruppen
- Darstellungstheorie
- Endliche Gruppen vom Lie-Typ
- Hyperebenenarrangements
- Spiegelungsarrangements

- 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 on
MathSciNet.

- Recent preprints on the
ArXiv.

- Latest preprints:

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

A construction of pseudo-reductive groups with non-reduced root system

with Michael Bate, Damian Sercombe and David I. Stewart

Abstract:

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type BC_n; these exist only in characteristic 2. We also give a formula for the dimensions of their irreducible modules.

math.GR/2204.09540

Inductive Freeness of Ziegler's Canonical Multiderivations

with Torsten Hoge

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 (A'',k). The aim of this paper is to prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if A is inductively free, then so is the multiarrangement (A'',k). In a related result we derive that if a deletion A' of A is free and the corresponding restriction A'' is inductively free, then so is (A'',k) -- irrespective of the freeness of A. In addition, we show counterparts of the latter kind for additive and recursive freeness.

math.CO/2204.09540

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

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

with T. Abe, C. Stump and M. Yoshinaga

Abstract:

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups.

math.DG/1809.05026

**
Former and current PhD students:**

Simon Goodwin
(University of Birmingham)

Michael Bate
(University of York)

Russell Fowler (Npower)

Glenn Ubly (NHS)

Sebastian Herpel (Barmenia Allgemeine Versicherungs-AG)

Peter Mosch (Volkswohl Bund)

Nils Amend (INTER Krankenversicherung AG)

Anne Schauenburg (Aldi International Services)

Maike Gruchot (Barmenia Krankenversicherung AG)

Tilman Möller

Falk Bannuscher

Sören Böhm

Sven Wiesner

Lorenzo Giordani

Editorial Activity:

Advanced Studies: Euro-Tbilisi Mathematical Journal
.

**Computer Programs**