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

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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).

- Publications on
MathSciNet.

- Recent preprints on the
ArXiv.

- Latest preprints:

Relative complete reducibility and normalised subgroups

with M. Gruchot and A. Litterick

Abstract:

We study a relative variant of Serre's notion of G-complete reducibility for a reductive algebraic group G. We let K be a reductive subgroup of G, and consider subgroups of G which normalise the identity component K^o. We show that such a subgroup is relatively G-completely reducible with respect to K if and only if its image in the automorphism group of K^o is completely reducible. This allows us to generalise a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of G, as well as `rational' versions over non-algebraically closed fields.

math.GR/1810.12096

On the Invariants of the Cohomology of Complements of Coxeter Arrangements

with J.M. Douglass and G. Pfeiffer

Abstract:

We refine Brieskorn's study of the cohomology of the complement of the reflection arrangement of a finite Coxeter group W. As a result we complete the verification of a conjecture by Felder and Veselov that gives an explicit basis of the space of W-invariants in this cohomology ring.

math.RT/1810.02563

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

Relative GL(V)-complete reducibility

with M. Bate and M. Gruchot

Abstract:

Let K be a reductive subgroup of a reductive group G over an algebraically closed field k. Using the notion of relative complete reducibility, in previous work of Bate-Martin-Roehrle-Tange a purely algebraic characterization of the closed K-orbits in G^n was given, where K acts by simultaneous conjugation on n-tuples of elements from G. This characterization generalizes work of Richardson and is also a natural generalization of Serre's notion of G-complete reducibility. In this paper we revisit this idea, focusing on the particular case when the ambient group G is a general linear group, giving a representation-theoretic characterization of relative complete reducibility. Along the way, we extend and generalize several results from the aforementioned work of Bate-Martin-Roehrle-Tange.

math.GR/1806.03067

Arrangements of ideal type are inductively free

with M. Cuntz and A. Schauenburg

Abstract:

Extending earlier work by Sommers and Tymoczko, in 2016 Abe, Barakat, Cuntz, Hoge, and Terao established that each arrangement of ideal type A_I stemming from an ideal I in the set of positive roots of a reduced root system is free. Recently, Röhrle showed that a large class of the A_I satisfy the stronger property of inductive freeness and conjectured that this property holds for all A_I. In this article, we confirm this conjecture.

math.CO/1711.09760

The K(pi,1)-problem for restrictions of complex reflection arrangements

with Nils Amend

Abstract:

In 1962, Fadell and Neuwirth showed that the configuration space of the braid arrangement is aspherical. Having generalized this to many real reflection groups, Brieskorn conjectured this for all Coxeter groups. This follows from Deligne's seminal work from 1972, where he showed that the complexification of every real simplicial arrangement is a K(pi,1)-arrangement. In the 1980s Nakamura and Orlik-Solomon established this property for all but six irreducible unitary reflection groups. These outstanding cases were resolved in 2015 in stunning work utilizing Garside theory by Bessis. In this paper we show that the K(pi,1)-property extends to all restrictions of complex reflection arrangements with the possible exception of only 11 restrictions stemming from some non-real exceptional groups.

math.AT/1708.05452

Freeness of multi-reflection arrangements via primitive vector fields

with T. Hoge, T. Mano, and C. Stump

Abstract:

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.

math.GR/1703.08980

**
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 (Gothaer Lebensversicherung AG)

Nils Amend
(Universität Hannover)

Anne Schauenburg

Maike Gruchot

Tilman Möller

Falk Bannuscher

Editorial Activity:

Tbilisi Mathematical Journal
.

**Computer Programs**