Ruhr-Universität Bochum

Fakultät für Mathematik

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 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

Restrictions of aspherical arrangements

with Nils Amend and Tilman Möller

Abstract:

In this note we present examples of K(pi,1)-arrangements which admit a restriction which fails to be K(pi,1). This shows that asphericity is not hereditary among hyperplane arrangements.

math.AT/1803.03637

The topology of arrangements of ideal type

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 finite Coxeter groups. This in turn 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 this paper we study the K(\pi,1)-property for a certain class of subarrangements of Weyl arrangements, the so called arrangements of ideal type A_I. These stem from ideals I in the set of positive roots of a reduced root system. We show that the K(\pi,1)-property holds for all arrangements A_I if the underlying Weyl group is classical and that it extends to most of the A_I if the underlying Weyl group is of exceptional type. Conjecturally this holds for all A_I. In general, the A_I are neither simplicial, nor is their complexification fiber type.

math.AT/1801.07157

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

Counting chambers in restricted Coxeter arrangements

with Tilman Moeller

Abstract:

Solomon showed that the Poincare polynomial of a Coxeter group W satisfies a product decomposition depending on the exponents of W. This polynomial coincides with the rank-generating function of the poset of regions of the underlying Coxeter arrangement. In this note we determine all instances when the analogous factorization property of the rank-generating function of the poset of regions holds for a restriction of a Coxeter arrangement. It turns out that this is always the case with the exception of some instances in type E8.

math.CO/1706.09649

Inductive Freeness of Ziegler's Canonical Multiderivations for Reflection Arrangements

with T. 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 is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let A = A(W) be the reflection arrangement of a complex reflection group W. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction A'' of A to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if A'' itself is inductively free.

math.CO/1705.02767

On unipotent radicals of pseudo-reductive groups

with Michael Bate, Benjamin Martin, and David Stewart

Abstract:

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, let k' be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars \R_{k'/k}(G') of a reductive k'-group G'. We prove that the unipotent radical R_u(G_{\bar k}) of the extension of scalars of G to the algebraic closure \bar k of k has exponent e. Our main theorem is to give bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases.

math.GR/1704.04385

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**