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).

- 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:

Edifices: Building-like spaces associated to linear algebraic groups

with M. Bate and B. Martin

Abstract:

Given a semisimple linear algebraic k-group G, one has a spherical building X_G, and one can interpret the geometric realisation X_G(R) of X_G over the reals R in terms of cocharacters of G. The aim of this paper is to extend this construction to the case when G is an arbitrary connected linear algebraic group; we call the resulting object X_G(R) the spherical edifice of G. We also define an object V_G(R) which is an analogue of the vector building for a semisimple group; we call V_G(R) the vector edifice. The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on V_G(R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, V_G(R) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.

math.GR/2305.11770

Complete reducibility for Lie subalgebras and semisimplification

with M. Bate, S. Böhm, B. Martin, L. Voggesberger

Abstract:

Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra h of the Lie algebra Lie(G) of G and prove some results when h is solvable or char(k)= 0. We introduce the concept of a k-semisimplification h' of h; h' is a Lie subalgebra of Lie(G) associated to h which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that h' is unique up to Ad(G(k))-conjugacy in Lie(G). Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and H' a k-semisimplification of H, the Lie algebra Lie(H') is a k-semisimplification of Lie(H).

math.GR/2305.00841

Flag-accurate arrangements

with Paul Mücksch and Tan Nhat Tran

Abstract:

In [MR21], the first two authors introduced the notion of an accurate arrangement, a particular notion of freeness. In this paper, we consider a special subclass, where the property of accuracy stems from a flag of flats in the intersection lattice of the underlying arrangement. Members of this family are called flag-accurate. One relevance of this new notion is that it entails divisional freeness. There are a number of important natural classes which are flag-accurate, the most prominent one among them is the one consisting of Coxeter arrangements. This warrants a systematic study which is put forward in the present paper. More specifically, let A be a free arrangement of rank r. Suppose that for every d between 1 and r, the first d exponents of A -- when listed in increasing order -- are realized as the exponents of a free restriction of A to some intersection of reflecting hyperplanes of A of dimension d. Following [MR21], we call such an arrangement A with this natural property accurate. If in addition the flats involved can be chosen to form a flag, we call A flag-accurate. We investigate flag-accuracy among reflection arrangements, extended Shi and extended Catalan arrangements, and further for various families of graphic and digraphic arrangements. We pursue these both from theoretical and computational perspectives. Along the way we present examples of accurate arrangements that are not flag-accurate. The main result of [MR21] shows that MAT-free arrangements are accurate. We provide strong evidence for the conjecture that MAT-freeness actually entails flag-accuracy.

math.CO/2302.00343

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**