### Wissenschaftliche Arbeitsgebiete

- Algebraische Lie Theorie
- Algebraische Gruppen
- Endliche Gruppen vom Lie Typ
- Darstellungstheorie
- Hyperebenen- arrangements

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 04.07.2022, 14:15 Uhr, IC 03/647

Vortragender: Götz Pfeiffer (Galway)

Titel: "Falling Powers and the Algebra of Descents"

Abstract:

A finite Coxeter group of classical type A, B or D contains a chain of
subgroups of the same type. We show that intersections of conjugates
of these subgroups are again of the same type, and make precise in
which sense and to what extent this property is exclusive to the
classical types of Coxeter groups. As the main tool for the proof we
use Solomon’s descent algebra. Using Stirling numbers, we express
certain basis elements of the descent algebra as polynomials and
derive explicit multiplication formulas for a commutative subalgebra
of the descent algebra. This is joint work with Linus Hellebrandt.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 20.06.2022, 14:15 Uhr, IC 03/647

Vortragender: Gerhard Röhrle (RUB)

Titel: "Inductive Freeness of Ziegler's Canonical Multiderivations"

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), alo known as the Ziegler restriction. I'll report on recent joint work with Torsten Hoge where we prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness. Namely, 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. I shall discuss several consequences of the theorem for natural classes of inductively free arrangements. Time permitting I shall explain counterparts of the latter kind for the notion of additive and recursive freeness.

### Oberwolfach Seminar June 2022

"G-Complete Reducibility, Geometric Invariant Theory and Spherical Buildings"

5 June - 11 June, 2022

Summary:

The "Oberwolfach Seminar 2223b" is in a core area of algebraic group
theory and at the
interdisciplinary cross roads of algebra and representation theory on
the one hand, geometry
and geometric invariant theory on the other. The notion of G-complete
reducibility for
subgroups of a reductive algebraic group G was introduced by J-P. Serre
in the 1990s as a
natural generalization of the notion of complete reducibility in
representation theory (which
corresponds to the case where G is the general linear group). Since its
introduction, this notion has been widely studied, both as an
important concept in its own right, with applications to the structure of
linear algebraic groups, and also as a useful tool with applications in
representation theory,
geometric invariant theory, the theory of buildings, and number theory.

The aim of this Oberwolfach seminar is to introduce participants to
G-complete reducibility and explain some of its many applications across
pure mathematics
— participants will learn some rich and deep modern algebra, and leave
equipped with an
understanding of how this mathematics continues to be applied to solve a
diverse range of
problems, particularly in the theory of algebraic groups.

Please see the detailed program and a recommended reading list on the
website of the seminar.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 09.05.2022, 14:15 Uhr, via Zoom

Vortragender: Eirini Chavli (Stuttgart)

Titel: "Complex Hecke algebras are real"

Abstract:

Iwahori Hecke algebras associated with real reflection groups appear in the study of finite reductive groups. In 1998 Broué, Malle, and Rouquier generalized in a natural way the definition of these algebras to complex case. However, some basic properties of the real case are also true for Hecke algebras in the complex case. In this talk we will talk about these properties and their state of the art.

**Zugangsdaten**

https://ruhr-uni-bochum.zoom.us/j/62326053276?pwd=TlVPNzAxdkt2NzcycVNwTXhQSW9Ldz09

Meeting ID: 623 2605 3276

Passcode: arrsym22

### Vortrag im Oberseminar Lie-Theorie

Vortrag am Montag, 02.05.2022, 14:15 Uhr, via Zoom

Vortragender: Prof. Apoorva Khare (Indian Institute of Science, Bangalore)

Titel: "Higher order Verma modules, and a positive formula for all highest weight modules"

Abstract:

We study weights of highest weight modules $V$ over a Kac-Moody algebra $\mathfrak{g}$ (one may assume this to be $\mathfrak{sl}_n$ throughout the talk, without sacrificing novelty). We begin with several positive weight-formulas for arbitrary non-integrable simple modules, and mention the equivalence of several "first order" data that helps prove these formulas. We then discuss the notion of "higher order holes" in the weights, and use these to present two positive weight-formulas for arbitrary modules $V$. One of these is in terms of "higher order Verma modules", and we end by explaining BGG resolutions and Weyl-Kac type character formulas, for these modules in certain cases. (Joint with G.V.K. Teja and with Gurbir Dhillon.)

**Zugangsdaten**

https://ruhr-uni-bochum.zoom.us/j/6340579550?pwd=Mk5BQVZySW5JYVpCeXkyM2tFMWRqZz09

Meeting ID: 634 057 9550

Passcode: alt

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 02.05.2022, 16:15--17:45, on Zoom

Vortragender: Giovanni Paolini (Caltech)

Titel: "Dual Coxeter groups of rank three"

Abstract:

In this talk, I will present ongoing work aimed at understanding the noncrossing partition posets associated with Coxeter groups of rank three. In particular, I will describe the combinatorial and geometric techniques used to prove the lattice property and lexicographic shellability. These properties can then be used to solve several problems on the corresponding Artin groups, such as the K(π,1) conjecture, the word problem, the center problem, and the isomorphism between standard and dual Artin groups. Joint work with Emanuele Delucchi and Mario Salvetti.

**Zugangsdaten**

https://ruhr-uni-bochum.zoom.us/j/62326053276?pwd=TlVPNzAxdkt2NzcycVNwTXhQSW9Ldz09

Meeting ID: 623 2605 3276

Passcode: arrsym22

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 25.04.2022, 14-16 Uhr, IC 03/647

Vortragender: Dr. Paul Mücksch (MPI Bonn)

Titel: "On formality for hyperplane arrangements"

Abstract:

An arrangement of hyperplanes is called formal provided all linear dependencies among the defining linear forms of the hyperplanes are generated by ones corresponding to intersections of codimension two.
This notion turns out to be necessary at the one hand for the apshericity of the complement of a complex arrangement
due to work by Falk and Randell.
One the other hand it is also necessary for the freeness of the module of logarithmic vector fields thanks to a result by Yuzvinsky.

In joint work with T. Möller and G. Röhrle we extend the above line of results by showing that the combinatorial property of factoredness implies formality.
Furthermore, we study formality with respect to the standard arrangement constructions of restriction and localization and comment on the behavior of the stronger property of k-formality introduced by Brand and Terao.

### Vortrag im Oberseminar Lie-Theorie

Vortrag am Montag, 11.04.2022, 14:15 Uhr, IA 1/75

Vortragender: Timm Peerenboom (Bonn)

Titel: "The Affine Grassmannian in Type A "

Abstract:

The Affine Grassmannian associated to a reductive group is an infinite-dimensional analogue of classical (partial) flag varieties. In this talk I will introduce the Affine Grassmannian with its Schubert cell decomposition in type A examples. I will also state the Geometric Satake Equivalence which relates the geometry of the Affine Grassmannian with the representation theory of the Langlands dual group.

### Vortrag im Oberseminar Arrangements and Symmetries

Vortrag am Montag, 11.04.2022, 13-14 Uhr, via Zoom

Vortragender: Shuhei Tsujie (Hokkaido University of Education)

Titel: "MAT-free graphic arrangements and strongly chordal graphs"

Abstract:

Recently Cuntz and Mücksch introduced MAT-free arrangements based on the Multiple Addition Theorem (MAT) by Abe, Barakat, Cuntz, Hoge, and Terao. In this talk, we will focus on graphic arrangements. Stanley showed that a graphic arrangement is free if and only if the graph is chordal. We will show that a graphic arrangement is MAT-free if and only if it is strongly chordal. This is joint work with Tan Nhat Tran.

**Zugangsdaten**

https://ruhr-uni-bochum.zoom.us/j/62326053276?pwd=TlVPNzAxdkt2NzcycVNwTXhQSW9Ldz09

Meeting ID: 623 2605 3276

Passcode: arrsym22

### Mathematisches Forschungsinstitut Oberwolfach

Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications (online meeting)

24 January - 30 January 2021

Organizers:

Takuro Abe, Fukuoka

Alexandru Dimca, Nice

Eva-Maria Feichtner, Bremen

Gerhard Röhrle, Bochum

### Promotionen in 2021

Unser wissenschaftlicher Mitarbeiter Falk Bannuscher hat im November 2021 seine Dissertation erfolgreich verteidigt und seine Promotion zum "Dr. rer. nat." mit Bravour abgeschlossen.

Herr Bannuscher hat seine Dissertation zum Thema: "On G-complete reducibility over non-perfect fields" zum Begriff von Serre von G-vollständiger Zerlegbarkeit verfasst.

Der Lehrstuhl für Algebra und Zahlentheorie gratuliert Herrn Bannuscher zu dieser herausragenden Leistung herzlich und wünscht ihm für seine weitere Karriere alles Gute.

### Alexander von Humboldt Fellowship awarded to Dr Tran

We are very pleased to announce that Dr. Tan Nhat Tran has been awarded a competitive two-year Alexander von Humboldt Fellowship for Postdoctoral Researchers, hosted by Professor Gerhard Röhrle, with a research topic of 'Tutte polynomial and its applications to arrangement theory'.

Tutte polynomial is one of the most-studied invariants of graphs. This two-variable polynomial encodes a substantial amount of the combinatorial information of a graph, and specializes to several important graph polynomials (including the chromatic, flow and reliability polynomials). Significant features of the Tutte polynomial have also been shown in diverse areas of mathematics and physics, for instance, it appears as the Jones and homfly polynomials in knot theory, and as the Ising and Potts model partition functions in statistical mechanics.

From arrangement theory's viewpoint, the Tutte polynomial is unquestionably important because of the pervasiveness of its extensions from graphs to other objects that have richer combinatorial and topological properties. These extensions find applications to three broad types of arrangements: hyperplane, toric and subgroup arrangements. Dr. Tran's research focuses on seeking general frameworks to study arrangements and their Tutte-like polynomials globally that may reveal interesting connections between them.

### Promotionen in 2020

Unsere wissenschaftliche Mitarbeiterin Anne Schauenburg hat im August 2020 ihre Dissertation erfolgreich verteidigt und ihre Promotion zum "Dr. rer. nat." mit Bravour abgeschlossen.

Frau Schauenburg hat ihre Dissertation zum Thema: "Ideal Subarrangements of Real Reflection Arrangements and Notions of Freeness" auf einem aktuellen Gebiet der Hyperebenenarrangements verfasst.

Alle Mitglieder des Lehrstuhls für Algebra und Zahlentheorie gratulieren Frau Schauenburg zu dieser herausragenden Leistung herzlich und wünschen ihr für die weitere Karriere alles Gute.

### Workshop on Hyperplane Arrangements and Reflection Groups

Leibniz Universität Hannover, Germany · September 23 - 27, 2019

Organizers:

Michael Cuntz (Hannover), Gerhard Röhrle (Bochum) and Christian Stump (Bochum)

Confirmed speakers:

Takuro Abe (Kyushu University)

Alexandru Dimca (Nice University)

Matthew Douglass (NSF)

Eva-Maria Feichtner (Bremen University)

Misha Feigin (University of Glasgow)

Lukas Kühne (Hebrew University of Jerusalem)

Jean Michel (Université Denis Diderot)

Tilman Möller (Bochum University)

Paul Mücksch (Bochum University)

Bernhard Mühlherr (Giessen University)

Luis Paris (Université de Bourgogne)

Götz Pfeiffer (National University of Ireland)

Mario Salvetti (Università di Pisa)

Anne Shepler (University of North Texas)

Jiro Sekiguchi (Tokyo University of Agriculture and Technology)

Michele Torielli (Hokkaido University)

Masahiko Yoshinaga (Hokkaido University)

### Summer School - Perspectives in Linear Algebraic Groups

From Monday 19th August until Friday 23rd August, 2019 we will host a Summer School on
"Perspectives in Linear Algebraic Groups", on Ruhr-University Bochum Campus.

This summer school aims at introducing junior researchers to a variety of
topics in linear algebraic groups over arbitrary fields, with a focus on
pseudo-reductive groups and unusual behavior over fields which are not
algebraically closed.

The school will consist of four lecture series given over four days,
followed by a day of conference-style talks.

Lecture series will be given by Ben Martin (University of Aberdeen),
Bernhard Mühlherr (University of Giessen), Gopal Prasad (University of Michigan)
and Bertrand Rémy (Université Paris-Saclay). Additional talks will be given by
Philippe Gilles (Université Lyon 1, France), David Stewart (Newcastle University, UK),
Adam Thomas (Birmingham, UK) and Tomohiro Uchiyama (Soka University, Japan).

### Promotionen in 2019

Unsere wissenschaftlichen Mitarbeiter Maike Gruchot und Tilman Möller haben beide ihre Dissertationen erfolgreich verteidigt und ihre Promotion zum "Dr. rer. nat." mit Bravour abgeschlossen.

Frau Gruchot untersuchte in ihrer Dissertation zum Thema: "A Relative Variant of Complete Reducibility" eine Verallgemeinerung des Begriffes von Serre von G-vollständiger Zerlegbarkeit.

Herr Möller widmete sich in seiner Arbeit "Combinatorial properties of hyperplane arrangements and reflection arrangements" mehreren aktuellen Themen aus der Theorie der Hyperebenenarrangements.

Der Lehrstuhl für Algebra und Zahlentheorie gratuliert den beiden zu ihren herausragenden Leistungen recht herzlich und wünscht ihnen für ihre weitere Karriere alles Gute.

### Alexander von Humboldt Fellowship awarded to Dr Uchiyama

We are very pleased to announce that Dr Tomohiro Uchiyama has been awarded a competitive six-month Alexander von Humboldt Fellowship for Postdoctoral Researchers, hosted by Professor Gerhard Röhrle, with a research topic 'Complete reducibility, geometric invariant theory, and spherical buildings'.

Dr. Uchiyama's research focuses on subgroup structures of reductive algebraic groups (matrix groups). In particular, he studies Serre's notion of complete reducibility of subgroups of reductive groups via geometric invariant theory (a branch of algebraic geometry) and the theory of spherical buildings (highly symmetric combinatorial objects).
In this research project the recently proved 50-years-old center conjecture of Tits comes into play. He also investigates various relations between complete reducibility and pseudo-reductivity due to Conrad-Gabber-Prasad.

### Summer School – New Perspectives in Hyperplane Arrangements

In September 2018 Tilmann Möller, Paul Mücksch, Gerhard Röhrle, and Anne Schauenburg held a summer school directed focusing on new developments in the theory of hyperplane arrangements and related topics.

The school was aimed towards Ph.D. students and young Postdocs in the field. The international set of speakers and the participants came from all over the globe.

The school consisted of six series of minicourses on several topics followed by a day of conference-style talks from international experts. The summer school was funded by RUB Research School.

### Spring School on Complete Reducibility

In April 2018 we will host a spring school, aimed at introducing graduate students and junior researchers to the notion of G-complete reducibility, featuring a lecture series by Professor Ben Martin (Aberdeen), and a number of additional lectures.

### Sloan Visiting Professor Cohen

Professor Daniel Cohen from Louisiana State University in Baton Rouge, LA, USA is currently visiting the Department of Mathematics. Professor Cohen's visit to Bochum is supported by the Sloan Visiting Professor scheme (SVP) of the Mathematisches Forschungsinstitut Oberwolfach (MFO).

Professor Cohen's research focus is in topology, with connections to other areas such as combinatorics and geometric group theory. He is interested in hyperplane arrangements, configuration spaces, braid groups, and related objects. In addition to, and sometimes in relation with, these subjects, portions of his recent work have been on topological problems motivated by the motion planning problem from robotics.

Professor Cohen will present his work in two lectures during the week long visit. They are titled "Pure braid groups and direct products of free groups" and "Topological complexity of surfaces and their configuration spaces" which are held Monday 22.1.18, at 14:15 and Wednesday 24.1.18, at 14:15, respectively. Both venues take place in Wasserstrasse 221, Room 4/20. All are welcome to attend.

During his visit Professor Cohen is hosted by Professor Gerhard Röhrle.

### Alexander von Humboldt Fellowship awarded to Dr Litterick

We are very pleased to announce that Dr Alastair Litterick has been awarded a competitive two-year Alexander von Humboldt Fellowship for Postdoctoral Researchers, hosted by Professors Christopher Voll (Bielefeld University) and Gerhard Röhrle (Ruhr-University Bochum), with a research topic of 'Subgroup structure and generation properties in reductive groups'. Reductive groups are an important class of mathematical objects which lie at the intersection between algebra and geometry. Because of their multi-faceted nature, they are important throughout mathematics and beyond, arising naturally, for instance, in branches of theoretical physics such as string theory.

Dr Litterick's research focuses on algebraic properties of these groups. Although the groups themselves are easy to describe, several aspects of their subgroup structure remain mysterious. In particular, he is concerned with understanding subgroups which are themselves reductive group, and with subgroups containing elements of specified forms. Current approaches to these problems involve many diverse areas of pure mathematics, including algebraic geometry, computational algebra, geometric group theory, geometric invariant theory, Lie theory, cohomology theory, representation theory, and still more. Working in the diverse groups of Professors Voll and Röhrle, Dr Litterick intends to pursue all of these avenues towards the solutions of these problems.

## Archiv der laufenden Ankündigungen

Briefanschrift | Paketanschrift | Fon/Fax/Mail |
---|---|---|

Ruhr-Universität Bochum Fakultät für Mathematik Prof. Dr. Gerhard Röhrle IB 2/133, Fach 69 D-44780 Bochum |
Ruhr-Universität Bochum Fakultät für Mathematik Prof. Dr. Gerhard Röhrle IB 2/133, Fach 69 Universitätsstraße 150 D-44801 Bochum |
Tel.: +49(0)234-32-28304 Fax: +49(0)234-32-14025 E-Mail: gerhard.roehrle@ruhr-uni-bochum.de |