# Oberseminare

Below you can find all talks, presentations, and lectures of our seminars (Oberseminare).

## winter term 2024/2025

**Giulia Iezzi **(RWTH Aachen)

Seminar: Lie-Theorie

Vortrag am Montag, 18.11.2024, 16:15 Uhr, in IA 1/109**Titel**: "Linear degenerations of Schubert varieties via quiver Grassmannians"**Abstract**:

Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples. We provide a construction for realising smooth Schubert varieties as quiver Grassmannians and desingularizing non-smooth Schubert varieties. We then exploit this construction to define linear degenerations of Schubert varieties, giving a combinatorial description of the correspondance between their isomorphism classes and the B-orbits of certain quiver representations.

**Dmitriy Rumynin **(Warwick, zZt MPIM Bonn)

Seminar: Lie-Theorie

Vortrag am Montag, 04.11.2024, 16:15 Uhr, in IA 1/109**Titel**: "Disconnected Reductive Groups"**Abstract**:

A disconnected reductive group is a linear algebraic group whose connected component of the identity is a reductive group. If one is only interested in connected reductive groups, disconnected ones enter the picture as subgroups.

In this talk I will explain how to classify disconnected reductive groups up to an isomorphism. Time permitting, I will also briefly discuss the representation ring of such a group. The talk is based on joint work with Dylan Johnston and Diego Martin Duro.

**Matilde Maccan** (RUB)

Seminar: Lie-Theorie

Vortrag am Montag, 28.10.2024, 16:15 Uhr, in IA 1/109**Titel**: "Parabolic subgroup schemes in small characteristics"**Abstract**:

Any rational projective homogeneous variety can be written as a quotient of a semi-simple algebraic group by a so-called parabolic subgroup. In this talk we complete the classification of parabolic subgroup schemes (which can be non-reduced) and formulate it in a uniform way, independent of type and characteristic. The cases we focus on are of a base field of characteristic two or three. We will then move on to a few geometric consequences.

## summer term 2024

**Sven Wiesner** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 15.07.2024, 16:15 Uhr, in IA 1/135**Titel**: "Free multiderivations of connected subgraph arrangements"**Abstract**:

Recently Cuntz and Kühne introduced a new class of arrangements coming from undirected graphs. In their work they classified all graphs which give a free simple arrangement. In this talk I will discuss joint work with Paul Mücksch and Gerhard Röhrle, where we extend this result to the multiarrangement case by classifying all graphs that (for some multiplicity) give a free multiarrangement.

**Alexander Ivanov** (RUB)

Seminar: Lie-Theorie

Vortrag am Montag, 08.07.2024, 14:15 Uhr, in IA 1/75**Titel**: "An introduction to the Langlands correspondences"**Abstract**:

I will try to explain some basics of the Langlands program, with as few prerequisites as possible. More concretely, I will concentrate on the case of number fields (the original case, where Langlands program took its origin) and discuss in detail the one-dimensional case --that is, global class field theory. Then I will sketch the n-dimensional conjecture.

**Jakub Löwit** (IST)

Seminar: Lie-Theorie

Vortrag am Montag, 03.06.2024, 14:15 Uhr, in IA 1/75**Titel**: "On modular p-adic Deligne--Lusztig theory for GL_n"**Abstract**:

In 1976, Deligne--Lusztig realized the characteristic zero representation theory of finite groups of Lie type inside cohomology of certain algebraic varieties. This picture has two interesting generalizations. In one direction, one can replace finite groups by p-adic groups. In another direction, one can consider modular coefficients. After recalling the key players, I will discuss what happens in the p-adic case with modular coefficients for GL_n. In particular, I will explain how to deduce such results from the case of characteristic zero coefficients.

**Daniel Bath** (KU Leuven)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 29.04.2024, 16:15 Uhr, in IA 1/135**Titel**: "Bernstein—Sato polynomials of Hyperplane Arrangements in C^3"**Abstract**:

The roots of Bernstein—Sato polynomial of a hypersurface manages to simultaneously contain most classical singularity invariants. Alas, computing these roots is mostly infeasible. For hyperplane arrangements in C^3, one can hope the roots of its Bernstein—Sato polynomial are combinatorially determined. Alas, things are more subtle. Walther demonstrated two arrangements with the same intersection lattice but whose respective Bernstein—Sato polynomials differ by exactly one root. We will show this is the only pathology possible. For arrangements in C^3, we prove that all but one root are (easily) combinatorially determined. We also give several equivalent criterion for the outlier, -2 + (2/deg), to in fact be a root of the Bernstein—Sato polynomial. These involve local cohomology data of the Milnor algebra and the non-formality of the arrangement.

This is an application of a study of Bernstein—Sato polynomials for a larger class of C^3 divisors than just arrangements. We will discuss Bernstein—Sato polynomials at large, our general strategy for divisor class, our main results, and how the promised formula for hyperplane arrangements appears.

**David Schwein** (University of Bonn)

Seminar: Lie-Theorie

Vortrag am Montag, 15.04.2024, 14:15 Uhr, in IA 1/75**Titel**: "Tame supercuspidals at very small primes"**Abstract**:

Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem, solved when p is large. When p is small, the remaining supercuspidals are expected to have an arithmetic source: wildly ramified field extensions. In this talk I'll discuss ongoing work joint with Jessica Fintzen that identifies a second, Lie-theoretic, source of new (tame!) supercuspidals: special features of reductive groups at very small primes. We'll summarize some of these features and explain how they contribute to the construction of supercuspidals.

## winter term 2023/2024

**Johannes Schmitt** (Universität Siegen)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 22.01.2024, 14:00 Uhr, in IA 1/177**Titel**: "Symplectic reflections and quotient singularities"**Abstract:**

Let V be a finite-dimensional vector space over the complex numbers and let G < GL(V) be a finite group. It is classically known that the reflections contained in G control aspects of the geometry of the linear quotient V/G, namely its smoothness (Chevalley- Shephard-Todd) and the class group (Benson). In this talk, we restrict to symplectic vector spaces V and groups G < Sp(V) which leave the symplectic form invariant and study partial symplectic resolutions of V/G. Here we find similar phenomena. We describe how the existence of a (smooth) symplectic resolution is related to aptly named symplectic reflections in G by a theorem of Verbitsky, but also how this relationship is still not completely understood. We further show that the class group of a partial symplectic resolution is controlled by the symplectic reflections in G.

**Takuro Abe** (Rikkyo University)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 15.01.2024, 14:00 Uhr, in IA 1/177**Titel**: "Multi-Euler derivations"**Abstract**:

In the study of hyperplane arrangements, the research of the logarithmic derivation modules of them is one of the most important topics. Based on Kyoji Saito's primitive derivation, Terao's investigation of the multi-Coxeter arrangements and Yoshinaga's solving of the Edelman-Reiner conjecture, the study of logarithmic derivation modules of multi-reflection arrangements is also important. Among these researches, the existence of so called the universal derivations, defined by Wakamiko, has been a key of the research, and investigated by Stump, Roehrle, Terao, Yoshinaga, Wakamiko and so on. In this talk, we generalize this concept of universal derivations to arbitrary arrangements as the multi-Euler derivations, and give a criterion for a derivation to be multi-Euler.

**Vanthana Ganeshalingam** (University of Warwick)

Seminar: Lie-Theorie

Vortrag am Montag, 20.11.2023, 16 Uhr, in IA 1/109**Titel**: "Subgroup Structure of Reductive Groups"**Abstract**:

This talk will introduce the concept of G complete-reducibility (G c-r) originally thought of by Serre in the 90s. This idea has important connections to the open problem of classifying the subgroups of a reductive group G. I will explain the methodology of the classification so far and the main obstacle which is understanding the non-G-cr subgroups.

**Lorenzo Giordani **(RUB)

Seminar: Arrangements and Symmetries

Montag, 13.11.2023, 14:45 - 15:15 Uhr, in IA 1/177**Titel**: "Cohomology Rings of toric wonderful Models "**Abstract**:

One of the leading motifs in the theory of arrangements is to understand the interplay of algebraic, geometric and topological properties with combinatorial ones, meant as properties of the intersection structure of the arrangements. An approach in this direction, initiated by DeConcini and Procesi in the nineties, is the introduction of ``wonderful'' compactifications for complement spaces of subspace arrangements. The study of wonderful models has ensured the combinatorial nature of relevant topological invariants of complement spaces, namely: cohomology, rational homotopy type, and mixed Hodge structure. Projective wonderful models were extended to the case of toric arrangements of arbitrary codimension. Their cohomology has then been studied by DeConcini and Gaiffi in the ``well-connected'' case, in which properties holding naturally in the linear case are imposed. I will report on work in progress with Roberto Pagaria and Viola Siconolfi, where we remove the hypothesis of well-connectedness, and offer a different presentation for the cohomology ring of the model. Our approach is closer to the combinatorial picture: we adapt the notions of blowups for semilattices and nested set complex introduced by Feichtner and Kozlov to the poset of layers of toric arrangements.

**Sven Wiesner** (RUB)

Seminar: Arrangements and Symmetries

Montag, 13.11.2023, 14:15 - 14:45 Uhr, in IA 1/177**Titel**: "Free multiderivations on connected subgraph arrangements "**Abstract**:

Recently, Cuntz and Kühne introduced (a way to assign) a hyperplane arrangement A(G) associated with a given connected graph G and they classified all graphs G such that A(G) is a free arrangement. I will present joint work with Gerhard Röhrle and Paul Mücksch in which we generalize this result to the multiarrangement case. We give a complete list of graphs which possess at least one non-trivial multiplicity m such that the corresponding multiarrangement (A(G),m) is free.

**Torben Wiedemann** (Bielefeld/Giessen)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 06.11.2023, 14 -16 Uhr, in IA 1/177**Titel**: "Root Graded Groups"**Abstract**:

Let \Phi be a finite root system. A \Phi-graded group is a group G together with a family of subgroups (U_\alpha)_{\alpha \in \Phi} satisfying some purely combinatorial axioms. The main examples of such groups are the Chevalley groups of type \Phi, which are defined over commutative rings and which satisfy the well-known Chevalley commutator formula. We show that if \Phi is of rank at least 3, then every \Phi-graded group is defined over some algebraic structure (e.g. a ring, possibly non-commutative or, in low ranks, even non-associative) such that a generalised version of the Chevalley commutator formula is satisfied. A new computational method called the blueprint technique is crucial in overcoming certain problems in characteristic 2. This method is inspired by a paper of Ronan-Tits.

**Graham Denham** (University of Western Ontario)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 23.10.2023, 14:00 Uhr, in IA 1/177**Titel**: "Kirchhoff polynomials configuration hypersurfaces"**Abstract**:

A finite graph determines a Kirchhoff polynomial, which is a squarefree, homogeneous polynomial in a set of variables indexed by the edges. The Kirchhoff polynomial appears in an integrand in the study of particle interactions in high-energy physics, and this provides some incentive to study the motives and periods arising from the projective hypersurface cut out by such a polynomial.

From the geometric perspective, work of Bloch, Esnault and Kreimer (2006) suggested that the most natural object of study is a polynomial determined by a linear matroid realization, for which the Kirchhoff polynomial is a special case.

I will describe some joint work with Avi Steiner, Delphine Pol, Mathias Schulze and Uli Walther on the interplay between geometry and matroid combinatorics for this family of objects, and the relationship with logarithmic derivations on the associated hyperplane arrangements.

**Sean Cotner** (University of Michigan)

Seminar: Lie-Theorie

Vortrag am Montag, 16.10.2023, 16:00 Uhr, via Zoom**Titel**: "Hom schemes and complete reducibility"**Abstract**:

In order to study semisimplicity phenomena in modular representation theory, Serre introduced the notion of G-completely reducible (G-cr) subgroup of a reductive group G. Later, using results of Richardson, Bate--Martin--Röhrle put Serre's theory into an algebro-geometric context, showing that G-cr subgroups are related to closed G-orbits in G^n. In this talk, I will describe another natural and powerful geometric context for complete reducibility, based on new existence results for schemes of homomorphisms. Throughout, there will be many examples and pictures to illustrate the (initially strange-looking) geometry of Hom schemes.

Zoom Link: ruhr-uni-bochum.zoom.us/j/65802295776

Meeting-ID: 658 0229 5776

Passcode: 150617

## summer term 2023

**Nathan Chapelier** (University of Sydney)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 14.08.2023, 16:15 Uhr, in IB 1/135**Titel**: "The Shi variety of an affine Weyl group "**Abstract**:

In this talk I will introduce a geometrical object associated to an affine Weyl group W, called the Shi variety, that has the property of having its set of integral points in bijection with W. Then we will discuss some consequences obtained with the set of the irreducible components of this variety, in particular its connections in type A with a certain conjugacy class and Young's lattice.

**Laura Voggesberger** (RUB)

Seminar: Lie-Theorie

Vortrag am Montag, 22.05.2023, 16:15 Uhr, in IA 1/135**Titel**: "Semisimplification for subalgebras of Lie algebras"**Abstract**:

Let G be a connected reductive linear algebraic group over a field k. We introduce the concept of a k-semisimplification h′ of h for a Lie subalgebra h of the Lie algebra g = Lie(G) of G. Here h′ is a Lie subalgebra of 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 Bate, Martin and Röhrle. As in the subgroup case, we show that h′ is unique up to Ad(G(k))-conjugacy in 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).

**Thomas Gerber** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 24.04.2023, 16:15 Uhr, in IA 1/135**Titel**: "Atomic length on Weyl groups II: combinatorics "**Abstract**:

In recent joint work with Nathan Chapelier-Laget, we introduced the notion of atomic length for (finite and affine) Weyl groups, as a variant of the usual Coxeter length function. In this second talk, I will present various properties and interpretations of this statistic using root system combinatorics. We will also see how this gives natural extensions of the results of the previous talk.

**Thomas Gerber **(RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 17.04.2023, 16:15 Uhr, in IA 1/135**Titel**: "Atomic length on Weyl groups I: representation theory "**Abstract**:

In recent joint work with Nathan Chapelier-Laget, we introduced the notion of atomic length for (finite and affine) Weyl groups, as a variant of the usual Coxeter length function. In this first talk, I will present the representation-theoretic motivations for studying this statistic. More precisely, we will review some fundamental problems in modular representation theory of symmetric groups and Hecke algebras, which can be tackled by investigating partition combinatorics.

## winter term 2022/2023

**Lorenzo Giordani** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 06.02.2023, 14:15 Uhr, in IA 1/177**Titel**: "On the Combinatorics and Cohomology of Wonderful models for subspace arrangements"**Abstract**:

A classical problem in the theory of hyperplane arrangements is to understand to what extent the combinatorial information of the arrangement, encoded in the associated matroid or lattice of intersections, determines geometric proprieties of the complement space. P. Orlik, L. Solomon, E. Brieskorn et al. proved that the cohomology ring of the complement space is isomorphic to the so called Orlik-Solomon algebra, which is defined entirely in terms of the underlying matroid. In this seminar, we recall the results on the Orlik-Solomon algebra and present some constructions by C. De Concini and P. Procesi, including their "Wonderful model" and its proprieties. Using the model, they proved that the cohomology ring of the complement space is still determined by the combinatorial data when hyperplanes are substituted by subspaces of arbitrary codimension.

**Sven Wiesner **(RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 30.01.2023, 14:15 Uhr, in IA 1/177**Titel**: "Techniques from algebraic geometry applied to matroids"**Abstract**:

June Huh et al. proved longstanding conjectures about specific sequences associated to matroids which are combinatorial objects. They did so by associating a structure to these matroids on which tools from algebraic geometry can get deployed. In my talk I want to give a short overview about the structures involved and how they derived the results about the matroid from them.

**Prof. Dr. Michael Cuntz** (Hannover)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 09.01.2023, 14:15 Uhr, on Zoom**Titel**: "On arrangements of hyperplanes from connected subgraphs "**Abstract**:

We investigate arrangements of hyperplanes whose normal vectors are given by connected subgraphs of a fixed graph. These include the resonance arrangement and certain ideal subarrangements of Weyl arrangements. We characterize those which are free, simplicial, factored, or supersolvable. In particular, such an arrangement is free if and only if the graph is a cycle, a path, an almost path, or a path with a triangle attached to it. This is joint work with Lukas Kühne.

Zoom Link: ruhr-uni-bochum.zoom.us/j/61417572342

Meeting ID: 614 1757 2342

Passcode: ArrSym22

**Dr. David Stewart** (University of Newcastle)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 05.12.2022, 16:00 Uhr, in IA 1/109**Titel**: "A Prolog-assisted search for simple Lie algebras" (jt work with David Cushing and George Stagg)**Abstract**:

Prolog is a very unusual programming language, developed by Alain Colmerauer in one of the buildings on the way to the CIRM in Luminy. It is not fundamentally iterative in the way that, for example, GAP and Magma are. Instead it operates by taking a list of axioms as input, and responds at the command line to queries asking the language to achieve particular goals. It gained some notoriety by beating contestants on the game show Jeopardy in 2011. It is also the worlds fastest sudoku solver. I will describe some recent Prolog investigations to search for new simple Lie algebras over the field GF(2). We were able to discover some new examples in dimensions 15 and 31 and extrapolate from these to construct two new infinite families of simple Lie algebras.

**Laura Voggesberger** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 21.11.2022, 14:15 Uhr, in IA 1/177**Titel**: "Nilpotent Pieces in Lie Algebras of Exceptional Type in Bad Characteristic"**Abstract**:

This talk will be a trial run for my defense concerning certain structures in algebraic groups and their Lie algebras. In group theory, a big and important family of infinite groups is given by the algebraic groups. These groups and their structures are already well-understood. In representation theory, the study of the unipotent variety in algebraic groups — and by extension the study of the nilpotent variety in the associated Lie algebra — is of particular interest. Let G be a connected reductive algebraic group over an algebraically closed field k, and let Lie(G) be its associated Lie algebra. By now, the orbits in the nilpotent and unipotent variety under the action of G are completely known and can be found for example in a book of Liebeck and Seitz. There exists, however, no uniform description of these orbits that holds in both good and bad characteristic. With this in mind, Lusztig defined a partition of the unipotent variety of G in 2011. Equivalently, one can consider certain subsets of the nilpotent variety of Lie(G) called the nilpotent pieces. This approach appears in the same paper by Lusztig in which he explicitly determines the nilpotent pieces for simple algebraic groups of classical type. The nilpotent pieces for the exceptional groups of type G2 , F4 , E6 , E7 , and E8 in bad characteristic have not yet been determined. In my thesis, I have explored the cases for G2 , F4 , and E6, and will present them in this talk.

**Sven Wiesner** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 14.11.2022, 14:15 Uhr, in IA 1/177**Titel**: "Inductive Freeness of Ziegler's Canonical Multiderivations for Restrictions of Reflection Arrangements "**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 Hoge and Röhrle proved an analogue of Ziegler's theorem for the stronger notion of inductive freeness: If A is inductively free, then so is the free multiarrangement (A",k). In 2018 Hoge and Röhrle classified all reflection arrangements which admit inductively free Ziegler restrictions. I will talk about joint work with Torsten Hoge and Gerhard Röhrle where we extended this classification to restrictions of reflection arrangements.

**Dr. Paul Mücksch** (Kyushu University)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 24.10.2022, 14:15 Uhr, via Zoom**Titel**: "Topology of supersolvable oriented matroids"**Abstract**:

A central result in the topology of complex hyperplane arrangements, due to Falk, Randell and Terao, states that supersolvability of the intersection lattice of such arrangements implies that their complements are $K(\pi,1)$-spaces.

The homotopy type of the complement of a complexified real hyperplane arrangement can be modeled by a nice regular CW-complex introduced by Salvetti. The Salvetti complex can be constructed for any oriented matroid -- a combinatorial abstraction of a real hyperplane arrangement.

In my talk, I will present a novel combinatorial way to prove that supersolvability of the geometric lattice of an oriented matroid implies the asphericity of its Salvetti complex. In particular, this extends to the non-realizable case.

Zoom Link: ruhr-uni-bochum.zoom.us/j/61417572342

Meeting ID: 614 1757 2342

Passcode: ArrSym22

**Avi Steiner** (Universität Mannheim)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 17.10.2022, 14:15 Uhr, IA 1/177**Titel**: "Symmetrizing" logarithmic derivations with respect to matroid duality"**Abstract**:

Of interest to people who study both hyperplane arrangements and commutative algebra are the homological properties of the module of logarithmic derivations of a hyperplane arrangement A. I will introduce the "ideal of pairs", which is a sort of "symmetrization" of this module of logarithmic derivations with respect to matroid duality. This is an ideal which simultaneously "sees" many of the homological properties of both the arrangement and its dual.

## summer term 2022

**Prof. Dr. Götz Pfeiffer** (Galway)

Seminar: Arrangements and Symmetries

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

**Prof. Dr. Gerhard Röhrle** (RUB)

Seminar: Arrangements and Symmetries

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

**Eirini Chavli** (Stuttgart)

Seminar: Arrangements and Symmetries

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

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym22

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

Seminar: Lie-Theorie

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

ruhr-uni-bochum.zoom.us/j/6340579550

Meeting ID: 634 057 9550

Passcode: alt

**Giovanni Paolini **(Caltech)

Seminar: Arrangements and Symmetries

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

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym22

**Dr. Paul Mücksch** (MPI Bonn)

Seminar: Arrangements and Symmetries

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

**Timm Peerenboom **(Bonn)

Seminar: Lie-Theorie

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

**Shuhei Tsujie** (Hokkaido University of Education)

Seminar: Arrangements and Symmetries

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

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym22

## winter term 2021/2022

**Norihiro Nakashima** (Nagoya Institute of Technology, Japan)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 24.01.2022, **13:15 Uhr **, via zoom**Titel**: "Dimensions for extended Shi and Catalan arrangements to be hereditarily free"**Abstract**:

A central arrangement is said to be hereditarily free if all restriction arrangements are free. Several investigations are interested in hereditarily free arrangements. Recently, Hoge and Röhrle proved that the finite complex reflection arrangements are hereditarily free. In this talk, we show that the cone of the extended Catalan arrangement of type A is always hereditarily free, while we determine the dimension that the cone of the extended Shi arrangement of type A is hereditarily free. For this purpose, using digraphs, we define a class of arrangements which contains the extended Shi and Catalan arrangements, and we characterize the freeness for the cone of this arrangement by graphical conditions. We also define contraction to prove that the class of arrangements are closed under restriction. The contraction is different from ordinary vertex contraction on digraphs. This is a joint work with Shuhei Tsujie.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym21

**Prof. Vyjayanthi Chari **(University of California, Riverside)

Seminar: Lie-Theorie

Vortrag am Montag, 13.12.2021, 16:15 Uhr, Raum IA 1/109 and via zoom**Titel**: " Quantum affine algebras and Macdonald polynomials "**Abstract**:

We explain a connection between finite dimensional representations of quantum affine algebras and indecomposable modules for the Borel subalgbera of an affine Lie algebra. We shall see that the characters of these modules are given by specialized Macdonald polynomials. Other connections with Demazure modules will also be discussed.**Zugangsdaten/Zoom Meeting**

ruhr-uni-bochum.zoom.us/j/64927423608

Meeting ID: 649 2742 3608

Passcode: math

**Prof. Dr. Gerhard Röhrle** (RUB)

Seminar: Lie-Theorie

Vortrag am Montag, 22.11.2021, 16:15 Uhr, Raum IA 1/109**Titel**: " OVERGROUPS OF REGULAR UNIPOTENT ELEMENTS IN REDUCTIVE GROUPS "**Abstract**:

There is a long and remarkable history of the study of the subgroup structure of reductive algebraic groups. This in particular involves overgroups of special elements. I shall report on recent joint work with Michael Bate and Ben Martin where we study reductive subgroups H of a reductive linear algebraic group G such that H contains a regular unipotent element of G. We show that under suitable hypotheses, such subgroups are G-irreducible in the sense of Serre; this means such H are not contained in a proper parabolic subgroup of G. This work generalizes previous results of Malle, Testerman and Zalesski. Time permitting I shall indicate analogous results for Lie algebras and for finite groups of Lie type.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/64927423608

Meeting ID: 649 2742 3608

Passcode: math

**Galen Dorpalen-Barry** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 15.11.2021, 14:15 Uhr, Raum IC 03/449**Titel**: " A Short Introduction to Cones of Hyperplane Arrangements (Part II) "**Abstract**:

In this two-part series we introduce some material relating to cones of hyperplane arrangements. This is the second talk of this series. In the first half of this talk, we will introduce some useful tools from commutative algebra (initial forms, filtered rings, Gröbner bases, etc). In the second half, we will use everything we’ve learned so far to introduce the Varchenko-Gel’fand ring and use it to study hyperplane arrangements and their cones.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym21

**Galen Dorpalen-Barry** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 8.11.2021, 14:15 Uhr, Raum IC 03/449**Titel**: " A Short Introduction to Cones of Hyperplane Arrangements (Part I) "**Abstract**:

In this two-part series we introduce some material relating to cones of hyperplane arrangements. This is the first talk of this series. In this talk we will introduce (cones of) hyperplane arrangements and use them to motivate the study of oriented matroids. Along the way, we will point out some recent results related to cones of hyperplane arrangements and pick up tools for proving the theorems we will encounter during the second talk of this series.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym21

**Sarah Rees** (University of Newcastle, UK)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 18.10.2021, 14:15 Uhr, via zoom**Titel**: " Rewriting in Artin groups and their relations "**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/62326053276

Meeting ID: 623 2605 3276

Passcode: arrsym21

## summer term 2021

**Dr. Tan Nhat Tran **(RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 19.07.2021, 16:15 Uhr, via zoom**Titel**: "Arrangements arising from digraphs and freeness of arrangements between Shi and Ish"**Abstract**:

To a given vertex-weighted digraph (directed graph) we associate an arrangement analogous to the notion of Stanley's $\psi$-graphical arrangements and study it from perspectives of combinatorics and freeness. Our arrangement unifies several arrangements in literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira.

It was shown that the arrangements between Shi and Ish all share the same characteristic polynomial with all nonnegative integer roots, thus raising the natural question of their freeness. We introduce two operations on the vertex-weighted digraphs and prove that subject to certain conditions on the weight $\psi$, the operations preserve the characteristic polynomials and freeness of the associated arrangements. In particular, by applying a sequence of these operations to the Shi arrangement, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone. Notably, all of the arrangements between Shi and Ish appear as the members in the operation sequence, thus giving a new insight into how they naturally arise and interpolate between Shi and Ish.

This is joint work with T. Abe (Kyushu) and S. Tsujie (Hokkaido)**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

**Sven Wiesner** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 12.07.2021, 16:15 Uhr, via zoom**Titel**: "On inductively free and additionally free arrangements"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

**Prof. Takuro Abe** (Kyushu University)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 12.07.2021, 12:00 Uhr, via zoom**Titel**: "Logarithmic vector fields and differential forms revisited"**Abstract**:

Logarithmic vector fields and logarithmic differential forms are known to be dual to each other, so their behaviors are similar. For example, it is free if the other is free. However, though they are similar, they are very different too. For example, if we delete one hyperplane from a free arrangement, then the projective dimension of the logarithmic vector field is at most one, but that of logarithmic differential forms can be larger as we want. We give a way to understand these differences in a uniform way, and give several applications of this viewpoint by solving several problems. This is a joint work with Graham Denham.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 14.06.2021, 16:15 Uhr, via zoom**Titel**: "Towards the Linearity of Complex Braid Groups"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

**Dr. Alexander Trost **(RUB)

Seminar: Arrangements and Symmetries

Vortrag (2/2) am Montag, 07.06.2021, 14:15 Uhr, via zoom**Titel**: "STRONG BOUNDEDNESS OF S-ARITHMETIC, SPLIT CHEVALLEY GROUPS - SANDWICH THEOREMS, COMPACTNESS AND BAD PRIMES"**Abstract**:

Infoblatt**Zugangsdaten**

Zoom link: ruhr-uni-bochum.zoom.us/j/61415157745

Password: math

**Dr. Alexander Trost** (RUB)

Seminar: Lie-Theorie

Vortrag (1/2) am Montag, 31.05.2021, 14:15 Uhr, via zoom**Titel**: "STRONG BOUNDEDNESS OF S-ARITHMETIC, SPLIT CHEVALLEY GROUPS - SANDWICH THEOREMS, COMPACTNESS AND BAD PRIMES"**Abstract**:

Infoblatt**Zugangsdaten**

Zoom link: ruhr-uni-bochum.zoom.us/j/61415157745

Password: math

**Lucien Hennecart** (Université Paris-Saclay)

Seminar: Lie-Theorie

Vortrag am Montag, 17.05.2021, 14:15 Uhr, via zoom**Titel**: "Cuspidal functions and Lusztig sheaves for affine quivers"**Abstract**:

In this talk, we will be interested in the Hall algebra and Lusztig sheaves of affine quivers. Such quivers have a well-understood representation theory which allows to describe explicitly their stack of representations. I will explain how to use this geometry to answer the questions of the description of cuspidal functions and of the microlocal characterization of Lusztig sheaves, a category of perverse sheaves defined by Lusztig to obtain the canonical basis of quantum groups. These questions can be generalized to arbitrary quivers, for which we can formulate conjectures.**Zugangsdaten**

Zoom link: ruhr-uni-bochum.zoom.us/j/61415157745

Password: math

**Dr. Tan Nhat Tran** (RUB)

Seminar: Arrangements and Symmetries

Vortrag (3/3) am Montag, 17.05.2021, 16:15 Uhr, via zoom**Titel**: "CHARACTERISTIC QUASI-POLYNOMIALS OF INTEGRAL HYPERPLANE ARRANGEMENTS"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

**Dr. Tan Nhat Tran** (RUB)

Seminar: Arrangements and Symmetries

Vortrag (2/3) am Montag, 10.05.2021, 16:15 Uhr, via zoom**Titel**: "CHARACTERISTIC QUASI-POLYNOMIALS OF INTEGRAL HYPERPLANE ARRANGEMENTS"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

**Dr. Tan Nhat Tran** (RUB)

Seminar: Arrangements and Symmetries

Vortrag (1/3) am Montag, 03.05.2021, 16:15 Uhr, via zoom**Titel**: "CHARACTERISTIC QUASI-POLYNOMIALS OF INTEGRAL HYPERPLANE ARRANGEMENTS"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym21

## winter term 2020/2021

**Prof. Dr Volkmar Welker** (Marburg)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 08.02.2021, 14:15 Uhr, via zoom**Titel**: "Relative Arrangements"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Professor Aner Shalev** (Hebrew University of Jerusalem)

Seminar: Lie-Theorie

Vortrag am Montag, 01.02.2021, 14:15 Uhr, via zoom**Titel**: "Random Generation: from Groups to Algebras"**Abstract**:

There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, with emphasis on finite simple groups. In this talk, based on a recent joint work with Damian Sercombe, we study similar notions for finite and profinite associative algebras. Let $A$ be a finite associative, unital algebra over a (finite) field $k$. Let $P(A)$ be the probability that two random elements of $A$ will generate $A$ as a unital $k$-algebra. It is known that, if $A$ is simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result for larger classes of finite associative algebras. For $A$ simple, we estimate the growth rate of $P(A)$ and find the best possible lower bound for it. We also study the random generation of $A$ by two special elements. Finally, we let $A$ be a profinite algebra over $k$. We show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth. Related quantitative results are also obtained.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/99572000333

Meeting ID: 995 7200 0333

Password: math

**Prof. Masahiko Yoshinaga** (Hokkaido University)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 18.01.2021, 12:15 Uhr, via zoom**Titel**: "A geometric realization of combinatorial reciprocity of order polynomials"**Abstract**:

The Euler characteristic of topological space can be considered as a generalization of the cardinality of a finite set. In previous work with Hasebe and Miyatani (2017), we generalized Stanley's combinatorial reciprocity for order polynomials to an equality of Euler characteristics of certain spaces of homomorphisms of posets. In this talk, we discuss recent development of geometric realization of the combinatorial reciprocity. The main result asserts that certain spaces of poset homomorphisms are actually homeomorphic which clearly implies the Euler characteristics. The proof is based on the detailed analysis of upper semicontinuous functions on metrizable topological spaces. This is joint work with Taiga Yoshida.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Dr. Alexander Sistko** (Manhattan College, New York)

Seminar: Lie-Theorie

Vortrag am Montag, 18.01.2021, 14:15 Uhr, via zoom**Titel**: "On quiver representations over the field with one element"**Abstract**:

To any quiver, we can associate its category of finite-dimensional (nilpotent) representations over the field with one element. This category shares many basic properties with its analog over a field: in particular, a version of the Krull-Schmidt Theorem is satisfied. Inspired by the classical Tame-Wild Dichotomy for finite-dimensional algebras, we discuss a stratification of quivers based on the growth of their indecomposable F1-representations. In particular, we classify all quivers of bounded representation type over F1 and provide a functorial interpretation for unbounded quivers. As a consequence, we develop a general framework for interpreting F1-representations as certain quiver maps, which allows for a more combinatorial description of the Ringel-Hall algebras associated to these categories.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/99572000333

Meeting ID: 995 7200 0333

Password: math

**Dr. Paul Mücksch** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 18.01.2021, 16:15 Uhr, via zoom**Titel**: "On Yuzvinsky's lattice sheaf cohomology for hyperplane arrangements "**Abstract**:

In my talk, I will establish the exact relationship between the cohomology of a certain sheaf on the intersection lattice of a hyperplane arrangement introduced by Yuzvinsky and the cohomology of the coherent sheaf on punctured affine space respectively projective space associated to the derivation module of the arrangement. I will derive a Künneth formula connecting the cohomology theories, answering a question posed by Yoshinaga. This, in turn, gives a new proof of Yuzvinsky’s freeness criterion and yields a stronger form of the latter.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Dr. Alex Malcolm** (University of Bristol, UK)

Seminar: Lie-Theorie

Vortrag am Montag, 11.01.2021, 14:15 Uhr, via zoom**Titel**: "Finite simple groups, prime order elements and width"**Abstract**:

The generation of finite simple groups has been a thriving area of research for many years. Since it was established that each is generated by a pair of elements, many interesting refinements have followed: for instance, determining the existence of generating pairs of prescribed orders.

More recently the notion of width has provided an additional perspective on generation, measuring how efficiently a chosen subset generates a group. For example we may ask, can every element be written as a product of at most 2, or perhaps 3, elements from a fixed conjugacy class? Answering such questions relies on a range of tools involving subgroup structure and character theory.

In this talk we will examine the width of finite simple groups with respect to elements of a fixed prime order. We will report on sharp bounds for particular families, and answer questions concerning Lie-type groups of unbounded rank.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/99572000333

Meeting ID: 995 7200 0333

Passwort: math

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 11.01.2021, 16:15 Uhr, via zoom**Titel**: "Lectures on Garside Theory"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 21.12.2020, 16:15 Uhr, via zoom**Titel**: "Lectures on Garside Theory"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Prof. Alexander Premet** (University of Manchester)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 14.12.2020, 14:15 Uhr via zoom**Titel**: "Modular representations of Lie algebras and Humphreys' conjecture"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/99572000333

Meeting ID: 995 7200 0333

Password: math

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 14.12.2020, 16:15 Uhr, via zoom**Titel**: "Lectures on Garside Theory"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Assistant Professor Jethro van Ekeren** (Universidade Federal Fluminense (UFF)

Seminar: Lie-Theorie

Vortrag am Montag, 07.12.2020, 14:00 Uhr via zoom**Titel**: "Singular support of the Ising model and a new modular Nahm sum"**Abstract**: (joint work with G. E. Andrews and R. Heluani) As part of an ongoing project to understand chiral homology of elliptic curves with coefficients in a vertex algebra V, we are led to study the associated graded algebra of V with respect to its Li filtration. The spectrum of this algebra is known as the singular support of V. For boundary Virasoro minimal models, i.e., those of type (2, p), p odd, the singular support is known to be isomorphic to an arc space. For the Ising model this is already not the case, and we show that its singular support is instead a ''differential hypersurface'' in an arc space, that is, it is defined by the vanishing of a single differential polynomial and all its derivatives. We obtain this result as a corollary of a new q-series identity of Rogers-Ramanujan type, which at the same time yields a new example of a modular Nahm sum.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/99572000333

Meeting ID: 995 7200 0333

Password: math

**René Marczinzik**, Universität Stuttgart

Seminar: Arrangements and Symmetries

Montag, 7. Dezember 2020, 16:15-17:45**Title**: Distributive lattices and Auslander regular algebras**Abstract**: We show that the incidence algebra of a finite lattice L is Auslander regular if and only if L is distributive. As an application we show that the order dimension of L coincides with the global dimension of its incidence algebra when L has at least two elements and we give a categorification of the rowmotion bijection for distributive lattices. At the end we discuss the Auslander regular property for other objects coming from combinatorics. This is joint work with Osamu Iyama.

**Lukas Kühne** (Max-Planck-Institut Leipzig)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 30. November 2020, 16:15-17:30, via Zoom**Titel**: "The Resonance Arrangement"**Abstract**: The resonance arrangement is the arrangement of hyperplanes which has all nonzero 0/1-vectors in R^n as normal vectors. It is also called the all-subsets arrangement. Its chambers appear in algebraic geometry, in mathematical physics and as maximal unbalanced families in economics.

In this talk, I will present a universality result of the resonance arrangement. Subsequently, I will report on partial progress on counting its chambers. Along the way, I will review some of the combinatorics of general hyperplane arrangements. If time permits I will also touch upon the related threshold arrangement which encodes Boolean functions that are linearly separable.**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Theo Douvropoulos, **University of Massachusetts Amherst

Seminar: Arrangements and Symmetries

Montag, 23. November 2020, 16:00-17:30, per Zoom**Title: **Recursions and proofs in Coxeter-Catalan combinatorics**Abstract**: The noncrossing partition lattice NC(W) associated to a finite Coxeter group W has become a central object in Coxeter-Catalan combinatorics during the last 25 years. We focus on two recursions on the simple generators of W; the first due to Deligne (and rediscovered by Reading) determines the chain number of NC(W) and the second, more general, due to Fomin-Reading recovers the whole zeta polynomial. The resulting formulas have nice product structures and are key players in the field, but are still not well understood; in particular, they are derived by the (case-free) recursions separately for each type.

A uniform derivation of the formulas from these recursions requires proving certain identities between the Coxeter numbers and invariant degrees of a group and those of its parabolic subgroups. In joint work with Guillaume Chapuy, we use the W-Laplacian (for W of rank n, this is an associated nxn matrix that we introduced in earlier work and which generalizes the usual graph Laplacian) to prove the required identities for the chain number of W. We give a second proof by using the theory of multi-reflection arrangements and the local-to-global identities for their characteristic polynomials. This latter approach is in fact applicable to the study of the whole zeta polynomial of NC(W) although it, currently, falls short of giving a uniform derivation of Chapoton's formula for it.

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 16.11.2020, 16:15 Uhr, in HIA

zusätzlich online via zoom**Titel**: "Lectures on Garside Theory"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 09.11.2020, 16:15 Uhr, in HIA

zusätzlich online via zoom**Titel**: "Lectures on Garside Theory"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Dr. Damian Sercombe** (RUB)

Seminar: Lie-Theorie

Vortrag am Montag, 02.11.2020, 14:15 Uhr, in HZO 70

zusätzlich online via zoom**Titel**: "Maximal connected subgroups of maximal rank in reductive k-groups"**Abstract**: Let k be any field. Let G be a connected reductive algebraic k-group. Associated to G is an invariant that is called the index of G. Tits showed that, up to k-anisotropy, the k-isogeny class of G is uniquely determined by its index. Moreover, for the cases where G is absolutely simple, Tits classified all possibilities for the index of G. Let H be a connected reductive k-subgroup of maximal rank in G. We introduce an invariant of the pair H**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/94548565437

Meeting-ID: 945 4856 5437

Passwort: 415419

**Dr. Georges Neaime** (RUB)

Seminar: Arrangements and Symmetries

Vortrag am Montag, 02.11.2020, 16:15 Uhr, in HIB

zusätzlich online via zoom**Titel**: "Lectures on Garside Theory"**Abstract**:

Infoblatt**Zugangsdaten**

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

## summer term 2020

**Prof. Dr. Michael Cuntz** (Leibniz Universität Hannover)

Seminar. Arrangements and Symmetries

Online-Vortrag (Zoom) am Montag, 13.7.2020, 14:15 Uhr**Titel**: "A greedy algorithm to compute arrangements of lines"**Abstract**:

We present a greedy algorithm optimizing arrangements of lines with respect to a property and apply this algorithm to the case of simpliciality. An implementation produces a database with many surprising examples.

Zugangsdaten

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Dr. Georges Neaime** (Universität Bielefeld)

Seminar. Arrangements and Symmetries

Online-Vortrag (Zoom) am Montag, 18.5.2020, 14:15 Uhr**Titel**: "Garside theory and the $K(\pi,1)$ conjecture"**Abstract**:

Garside theory was developed in order to better understand Artin groups and their generalizations. Based on the work of Bessis for complex braid groups and of Paris on Artin groups, as well as recent inventions by McCammond--Sulway and Paolini--Salvetti for affine Artin groups, we provide additional evidence of the link between Garside theory and the topology of complements of hyperplane arrangements. Actually, the theory provides a proof in full generality of the K(\pi,1) conjecture for complex braid groups, and for spherical and affine Artin groups.

Zugangsdaten

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

**Herr Florian Kranhold **(Mathematisches Institut der Universität Bonn)

Seminar. Arrangements and Symmetries

Online-Vortrag (Zoom) am Montag, 11.5.2020, 14:15 Uhr**Titel**: "Gekoppelte Konfigurationen"**Abstract**:

Die Räume geordneter Konfigurationen in der komplexen Ebene haben sehr bekannte Eigenschaften und Anwendungen: Sie sind Komplemente eines komplexen Hyperebenen-Arrangements, klassifizieren die reinen Zopfgruppen, haben eine einfache Zellzerlegung und sind homotopieäquivalent zu den Komponenten der kleinen 2-Kuben-Operade, deren Homologie Poisson-Algebren klassifiziert. Diese Operade wirkt auf vielen interessanten Räumen, zum Beispiel auf den Modulräumen Riemannscher Flächen mit genau einer Randkurve. Möchte man Modulräume von Flächen mit mehreren Randkurven betrachten, ist eine Färbung der 2-Kuben-Operade naheliegend. Um diese Wirkung nun in verschiedenen simplizialen Modellen betrachten zu können, muss eine spezielle Unteroperade betrachtet werden. Deren Komponenten sind Konfigurationsräume mit einer speziellen Kopplungsbedingung: Einige Punkte haben stets den gleichen Realteil. Wir haben einige Eigenschaften dieser Räume verstanden: Auch sie lassen sich als Komplement eines Hyperebenen-Arrangements schreiben, haben eine vergleichsweise einfache Zellzerlegung und ihre Homologie kann mithilfe diskreter Morsetheorie berechnet werden. Ein großer Unterschied ist das Fehlen von Fadell-Neuwirth-Faserungen, weswegen die Asphärizität dieser Räume nach wie vor eine offene Frage ist.

Zugangsdaten

ruhr-uni-bochum.zoom.us/j/92406796238

Meeting-ID: 924 0679 6238

Passwort: ArrSym20

## winter term 2019/2020

**Prof. Dr. Eric Opdam** (Universiteit van Amsterdam, NL)

Kolloquium Algebra, Geometrie und Kombinatorik

Mittwoch, den 29.01.20 um 16 st in IA 01/473

**Titel:** "Square integrable hypergeometric functions for root systems"

**Abstract: **Classifying the square integrable solutions of the system of hypergeometric equations for root systems is relevant to understanding the discrete series for real symmetric spaces. We will discuss this connection and explain some aspects of this classification.

**Prof. Dr. Sergey Mozgovoy** (Trinity College Dublin)

Seminar: Lie-Theorie

Montag, den 13.01.20 um 14 ct in IA 1/135**Titel**: "Commuting matrices and Higman's conjecture"**Abstract**: Higman's conjecture states that the number of conjugacy classes in the group of upper triangular matrices over F_q is polynomial in q. It can be also formulated as a problem of counting commuting upper triangular matrices over a finite field. I will introduce a generalisation of this problem in terms of quiver representations and prove relations between various counting invariants that arise. In particular, I will show that the original conjecture is equivalent to polynomial-count of certain absolutely indecomposable quiver representations.

**Dr. Gleb A. Koshevoy** (CEMI, Russian Academy of Sciences, Moskau)

Seminar: Lie-Theorie

Montag, den 02.12.19 um 14 ct in IA 1/135**Titel**: "Cubillages of ciclic zonotopes and higher Auslander-Reiten theory"**Abstract**: Cubillages of cyclic zonotopes studied by Kapranov and Voevodskii in relations to higher Bruhat orders, Zamolodchikov equations, and polycategories. Combinatorics of two-dimensional cubillages related to quasi-commuting collections of qunatum determinants due to Leclerc and Zelevinsky, cluster algebras, and Auslander-Reiten theory . For an odd integer $ r> 0$ and an integer $n > r$, we introduce a notion of weakly $r$-separated subsets of $[n] = \{1, 2, \ldots n\}$. When $r =1$, this corresponds to the concept of weak separation introduced and studied by Leclerc and Zelevinsky. We extend results due to Leclerc-Zelevinsky, and develop a geometric approach to combinatorics maximal weakly $r$-separated collections. From this we get a combinatorical view point to the higher Auslander-Reiten theory due to Iyama and higher cluster categoris due to Oppermann and Thomas. This is a joit work with V.Danilov and A.Karzanov

**Dr. Jens Eberhardt **(MPI, Bonn)

Seminar: Lie-Theorie

Montag, den 25.11.19 um 14 ct in IA 1/135**Titel**: "Motives in Geometric Representation Theory"**Abstract**:

Categories of representations arising in Lie theory can often be modeled geometrically in terms of constructible sheaves on certain spaces, as for example on the flag variety, affine Grassmannian or the nilpotent cone.

Recent developments in the theory of motives allow to consider so called "motivic sheaves", an algebro-geometric analogue of constructible sheaves. In this talk we will explain how one can practically work with motivic sheaves (using Grothendieck's six functor formalism) and apply them in representation theory. We will show how motivic sheaves can be used to model Category O associated to a reductive complex Lie algebra, modular Category O associated to a split reductive group over a finite field and categories of representations of convolution algebras, such as the graded affine Hecke algebra and KLR-algebras. We also will explain how more "exotic" versions of motivic sheaves provide exciting new opportunities in geometric representation theory.

**Dr. Jenny August** (MPI, Bonn)

Seminar: Lie-Theorie & Arrangements and Symmetries

Montag, den 18.11.19 um 14 ct in IA 1/135**Titel**: "Contraction Algebras, Hyperplane Arrangements and K(pi,1)"**Abstract**:

Contraction algebras are a class of finite dimensional algebras used to study minimal models in geometry. While they are very useful in this area, this talk will instead focus on their connection to simplicial hyperplane arrangements. I will explain how each contraction algebra has an associated hyperplane arrangement, which in special cases is an ADE root system, and further, I will describe how this arrangement controls all the homological information of the algebra. In particular, we show the space of stability conditions of the algebra is the universal cover of the complexified complement and thus, as this space is known to be contractible, we obtain a new homological proof of the K(pi,1) theorem for finite type ADE braid groups.

**Professor Wim H. Hesselink** (Bernoulli Institute, University of Groningen, NL)

Seminar: Lie-Theorie

Montag, den 04.11.19 um 14 ct in IA 1/135**Titel**: "Nilpotent conjugacy classes for G2 and classical groups"**Abstract**:

The Mumford-Kempf instability theory is sketched. It is applied to the Lie algebra of the group G2. The nullcone of G2 is shown to have five strata. If char(K) differs from 3, each of the strata is a single orbit. If char(K) = 3, one stratum splits into two orbits. The nullcone has singularities in the points of the nonregular orbits. Cross sections are used to prove this and to analyse the singularities. Are the singularities different in different orbits? The answer is yes, except for characteristic 2. A measure of singularity is introduced to prove this.

If time permits: Classical nilpotency in characteristic 2, revisiting a paper of 40 years ago. The paper covers 8 cases: the orthogonal case and the symplectic case, the group and the Lie algebra, chararacteristic 2 and different from 2. Conjugacy is translated into isomorphy between modules with forms over the ring of formal power series. A new way is presented to determine and distinguish the indecomposable form modules. An unconvincing proof about compositions of indecomposables must be repaired.

**Dr. Travis Schedler** (Imperial College, London)

Seminar: Lie-Theorie

Montag, den 07.10.19 um 14 ct in IA 1/135**Titel**: "Symplectic resolutions of Hamiltonian reductions "**Abstract**:

Given a symplectic representation of a reductive group, one considers the Hamiltonian reduction, in physics called “Higgs branch” varieties. This includes quiver and toric hyperkähler varieties. I will discuss the question of existence of symplectic resolutions of these, and how one might go about constructing them via geometric invariant theory. This is joint work with Gwyn Bellamy, and heavily uses work of Herbig, Schwarz, and Seaton.

## summer term 2019

**Professor Dr. Götz Pfeiffer**, National University of Ireland, Galway, EI

Seminar: Lie-Theorie

Montag, den 08.07.19 um 16 ct in IA 1/181**Titel**: "Bisets and the Double Burnside Algebra of a Finite Group "**Abstract**:

The double Burnside group $B(G, H)$ of two finite groups $G, H$ is the Grothendieck group of the category of finite $(G, H)$-bisets. Certain bisets encode relationships between the representation theories of $G$ and $H$. Bouc's biset category provides a framework for studying such relationships, it has finite groups as objects, and $B(G, H)$ as morphisms between $G$ and $H$, with composition induced by the tensor product of bisets. The endomorphism ring $B(G, G)$ is called the double Burnside ring of $G$. In contrast to the (ordinary) Burnside ring $B(G)$, the double Burnside ring $B(G, G)$ of a nontrivial group $G$ is not commutative. In general, little more is known about the structure of $B(G, G)$. In the talk I'll describe a relatively small faithful matrix representation of the rational double Burnside algebra $\mathbb{Q}B(G,G)$ for certain finite groups $G, based on a recent decomposition of the table of marks of the direct product $G \times G$, exhibiting the cellular structure of the algebra $\mathbb{Q}B(G, G)$. This is joint work with Sejong Park.

**Dr. Mohamed Barakat** (Universität Siegen)

Seminar: Lie-Theorie

Montag, den 08.07.19 um 14 ct in IA 1/53**Titel**: "Chevalley’s Theorem on constructible images made constructive "**Abstract**:

Chevalley proved that the image of an algebraic morphism between algebraic varieties is a constructible set. Examples are orbits of algebraic group actions. A constructible set in a topological space is a finite union of locally closed sets and a locally closed set is the difference of two closed subsets. Simple examples show that even if the source and target of the morphism are affine varieties the image may neither be affine nor quasi-affine. In this talk I will present an Gröbner-basis-based algorithm which computes the constructible image of a morphism of affine spaces, along with applications to Terao’s freeness conjecture.

**Dr. Paul Mücksch** (RUB)

Seminar: Arrangements and Symmetries

Montag, den 29.04.2019, 14:15 in IA 1/53**Titel**: "MAT-freie Spiegelungsarrangements"**Zusammenfassung**:

Die algebraische Eigenschaft der Freiheit eines Hyperebenenarrangements mit seiner Kombinatorik zu verbinden ist ein wichtiges Problem in der Theorie der Hyperebenenarrangements. Hinreichende kombinatorische Bedingungen liefert Terao's Addition-Deletion Theorem. Dies motiviert die Klasse der induktiv freien Arrangements.

Motiviert durch das Multiple Addition Theorem (kurz MAT) von Abe, Barakat, Cuntz, Hoge und Terao werde ich die neue Klasse der MAT-freien Arrangements einführen. Erst kürzlich konnten Abe und Terao eine Verallgemeinerung des MAT, das Multiple Addition Theorem 2 (MAT2) zeigen. Mit Hilfe des MAT2 lässt sich wiederum die Klasse der MAT2-freien Arrangements definieren.

In meinem Vortrag werde ich eine Klassifikation aller Spiegelungsarrangements, die diesen neuen Freiheitsbegriffen genügen, vorstellen.

Außerdem möchte ich Beziehungen zu bekannten Freiheitsklassen kommentieren und damit verbundene Probleme vorstellen. Dies ist eine gemeinsame Arbeit mit Michael Cuntz (Hannover).

## winter term 2018/2019

**Prof. Dr. Cheryl Praeger** (Perth, Australia)

Seminar: Lie-Theorie

Dienstag, den 18.12.18 um 14 ct in IA 1/135**Titel**: "Finding involution centralisers efficiently in classical groups of odd characteristic"**Abstract**:

Bray's involution centraliser algorithm plays a key role in recognition algorithms for classical groups over finite fields of odd order. It has always performed faster than the time guaranteed/justified by complexity analyses. Work of Dixon, Seress and I published this year gives a satisfactory analysis for SL(n,q). And we are slowly making progress with the other classical groups. The "we" are Colva Roney-Dougal, Stephen Glasby and me - and we have conquered the unitary groups so far.

**Prof. Dr. Michael Cuntz** (Hannover)

Seminar: Arrangements and Symmetries

Dienstag, den 11.12.18 um 14 ct in IB 1/103**Titel**: "Klassifikation der Weyl-Gruppoide"**Abstract**:

Die Klassifikation der endlichen Weyl-Gruppoide (das sind gewisse simpliziale Arrangements in einem Gitter) beruht auf Rechnungen mit dem Computer. In diesem Vortrag möchte ich über Fortschritte berichten, die zu einem kürzeren Beweis führen. Die neuen Techniken können ferner zur Klassifikation größerer Klassen von Arrangements verwendet werden.

**Professor Dmitriy Rumynin**, University of Warwick, UK

Seminar: Lie-Theorie

Dienstag, den 13.11.18, von 14 - 16 Uhr, IA 01/131**Titel**: "Kac-Moody Groups: representations, localisation, duality"**Abstract**:

We will look at representation theory of a complete Kac-Moody group G over a finite field. G is a locally compact totally disconnected group, similar, yet slightly different to the group of points of a reductive group scheme over a local field. After defining the group we discuss localisation of its category of smooth representations. We also discuss homological duality for this category.

**Prof. Dr. J. M. Douglass** (NSF, Washington, DC, USA)

Seminar: Lie-Theorie

Dienstag, den 09.10.18, von 14 - 16 Uhr in der Wasserstraße 221, Raum 4/20**Titel**: " A factorization of the T-equivariant K-theory of flag varieties"**Abstract**:

Let G be a reductive, complex, algebraic group, B a Borel subgroup, T is a maximal torus in B, and P is a parabolic subgroup containing B. Then G/B is the "flag variety" of G and the projection from G/B to G/P is a G-equivariant fibre bundle with fibre P/B. As smooth varieties, G/B is locally isomorphic to the product G/P x P/B. The quotient P/B may be canonically identified with the flag variety of the Levi subgroup of P containing T and the "factorization" G/B = G/P x P/B may be viewed as a geometric incarnation of the factorization W = W^P x W_P, where W is the Weyl group of (G,T), W_P is the Weyl group of (P,T), and W^P is a set of left coset representatives of W_P in W. In this talk I will describe a factorization of the T-equivariant K-theory of G/B as a tensor product of the T-equivariant K-theory of G/P and the T-equivariant K-theory of P/B. The factorization theorem can be described in terms that make sense for any generalized cohomology/homology theory and the factorization in equivariant K-theory leads immediately to a uniform, geometric construction of corresponding factorizations in K-theory, equivariant cohomology, and ordinary cohomology.

## summer term 2018

**Dr. Paul Mücksch** (RUB)

Seminar. Arrangements and Symmetries

Donnerstag, den 26.7.18 von 14 - 16 Uhr in der Wasserstraße 221 Raum 4/20**Titel**: "New characterizations of freeness of hyperplane arrangements"**Abstract**:

New characterizations of freeness of hyperplane arrangements This talk is a report on recent work by Anna Maria Bigatti, Elisa Palezzato, and Michele Torielli. In their article (arXiv:1801.09868) the authors investigate two commutative al- gebraic invariants of a hyperplane arrangement. They are the generic initial ideal and the sectional matrix of the Jacobian ideal of the arrangement. Starting from a classic characterization of freeness by Terao they derive charac- terizations in terms of the generic initial ideal and the sectional matrix. Further- more, under the assumption that the arrangement in question is free, the generic initial ideal is completely determined by the exponents of the arrangement and vice versa. Nonetheless, thinking of Teraos conjecture, there are non-free lattice equivalent arrangements having different generic initial ideals.

**Prof Alexander Varchenko **(University of North Carolina at Chapel Hill, zzt. MPI Bonn)

Seminar: Arrangements and Symmetries

Montag, den 18.06.2018, 16:15 in NA 2/64**Title**: Critical points of master functions and integrable hierarchies**Abstract**: Critical points of master functions are non-isolated and come in "populations". I will discuss how the populations are related to integrable hierarchies and to representations of the affine Lie algebras.

**Christof Geiss** (zzt. Universität Bonn)

Seminar: Lie-Theorie

Montag, den 18.06.2018, 14:15 in NA 2/64**Title**: "Crystal graphs and semicanonical functions for symmetrizable Cartan matrices"**Abstract**: In joint work with B. Leclerc and J. Schröer we propose a 1-Gorenstein algebra H, defined over an arbitrary field K, associated to the datum of a symmetrizable Cartan Matrix C, a symmetrizer D of C and an orientation $\Omega$. The H-modules of finite projective dimension behave in many aspects like the modules over a hereditary algebra, and we can associate to H a generalized preprojective algebra $\Pi$. If we look, for K algebraically closed, at the varieties of representations of $\Pi$ which admit a filtration by generalized simples, we find that the components of maximal dimension provide a realization of the crystal $B_C(-\infty)$. For K being the complex numbers we can construct, following ideas of Lusztig, an algebra of constructible functions which contains a family of "semicanonical functions", which are naturally parametrized by the above mentioned components of maximal dimensions. Modulo a conjecture about the support of the functions in the "Serre ideal" those functions would yield a semicanonical basis of the enveloping algebra U(n) of the positive part of the Kac-Moody Lie algebra g(C).

**Daniel Kalmbach**, Universität zu Köln

Seminar: Lie-Theorie

Montag, den 23.04.2018, 14:15 in NA 2/64**Title**: A Linear formula for the Schützenberger involution**Abstract**: "The Schützenberger involution is a piecewise-linear function which was originally defined on Young tableaux. Its generalization to semi-standard Young-tableaux can be equivalently described by the action of the Bender-Knuth involutive operators translated into the language of Gelfand-Tsetlin patterns. A different approach is to define an automorphism on the generators of the quantum enveloping algebra U(g), which under a suitable parametrization of Lusztig’s basis in U(g) by Gelfand-Tsetlin patterns, acts as the Schützenberger involution. This was done by A. Berenstein and A. Zelevinsky. We show that by a good choice of parametrization of the canonical basis, we can give an explicit linear formula for the Schützenberger involution."

## winter term 2017/2018

**Prof. Daniel C. Cohen** (Louisiana State University, Baton Rouge, LA, USA)

Seminar: Arrangements and Symmetries

Mittwoch, den 24.1.18, 14:15 Uhr, Wasserstrasse 221, Raum 4/20**Title**: Topological complexity of surfaces and their configuration spaces**Abstract**:

Topological complexity is a numerical homotopy-type invariant introduced by M. Farber about 15 years ago, motivated by the motion planning problem from robotics. For a given space, this invariant provides a measure of the complexity of navigation in the space. Computing this invariant is sometimes easy, sometimes hard. I'll attempt to illustrate this, with surfaces and their configuration spaces.

**Prof. Daniel C. Cohen** (Louisiana State University, Baton Rouge, LA, USA)

Seminar: Arrangements and Symmetries

Montag, den 22.1.18, 14:15 Uhr, Wasserstrasse 221, Raum 4/20**Title**: Pure braid groups and direct products of free groups**Abstract**:

I'll discuss some properties and invariants of fundamental groups of complements of arrangements, largely in the context of the above classes. By the end of the talk, I should be able to pose a question we might discuss during the week.

**Balazs Elek, Cornell University**

Seminar: Lie-Theorie

Montag, den 08.01.2018, 14:15 in NA 2/24**Title**: Kirillov-Reshetikhin crystals and the Cactus group**Abstract**: tba

**Dr Michael Bate** (University of York)

Seminar: Lie-Theorie

Dienstag, den 05.12.2017, 15:45 in NA 1/58**Title**: Orbit closures and Invariants**Abstract**:

Following Ben's talk, I'll also talk about some work which aims to find the correct formulation in positive characteristic of classical results about algebraic groups and invariants in characteristic zero. I'll concentrate on a result of Luna from the 1970s which rests in part on his celebrated "Etale Slice Theorem". The Slice Theorem fails in positive characteristic, but we can still do something using the notion of G-complete reducibility. I'll focus on illustrative examples to motivate the results and to give an idea of the techniques used in the proof. This is joint work with Harry Geranios and Ben Martin.

**Prof Benjamin Martin** (University of Aberdeen)

Seminar: Lie-Theorie

Dienstag, den 05.12.2017, 14:15 in NA 1/58**Title**: Generic stabilisers of actions of reductive groups**Abstract**:

Actions of a topological or algebraic group G on a manifold or variety V play an important part in geometry. A fundamental problem is to understand the behaviour of the stabilisers G_v for v in V. Typically one finds that for generic v in V, the stabilisers are closely related - for instance, they are all conjugate or are all isomorphic to each other. If G is a linear algebraic group over a field k of characteristic p>0, however, then we can have more complicated behaviour. To understand what is going on, the notion of G-complete reducibility turns out to be very helpful. I will discuss work of Richardson in characteristic zero and some more recent work in positive characteristic.

**Dr. Gwyn Bellamy**, University Glasgow

Seminar: Lie-Theorie

Freitag, den 01.12.2017, 14:15 in NA 3/24**Titel**: Symplectic resolutions of quotient singularities**Abstract**:

In this talk I will describe progress on a program, joint with Schedler, to classify those symplectic quotient singularities that admit symplectic resolutions I will explain how one can use the representation theory of symplectic reflection algebras in order to do this. I will also explain how one can use these algebras, combined with general theory developed by Namikawa, to compute the nef and movable cones of the minimal models of these quotient singularities. As a consequence, one can explicitly count the number of minimal models. Finally, I will describe a number of interesting problems in the field that are still open.

**Prof. Dr. Markus Reineke**, RUB

Seminar: Lie-Theorie

Montag, den 13.11.2017, 14:15 in NA 2/24**Titel**: Trägergarben für lineare Entartungen von Fahnenmanigfaltigkeiten**Abstract**: In gemeinsamer Arbeit mit G. Cerulli Irelli, X. Fang, E. Feigin und G. Fourier wurde eine flache Familie so genannter linearer Entartungen von Fahnenmannigfaltigkeiten konstruiert. Im Vortrag wird das Verhalten der Kohomologie dieser Räume mittels des Konzepts der Trägergarben untersucht.

**Jun.-Prof. Dr. Deniz Kus**, RUB

Seminar: Lie-Theorie

Donnerstag, den 06.11.2017, 14:15 in NA 2/24**Titel**: "Lattice path enumeration in representation theory"**Abstract**: "Counting lattice paths is a classical topic in combinatorics which has applications in many fields of mathematics, as they encode various combinatorial objects and their properties. In this talk we will explain the connection to the representation theory of affine Lie algebras, ecpecially the relationship to maximal indecomposable highest weight modules. We introduce the notion of Demazure flags (a more general version of Jordan-Hölder series) and determine the graded multiplicities in these flags. It turns out that a suitable combinatorial model is given by certain lattice paths. "

**Prof. Dr. Michael Cuntz** (Hannover)

Seminar: Arrangements and Symmetries

Donnerstag, den 12.10.2017, 14:15 in NA 2/64**Titel**: "Frieze patterns over subsets of the complex numbers"**Zusammenfassung**: "Frieze patterns were introduced by Conway and Coxeter as certain arrays of positive integers with a condition on subdeterminants. They are closely related to cluster algebras, since every such pattern may be viewed as a specialization of cluster variables in type A, and they are in bijection with triangulations of a convex polygon by non-intersecting diagonals. Generalizing classical friezes leads to many interesting observations. In this talk, we consider frieze patterns with entries in an arbitrary ring. In this general setting, the combinatorics seem to get very complicated. However, for instance certain rings of integers produce new rules and transformations, as well as recursive constructions. This is a joint work with Thorsten Holm. "

## summer term 2017

**Dr. Adam Thomas** (Bristol)

Seminar: Lie-Theorie

Montag, den 25.09.2017, 14:15 in NA 2/64**Titel**: "Complete Reducibility: The Good, the Bad and the Ugly"

**Melvin Dauter**, RUB

Seminar: Lie-Theorie

Montag, den 24.07.2017, 14:15 in NA 2/64**Titel**: " Beispiele und Anwendungen reduktive Paare bei linearen algebraischen Gruppen "**Zusammenfassung**: "Reduktive Paare (G,H) linearer algebraischer Gruppen können verwendet werden, um gewisse Eigenschaften einer algebraischen Gruppe G auf die Untergruppe H zu übertragen. Wir werden sehen, wie Richardson den Begriff benutzt, um zu zeigen, dass eine halbeinfache Gruppe G in guter Charakteristik nur endlich viele unipotente Konjugationsklassen besitzt. Als weitere Anwendung werden wir G-vollständige Zerlegbarkeit untersuchen. Dabei werden wir auch kurz auf Fragen hinsichtlich der Existenz reduktiverPaare eingehen."

**Professor Dr. Gerhard Röhrle**, RUB

Seminar: Lie-Theorie

Montag, den 17.07.2017, 14:15 in NA 2/64**Title**: " Freeness of multi-reflection arrangements for complex reflection groups "**Abstract**: "In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free. In this overview on joint work with T. Hoge, T. Mano, and C. Stump, 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. In a second step we extend our results further to all imprimitive irreducible unitary reflection groups (the bulk of which are not well-generated!). 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. I shall try to explain our result in detail and outline how we generalized Yoshinaga's approach to Terao's result for Coxeter groups mentioned above making use of recent developments of flat systems of invariants due to Kato, Mano and Sekiguchi."

**Professor Graham Denham** (University of Western Ontario, Kanada)

Seminar: Lie-Theorie

Montag, den 26.06.2017, 14:15 in NA 2/64**Title**: "Critical points, matroids, and log-concave sequences"**Abstract**: "It is well-known that complex hyperplane arrangements can be conveniently resolved to normal crossing divisors with the help of the permutohedral toric variety. The cohomology algebras of the resulting wonderful compactifications are not only matroid invariants, but Adiprasito, Huh and Katz (2015) found that Hodge-theoretic constraints imposed on them by complex geometry persist for arbitrary matroids. The maximal likelihood variety of a complex arrangement captures the set of critical points of all rational functions with poles and zeros on the arrangement. Its bidegree (as a biprojective variety) encodes a combinatorially significant sequence of integers, the h-vector of the broken circuit complex. I will describe work in progress with Federico Ardila and June Huh in which we construct a combinatorial analogue of the maximal likelihood variety for arbitrary (nonrealizable) matroids. In particular, this leads to a proof that the h-vector of the broken circuit complex is a log-concave sequence. "

**Dr. Tomohiro Uchiyama** (National Taiwan University, National Center for Theoretical Sciences)

Seminar: Lie-Theorie

Montag, den 19.06.2017, 16:45 in NA 2/64**Title**: "Complete reducibility, geometric invariant theory, and spherical buildings"**Abstract**: "In this talk, I will talk about Serre's notion of complete reducibility for subgroups of reductive algebraic groups (matrix groups). Serre's notion of complete reducibility nicely generalizes completely reducible representations and it is useful to study the subgroup structure of reductive groups in positive characteristic. I will explain how to use geometric invariant theory (a branch of algebraic geometry) and Tits' spherical buildings (highly symmetrical combinatorial objects) to study complete reducibility. The recently proved 50-years-old center conjecture of Tits in spherical buildings comes into play. No background in algebraic groups or algebraic geometryis necessary."

**Prof. Gleb Koshevoy**, Russian Academy of Sciences, Moskau

Seminar: Lie-Theorie

Montag, den 22.05.2017, 14 ct in NA 2/64**Title**: "Combinatorics of crystals and Toda systems"

**Dr. Wassilij Gnedin**, RUB

Seminar: Lie-Theorie

Montag, den 15.05.2017, 14:00 in NA 2/64**Title**: "Tame categories of Harish-Chandra modules"

**Professor Masahiko Yoshinaga**, Hokkaido University, Sapporo

Seminar: Lie-Theorie

Montag, den 08.05.2017, 14:00 in NA 2/64**Title**: "The characteristic polynomial of Linial arrangement"**Abstract**:

The (m-th extended) Linial arrangement is a certain finite truncation of affine Weyl arrangement associated to a root system. Postnikov and Stanley (2000) conjectured that the roots of the characteristic polynomial of Linial arrangement have the same real part. We will report that the application of Ehrhart theory and Eulerian polynomials enables us to make progress on the conjecture. This talk is based on the following two preprints.

https://arxiv.org/abs/1501.04955

https://arxiv.org/abs/1610.07841

## winter term 2016/2017

**Dr. Ulrich Thiel**, Universität Stuttgart

Seminar: Lie-Theorie

Mittwoch, den 08.03.2017, 16:00 in NA 2/64**Title**: "Hyperplane arrangements associated to symplectic quotient singularities"**Abstract**: To any symplectic reflection group there is an associated symplectic singularity. Namikawa constructed a hyperplane arrangement encoding certain geometric information of this singularity. In the special case of the symplectic reflection group defined by an ordinary complex reflection group we show that this hyperplane arrangement has a much more accessible representation-theoretic description via blocks of restricted rational Cherednik algebras, namely it equals the so-called Calogero-Moser locus which is quite interesting by itself. This result allows us on the one hand to explicitly compute Namikawa's geometrically defined hyperplane arrangement in many cases (in particular for many exceptional groups) and on the other hand it implies several, so far unknown, general properties of the Calogero-Moser locus. It is an interesting question whether properties of these hyperplane arrangements encode any further information and if they yield some new examples of hyperplane arrangements. This is joint work with G. Bellamy and T. Schedler.

**Falk Bannuscher**, RUB

Seminar: Lie-Theorie

Montag, den 23.01.2017, **16:00** in NA 2/**64****Title**: "Konjugationsklassen halbeinfacher algebraischer Gruppen und Lie Algebren"**Abstract**: In der Gruppe der invertierbaren Matrizen, über einem algebraisch abgeschlossenen Körper, gibt es nur endlich viele Konjugationsklassen unipotenter Matrizen. Im Vortrag befassen wir uns damit, inwieweit sich dieses Resultat auf Untergruppen verallgemeinern lässt. Mit Hilfe von reduktiven Paaren werden wir diese Frage partiell für halbeinfache algebraische Gruppen beantworten.

**Lukas Kühne**, Universität Bonn

Seminar: Lie-Theorie

Montag, den 16.01.2017, 14:00 in NA 2/24**Title**: Heavy hyperplanes in multiarrangements and their freeness**Abstract**: One of the central topics among the theory of hyperplane arrangements is their freeness. Terao's conjecture tries to link the freeness with the combinatorics of an arrangement. One of the few categories of arrangements which satisfy this conjecture consists of 3-dimensional arrangements with an unbalanced Ziegler restriction. This means that the arrangement contains a lot of hyperplanes intersecting in one single line In this talk, we generalize this result to arbitrary dimensional arrangements in terms of flags by introducing unbalanced multiarrangements. For that purpose, we generalize several freeness criteria for simple arrangements, including Yoshinaga's freeness criterion, to unbalanced multiarrangements. This is joint work with Takuro Abe.

**Dr. Xin Fang**, Universität Köln

Seminar: Lie-Theorie

Montag, den 19.12.2016, 14:00 in NA 2/24**Title**: "Toric degenerations of flag varieties and applications"**Abstract**: In this talk I will explain a general framework to construct toric degenerations of flag varieties via birational sequences and Newton-Okounkov bodies. If time permits, I plan to apply these constructions to determine the Gromov widths of coadjoint orbits.

**Dr. Christian Stump**, Freie Universität Berlin

Seminar: Lie-Theorie

Montag, den 12.12.2016, 14:00 in NA 2/24**Title**: "What are Coxeter elements in reflection groups?"**Abstract**: In this talk, I aim to provide a conceptual reason why any two reflections in the symmetry group of a regular pentagon form a Coxeter system. I will do so by providing a conceptual definition of Coxeter elements in finite (well-generated) reflection groups. The main ingredient is to study properties of the Galois group of the field of definition. This is joint work with Vic Reiner and Vivien Ripoll.

**Prof. Dr. Eamonn O'Brien**, University of Auckland

Seminar: Lie-Theorie

Montag, den 05.12.2016, 14:00 in NA 2/24**Title**: "Effective algorithms for matrix groups"**Abstract**: How can we compute effectively with a matrix group whose entries lie in a finite field? We identify some inherent challenges, and outline a practical model which exploits randomness, geometry and detailed knowledge of the group structure.

**Dr. Alistair Litterick**, Universität Bielefeld

Seminar: Lie-Theorie

Montag, den 28.11.2016, 14:00 in NA 2/24**Title**: "Subgroup Structure of Reductive Groups"**Abstract**: A long-standing program seeks to understand the subgroup structure of reductive groups over algebraically closed fields. This program began in earnest with Dynkin in the 1950s, and continues to this day through work of Liebeck, Seitz, Saxl, Stewart, Testerman, Thomas, myself, and numerous others besides. We will discuss this ongoing effort, with a focus on reductive subgroups of exceptional simple algebraic groups and the notion of G-complete reducibility due to Serre, which provides a link with representation theory and streamlines the study of subgroup structure.

**Prof. Dr. Christopher Voll**, Universität Bielefeld

Seminar: Lie-Theorie

Montag, den 21.11.2016, 15:00 in NA 2/24**Title**: "Submodule zeta functions -- polynomiality and nonnegativity"**Abstract**: Given a free module M of finite rank over the ring of integers of a number field K, together with a set A of linear operators on M, the associated submodule zeta function enumerates A-invariant submodules of M of finite additive index. Given, in addition, a grading on M, the associated graded submodule zeta function enumerates submodules which are homogeneous with respect to the grading. Submodule zeta functions -- graded or otherwise -- satisfy natural Euler product decompositions: the respective factors are rational functions, indexed by the finite places of K. We discuss a number of results illustrating what seem to be quite general polynomiality and nonnegativity properties satisfied by the coefficients of these rational functions. Some of them are due to Rossmann, whilst others are the outcome of joint work with Lee.

**Dr. Angela Carnevale**, Universität Bielefeld

Seminar: Lie-Theorie

Montag, den 21.11.2016, 14:00 in NA 2/24**Title**: "Orbit Dirichlet series and multiset permutations"**Abstract**: We study Dirichlet series enumerating orbits of products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups. We interpret Euler factors of such Dirichlet series in terms of generating polynomials for statistics on multiset permutations. As applications, we establish local functional equations, determine the (global)abscissae of convergence and exhibit natural boundaries. This is joint work with Christopher Voll.

**Arik Wilbert**, Universität Bonn

Seminar: Lie-Theorie

Montag, den 14.11.2016, 14:00 in NA 2/24**Title**: "Two-row Springer fibers in types C & D: Topology, Representation Theory & Combinatorics "**Abstract**: tba

**Dr. Hans Franzen**, RUB

Seminar: Lie-Theorie

Montag, den 07.11.2016, 14:00 in NA 2/24**Title**: "The value of the Kac polynomial at I"**Abstract**: We establish a formula for the value of the Kac polynomial at one in terms of Kac polynomials, evaluated at one, of the universal (abelian) covering quiver by applying torus localization methods to quiver varieties introduced by Hausel--Letellier--Rodriguez-Villegas.

## summer term 2016

**Dr. Hery Randriamaro**, Universität Antananarivo, Madagaskar

Seminar: Lie-Theorie

Montag, den 04.07.2016, 14:00 in NA 2/64**Title**: "The Varchenko Determinant of a Coxeter Arrangement"**Abstract**: The Varchenko determinant is a matrix determinant defined on hyperplane arrangements. The formula of this determinant is very beautiful, only it is impossible to compute it from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the Coxeter arrangements. From this explicit one, the Varchenko determinant associated to any finite Coxeter group becomes computable. This a joint work with Goetz Pfeiffer.

**Prof. Dr. Gerhard Roehrle**, RUB

Seminar: Lie-Theorie

Montag, den 27.06.2016, 14:00 in NA 2/64**Title**: "Serre's notion of complete reducibility and GIT"**Abstract**: In the talk we outline Serre's notion of G-complete reducibility for subgroups of the reductive group G and show how methods from geometric invartiant theory can be employed to study this notion and to shed some light on the geometric nature of this concept.

**Mikaël Cavallin**, Technische Universität Kaiserslautern

Seminar: Lie-Theorie

Montag, den 20.06.2016, 14:00 in NA 2/64**Title**: "On the natural embedding of SO(V) in SL(V)"**Abstract**: Let V be a finite-dimensional vector space over an algebraically closed field K having characteristic p greater than or equal to 0. In this talk, we show how the natural embedding of X=SO(V) in Y=SL(V) can be used in order to determine the structure of certain Weyl modules for X. In addition, we see how this question relates to the problem of determining irreducible KY-modules on which X acts with exactly two composition factors.

**Dr. Giovanni Cerulli-Irelli**, Universität Rom I

Seminar: Lie-Theorie

Montag, den 13.06.2016, 14:00 in NA 2/64**Title**: "Quiver Grassmannians of Dynkin type"**Abstract**: Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver representations. In case the quiver is an orientation of a simply laced Dynkin diagram, we call them of Dynkin type. In this introductory talk I will present some results concerning the geometry of those projective varieties, which are based on techniques developed in collaboration with M. Reineke and E. Feigin. In particular I will show that the generic quiver Grassmannians have positive Euler characteristic, confiriming a conjecture by S. Fomin and A. Zelevinsky.

**Dr. Oliver Goodbourn**, RUB

Seminar: Lie-Theorie

Montag, den 06.06.2016, 14:00 in NA 2/64**Title**: "Reductive pairs from representations of algebraic groups"**Abstract**: Reductive pairs are a class of nice embeddings of reductive algebraic groups. They have been used to salvage some good behaviour observed in characteristic 0 in the positive characteristic case, for instance in work of Bate, Herpel, Martin and Röhrle on G-complete reducibility, and in providing uniform proofs of otherwise technical results. I will discuss work into determining when we get reductive pairs arising from representations of an algebraic group, including complete pictures for simple and Weyl modules for SL_2 in arbitrary characteristic.

**Dr. Magdalena Boos**, RUB

Seminar: Lie-Theorie

Montag, den 30.05.2016, 14:00 in NA 2/64**Title**: "Finiteness criteria for parabolic conjugation"**Abstract**: Motivated by the study of commuting varieties we consider a parabolic upper-block subgroup P of $\mathrm{GL}_n(\mathhb{C})$ and study its conjugation-action on the variety of nilpotent matrices in Lie(P). The main question posed in this talk is "For which P does the mentioned action only admit a finite number of orbits?" In order to approach such finiteness criterion, we make use of methods from Representation Theory of finite-dimensional algebras, for example covering techniques and Delta-filtrations. The talk will give an overview of the current status of results and conjectures. (This is work in progress, joint with M. Bulois)

**Prof. Dr. Gerhard Röhrle**, RUB

Seminar: Lie-Theorie

Montag, den 23.05.2016, 14:00 in NA 2/64**Title**: "Cocharacter-closure and the rational Hilbert-Mumford Theorem"**Abstract**: I shall introduce the notion of cocharacter-closure and will explain how this leads to a rational version of the celebrated Hilbert-Mumford Theorem from geometric invariant theory. We will illustrate with some examples how this concept differs from the usual Zariski-closure and discuss some applications. This reports on joint work with M. Bate, S. Herpel and B. Martin.

**Prof. Dr. Markus Reineke**, RUB

Seminar: Lie-Theorie

Montag, den 09.05.2016, 14:00 in NA 2/64**Title**: "Linear degenerations of flag varieties"**Abstract**: Linear degenerations of SL(n)-flag varieties are constructed by relaxing the containment condition for the subspaces in a flag. We will discuss characterizations of flatness, irreducibility, normality, and other geometric properties of the resulting degenerations, in terms of linear algebra data. The underlying methods, quiver Grassmannians and PBW degenerations of representations, will be introduced. This is a report on recent joint work with G. Cerulli Irelli, X. Fang, E. Feigin and G. Fourier.

**Marcel Maslovaric**, Georg-August-Universität Göttigen

Seminar: Lie-Theorie

Montag, den 25.04.2016, 14:00 in NA 2/64**Title**: "Variation of Geometric Invariant Theory and Birational Geometry"**Abstract**: Forming a quotient with respect to a group action on a variety via Geometric Invariant Theory depends on the choice of a stability parameter. The variation of this parameter, the birational geometry of the quotients and the line bundles on the quotients are closely related. In this talk we discover a class of quotients (producing so called Mori dream spaces) where this relation is fundamental. We will see that moduli of representations of a quiver belong to this class.

**Prof. Dr. Meinolf Geck**, Universität Stuttgart

Seminar: Lie-Theorie

Montag, den 18.04.2016, 14:00 in NA 2/64**Title**: "A new construction of semisimple Lie algebras"**Abstract**: We work out a remark of Lusztig which leads to a simplified construction of a semisimple Lie algebra from a root system.