Seminar Gruppen und Darstellungen
Sommersemester 2022
Organisatoren: Caroline Lassueur, Lucas Ruhstorfer
 Donnerstag, 28. April 2022:
17:0017:50: Georges Neaime [Universität Bielefeld und RuhrUniversität Bochum], Towards the Linearity of Complex Braid Groups [Abstract]Complex braid groups are a generalisation of Artin–Tits groups. They are attached to complex reflection groups, which are themselves a generalisation of finite Coxeter groups. It is an ongoing challenge to extend the theory of Artin–Tits groups to all complex braid groups. A part of this theory extension was established by Broué, Malle, and Rouquier in their seminal work. An important feature of spherical Artin groups is that they are linear groups, i.e., they admit a faithful linear representation of finite dimension. For the usual braid group, this property was shown to hold independently by Bigelow and Krammer. We seek to extend the theory of linearity to the context of complex braid groups, with a focus on the infinite families. Indeed, we will describe a definition of BMW and Brauer algebras, from which we can construct suitable linear representations and conjecture their faithfulness. We present a number of theorems and conjectures related to the structure of the aforementioned algebras, as well as properties of the relevant representations. We will finally propose a research programme for the sequel.  Donnerstag, 2. Juni 2022:
17:0017:50: Deniz Yilmaz [LAMFA, Amiens], TBA [Abstract]Let \(k\) be an algebraically closed field of positive characteristic \(p > 0\) and let \(F\) be an algebraically closed field of characteristic \(0\). In this talk we first show that the category of diagonal \(p\)permutation functors over \(F\) is semisimple, and we give a parametrization of its simple objects, together with a description of their evaluations. Next, to any pair \((G, b)\) of a finite group \(G\) and a block idempotent \(b\) of \(kG\), we associate a diagonal \(p\)permutation functor and find its decomposition as a direct sum of simple functors. This leads to a characterization of nilpotent blocks in terms of their associated functors. Finally, for such pairs \((G, b)\) of a finite group and a block idempotent, we introduce the notion of functorial equivalence over F and we prove a corresponding finiteness theorem: for a given finite pgroup \(D\), there is only a finite number of pairs \((G, b)\), where \(G\) is a finite group and \(b\) a block idempotent of \(kG\) with defect isomorphic to \(D\), up to functorial equivalence over \(F\). We also give a sufficient condition for two pairs \((G, b)\) and \((H, c)\) to be functorially equivalent over \(F\) in the situation of Broué’s abelian defect group conjecture. This is joint work with Serge Bouc  Donnerstag, 9. Juni 2022:
17:0017:50: Nicola Grittini [Università Cattolica del Sacro Cuore, Brescia] TBA [Abstract]The investigation of the field of values of certain irreducible characters is one of the main areas of interests in Character Theory of Finite Groups, especially for what concerns rationalvalued characters. In 2006, Navarro and Tiep proved that, in a group of even order, there always exists a nonprincipal rationalvalued irreducible character of odd degree. More in general, Navarro and Tiep proved that a similar property holds also for irreducible characters which are close to be rationalvalued. In fact, they proved that, for a finite group, given a prime number p, it is always possible to find a nonprincipal irreducible character having values in some pcyclotomic extension of the field of rational numbers and with degree coprime with p, provided that either 2 or p divide the order of the group.
Another, apparently unrelated problem in Group Theory concerns the study of the primes dividing the order of a Sylow normalizer. In 2002 Isaacs and Navarro proved that, for solvable groups, these primes can be deduced from the character table, while in 2016 Guralnick, Navarro and Tiep classified the groups with a Sylow normalizer of odd order. In this talk, we will see how a variant of the first problem is related to the second, and how sometimes the character table can provide some information on the primes dividing the order of a Sylow normalizer, even for nonsolvable groups. We will also discuss some of the techniques we used to prove the results we mentioned, particularly one involving isomorphisms between character triples.
(Joint work with M. A. Pellegrini)  Donnerstag, 23. Juni 2022:
17:0017:50: Norman MacGregor [University of Birmingham] TBA [Abstract]The respresentation type of an algebra is finite, tame, or wild, and a block algebra is tame precisely if its defect groups are dihedral, semidihedral, or quaternion. Several families of tame algebras were classified by Karin Erdmann, and we shall consider which occur as blocks of nite groups. We shall give some background of the relevant represenation theory, before outlining how Erdmann's list can be rened for blocks via a reduction using the Classification of Finite Simple Groups.  Dienstag, 19. Juli 2022:
16:0016:40: Michael Geline [Northern Illinois University] TBA [Abstract]Modular representation theory seeks to understand how global properties of finite groups are determined by local subgroups. The global properties of greatest interest concern the number and dimensions of irreducible representations. While these quantities seem to resist a satisfying local characterization, we shall show how rationality properties of representations do not.
16:5017:30: Emily Norton [University of Kent] TBA [Abstract]n open problem in the ℓmodular representation theory of finite reductive groups G(q) is to calculate certain numerical invariants called decomposition numbers, conveniently assembled into a matrix called the decomposition matrix. This is expected to depend only on the order d of q mod ℓ. For groups of types B and C, the rectangular matrices recording multiplicities of simple modules in Specht modules of level 2 cyclotomic Hecke algebras appear as submatrices of the decomposition matrix. In the "baby case" that d is suficiently large, LeclercMiyachi and BrundanStroppel showed that these Hecke algebra matrices are given by the nicest KazhdanLusztig polynomials in existence, KazhdanLusztig polynomials of Grassmannians. In this talk, I will discuss extending that result to certain square submatrices of the decomposition matrix of a finite reductive group of type B or C. This work in progress with Olivier Dudas should provide the first glimpse of chunks of the decomposition matrix in blocks of arbitrary defect (beyond the chunks given by Hecke algebras).
Wintersemester 2021/22
Organisatoren: Caroline Lassueur (RWTH Aachen), Lucas Ruhstorfer (TU Kaiserslautern)
Dieses Semester findet das Seminar als virtuelles Seminar zusammen mit dem Lehrstuhl für Algebra und Zahlentheorie der RWTH Aachen University statt.
Zur Anmeldung als externer Gast verwenden Sie bitte diesen Link.
 Donnerstag, 13. Januar 2022:
16:1517:00: Nicolas Jacon [Université de Reims ChampagneArdenne], On the computation of the Mullineux involution for symmetric groups and Hecke algebras [Abstract]The Mullineux problem is a classical topic in the modular representation theory of the symmetric group. The aim is to understand how the sign representation twist the irreducible representations of the symmetric group in positive characteristic. In this talk, we give a new approach for this problem by using the representations of affine Hecke algebras and ArikiKoike algebras.
17:0517:50 William Murphy [City, University of London], The first Hochschild cohomology of blocks of finite group algebras [Abstract]The Hochschild cohomology (the HH*) of finite dimensional algebras is a useful tool to help with classifying blocks of group algebras. For example, if k is a field and A, B are two symmetric kalgebras, then a stable equivalence of Morita type between A and B induces an isomorphism between the HH^n of A and B in degree n greater than zero. What is more, the graded ring HH* forms a Gerstenhaber algebra, and in particular the HH^1 forms a Lie algebra over k, whose structure is also preserved under stable equivalence of Morita type.
In this talk I will discuss a range of topics and questions regarding the HH^1 of blocks of finite group algebras. These will include results on blocks with a cyclic or low rank elementary abelian defect group, the use of a powerful computational result to calculate the dimensions of the HH^1 of blocks, calculation of the Lie algebra structure of the HH^1 of blocks of the Mathieu groups, and results on the HH^1 of twisted group algebras.
 Donnerstag, 20. Januar 2022:
17:1518:00: Georges Neaime [Universität Bielefeld], FINDET NICHT STATT / CANCELLED
 Donnerstag, 27. Januar 2022:
16:1517:00: Carolina Vallejo [Universidad Autónoma de Madrid], Groups with a 2generated Sylow 2subgroup [Abstract]We present a characterization of the groups possessing a 2generated Sylow 2subgroup in terms of their character theory. This talk is based on joint works with G. Navarro, N. Rizo and A. A. Schaeffer Fry.
17:0517:50 Claudio Marchi [University of Manchester], Picard groups for blocks with normal defect group [Abstract]Picard groups of algebras have extensively been studied in the past, but just recently people started looking at Picard groups of blocks of finite groups. These revealed themselves to be useful tools, for example for dealing with Donovan conjecture, but they’re also interesting in their own right, since they have the structure of a finite group. In this talk we will give an introduction to Picard groups and Picent of blocks, and then present joint work with Michael Livesey on blocks with normal defect groups, providing evidence to a conjecture on basic Morita equivalences.
 Donnerstag, 10. Februar 2022:
17:1518:00: Julian Brough [Bergische Universität Wuppertal], Characters of normalisers of \(d\)split Levi subgroups in \(Sp_{2n}(q)\) [Abstract]In a current ongoing programme to prove some of the local global counting conjectures a key ingredient requires an understanding of characters in the normalisers of \(d\)split Levi subgroups. In this talk I will present recent results in this direction for the symplectic groups.
Sommersemester 2021
Organisatoren: Caroline Lassueur, Lucas Ruhstorfer
Dieses Semester findet das Seminar als virtuelles Seminar zusammen mit der Arbeitsgruppe Algebra und Zahlentheorie der Bergischen Universität Wuppertal statt.
Zur Anmeldung als externer Gast verwenden Sie bitte diesen Link.
 Dienstag, 27. April 2021:
16:0016:40: Eugenio Giannelli [Università degli Studi di Firenze], On a conjecture of Malle and Navarro [Abstract]Recently I have been interested in studying the restriction of irreducible characters of finite groups to Sylow subgroups. In this talk I will discuss a nice application of this topic to a conjecture proposed by Malle and Navarro in 2012. This talk is based on joint work with Stacey Law, Jason Long and Carolina Vallejo.
17:0017:40: Noelia Rizo [Universidad del País Vasco], On the trivial intersection block conjecture [Abstract]In 1997 G. Navarro and W. Willems asked when the irreducible complex characters in a Brauer \(p\)block \(B_p\) coincide with those of a \(q\)block \(B_q\). This situation was later characterized for principal blocks by work of C. Bessenrodt, G. Navarro, J.B. Olsson and P.H. Tiep.
In 2008, the opposite situation, the trivial intersection case, that is, the case where \[{\rm Irr}({B_p(G)}) \cap {\rm Irr}({B_q(G)})=1\] for \(p \ne q\), was first studied by C. Bessenrodt and J. Zhang, where \(B_p(G)\) denotes the principal \(p\)block of \(G\). In particular, they prove that $G$ is nilpotent if and only if \({\rm Irr}({B_p(G)}) \cap {\rm Irr}({B_q(G)})=1\) for all \(p \ne q\). This result inspired Y. Liu, W. Willems, H. Xiong and J. Zhang who, in a recent work, have conjectured (and reduced to almost simple groups) that if \({\rm Irr}({B_p(G)}) \cap {\rm Irr}({B_q(G)})=1\), for \(p\neq q\), then then there is a Sylow \(p\)subgroup \(P\) of \(G\) and a Sylow \(q\)subgroup \(Q\) of \(G\) such that \([P,Q]=1\).

Dienstag, 11. Mai 2021 (Organisiert von Damiano Rossi, Wuppertal):
16:0016:40: VirgiliusAurelian Minuta [BabesBolyai University, ClujNapoca], An overview of Knörr lattices for finite groups [Abstract]A modern approach to the localglobal conjectures in the representation theory of finite groups consists in reducing them to some stronger statements about simple groups. Such reduction theorems can be obtained by utilizing the language of character triples and of the various relations between them. In particular, recent results of Britta Späth utilize the relations ≥, ≥ c and ≥ b between character triples.
We have introduced a categorical version of these relations, between socalled module triples and we have proved that they are consequences of a special type of group graded Morita equivalences, group graded Rickard equivalences and group graded derived equivalences, respectively. Furthermore, again motivated by the reduction theorems, we show that our relations between module triples are compatible with wreath products.
16:5017:30: Sofia Brenner [FriedrichSchiller Universität Jena], On Socles of Centers of Group Algebras [Abstract]In this talk, we consider the socle of the center of an algebra and address the question under which conditions it is an ideal in this algebra. To this end, we mainly focus on group algebras and discuss some approaches for \(p\)groups in greater detail. 
Dienstag, 25. Mai 2021 (Organisiert von Damiano Rossi, Wuppertal):
16:0016:40: Scott Harper [University of Bristol], The spread of a finite group [Abstract]After a brief review of what Knörr lattices are and the role they play in some of the local to global conjectures of block theory, we shall give constructions of some of the known positive height examples for $p$groups as well as a recent and elementary proof of why they cannot occur as sources of irreducible lattices (at least when the defect group of the relevant block is TI).
16:5017:30: Justin Lynd [University of Louisiana, Lafayette], Punctured groups for exotic fusion systems [Abstract]The fusion system of a finite group G at a prime p is a category whose objects are the subgroups of a fixed Sylow psubgroup S, and where the morphisms are the conjugation homomorphisms induced by the elements of G. The notion of a saturated fusion system is abstracted from this standard example and gives rise to exotic examples not realizable by a group in the above fashion. Given an exotic fusion system, one might want to ask: how close is it to being realizable? One way to approach this question is by trying to construct transporter categories, or localities, associated with a given exotic system. A punctured group for a fusion system is a transporter system with objects the full set of nonidentity subgroups of the underlying pgroup, and is perhaps the closest approximation to a group one can hope for in the case of an exotic fusion system. I plan to give background on these topics, look at some examples such as the BensonSolomon exotic fusion systems at the prime 2, and mention an application of the existence of a punctured group to the topology of classifying spaces. This is joint work with Ellen Henke and Assaf Libman. 
Dienstag, 8. Juni 2021:
16:0016:40: Michael Geline [Northern Illinois University, DeKalb], An overview of Knörr lattices for finite groups [Abstract]After a brief review of what Knörr lattices are and the role they play in some of the local to global conjectures of block theory, we shall give constructions of some of the known positive height examples for $p$groups as well as a recent and elementary proof of why they cannot occur as sources of irreducible lattices (at least when the defect group of the relevant block is TI).
16:5017:30: Andrei Marcus [BabesBolyai University, ClujNapoca], Extending basic Morita equivalences [Abstract]Let \((K, \mathcal{O}, k)\) be a \(p\)modular system where \(k\) is algebraically closed, let \(b\) be a \(G\)invariant block of the normal subgroup \(H\) of a finite group \(G\), having defect pointed group \(Q_\delta\) in \(H\) and \(P_\gamma\) in \(G\), and consider the block extension \(b\mathcal{O}G\). One may attach to \(b\) an extended local category \(E(b,H,G)\), a group extension \(L\) of \(Z(Q)\) by \(N_G(Q_\delta)/C_H(Q)\) having \(P\) as a Sylow \(p\)subgroup, and a cohomology class \([α] ∈ H^2(N_G(Q_\delta)/QC_H(Q), k^\times)\). We prove that these objects are invariant under the \(G/H\)graded basic Morita equivalences. Our methods lead us to alternative and unified proofs of the results of Külshammer and Puig (1990), Puig and Zhou (2012) on extensions of nilpotent blocks, and of Zhou (2016) on \(p'\)extensions of inertial blocks.
This is a joint work with T. Coconet ̧ and C.C. Todea. 
Dienstag, 29. Juni 2021 (Organisiert von Max Horn):
16:0017:00: Andrea Thevis [Universität des Saarlandes], Squaretiled surfaces, normal covers, and cylinder decompositions [Abstract]A translation surface is obtained by taking finitely many polygons in the Euclidean plane and gluing them along their edges by translations. If we restrict to gluing unit squares, we obtain a squaretiled surface, also known as origami. These surfaces can be decomposed into cylinders. In this talk, I will present two projects using cylinder decompositions of squaretiled surfaces as a main tool. Many interesting properties of squaretiled surfaces are closely related to finite index subgroups of SL(2, Z) called Veech groups. In the first project, we study when Veech groups are far from being a congruence group.
The second project deals with geminal surfaces. These surfaces consist in each direction out of two cylinders of equal height and circumference. Apisa and Wright conjecture that geminal surfaces are cyclic covers of the (2 × 2)torus. We discuss first steps towards proving this conjecture. This is joint work with Johannes Flake.  Do. 15. Juli / Fr. 16. Juli 2021: Meeting Representation Theory at the Villa Denis 2021.
Wintersemester 2020/21
Organisatoren: Caroline Lassueur, Lucas Ruhstorfer
 Donnerstag, 14. Januar 2021:
16:0016:45: Nadia Mazza [Lancaster University], Endotrivial modules for finite groups of Lie type [Abstract]In this talk, we will present an update on the classification of endotrivial modules for finite groups of Lie type using a range of techniques. We will see that in all but a few exceptions (as to be expected with these groups), the group of endotrivial modules is cyclic, generated by the stable isomorphism class of the syzygy of the trivial module.
17:0017:45: Matthew Gelvin, Dade groups and relative syzygies for finite groups [Abstract]If \(S\) if a finite \(p\)group and \(k\) a field of characteristic \(p\), Dade observed that there is a natural group structure on certain equivalence classes of endopermuation \(kS\)modules, now called the Dade group \(D(S)\). This fact is heavily dependent on the assumption that \(S$\) is a \(p\)group; naively attempting the same construction for an arbitrary finite group \(G\) and the class of endopermutation \(kG\)modules fails to yield a welldefined group operation. Lassueur was able to define a Dade group \(D(G)\) by considering only those endo\(p\)permutation \(kG\)modulescalled \emph{strongly capped}with the additional structural constraint that \(End(M)\) should have a single Sylowvertex component, which is isomorphic to the trivial module \(k\). This talk will detail joint work with Ergün Yalçin, which begins with a slight generalization of Lassueur's ideas by considering those endo\(p\)permutation \(kG\)modules \(M\) such that \(End(M)\) may contain multiple copies of \(k\) but no other nontrivial Sylowvertex components. This yields a second option for the definition of the Dade group \(D(G)\), but fortunately one that turns out to be isomorphic to Lassueur's. The motivation for this generalization stems from the desire to define tensor induction between different Dade groups; I will describe how our approach allows for tensor induction in some additional cases but still fails to form a general theory. The talk will conclude with a discussion of the subgroup of relative syzygies, which is possible thanks to the observation that if \(X\) is a finite \(G\)set with no Sylowfixedpoints then the corresponding relative syzygy module \(\Delta(X)\) is an endo\(p\)permutation module that satisfies our extra component hypothesis. We will briefly describe how the corresponding subgroup \(D^\Omega(G)\leq D(G)\) can be computed in terms of certain dimension functions of \(G\).
 Donnerstag, 28. Januar 2021:
16:0016:45: Jay Taylor [University of Southern California], Rationality Properties of Kawanaka Characters [Abstract]Let \(G\) be a finite group and let \(K\) be a subfield of the complex numbers. For a (complex, not necessarily irreducible) character \(\chi\) of \(G\) one can ask the following questions: What is the character field \(K(\chi) = K(\chi(g) \mid g \in G)\)?
 Is \(\chi\) realisable over \(K(\chi)\)?
17:0017:45: Mandi A. Schaeffer Fry [MSU Denver], The Inductive McKayNavarro Conditions in Type C for the Prime 2 [Abstract]The McKay conjecture is one of the major conjectures in the realm of the localglobal philosophy in character theory and posits a bijection between the set of irreducible characters of a group with \(p’\)degree and the corresponding set in the normalizer of a Sylow \(p\)subgroup. Navarro’s refinement suggests that the values of the characters should also be related, proposing that the bijection further commutes with certain Galois automorphisms. Recently, Navarro—Späth—Vallejo have reduced the McKay—Navarro conjecture to certain “inductive” conditions on finite simple groups. In this talk, I’ll discuss joint work with L. Ruhstorfer, in which we complete the proof of these inductive conditions in the case \(p=2\) for the symplectic groups.
 Donnerstag, 4. Februar 2021:
16:0016:45: Cesare Ardito [City, University of London], Classification of blocks and open conjectures [Abstract]Donovan’s conjecture predicts that given a pgroup D there are only finitely many Morita equivalence classes of blocks of group algebras with defect group D. While the conjecture is still open for a generic pgroup D, it has been proven in many special cases. A harder question is the one to actually classify Morita equivalence classes in a given situation, i.e. to have a complete list of blocks that can occur determined up to Morita equivalence. In this talk, I will introduce the topic, give the relevant definitions and then roughly describe the process of classifying blocks, with a focus on the methodology and the tools normally needed to obtain such a result. Then I will focus on some applications of having such a list, mentioning some (perhaps surprising) difficulties that can arise in verifying certain classic modular representation theory conjectures, even when a complete list of blocks is given.
17:0017:45: Patrick Serwene [City, University of London], Exotic and blockexotic fusion systems [Abstract]One of the main problems in the theory of fusion systems is the question whether a fusion system arises in the form of a finite group if and only if it arises in the form of a pblock of a finite group. There is a conjecture saying that a fusion system is induced by a group if and only if it is induced by a block. We present reduction theorems for this problem reducing it to blocks of quasisimple groups in certain cases. One of these reductions settles the conjecture for the family of Parker–Semeraro fusion systems. We discuss ongoing work concerning our strategy to prove the conjecture for some groups of Lie type.
Sommersemester 2020
Im Sommersemester 2020 wurde das Seminar im Wesentlichen durch das Meeting Representation Theory at the Villa Denis 2020 ersetzt.
 Donnerstag, 17. September 2020:
Britta Späth [Bergische Universität Wuppertal], Extensions of cuspidal characters
Laura Voggesberger [TU Kaiserslautern], Nilpotent pieces in Lie algebras of algebraic groups
Emil Rotilio [TU Kaiserslautern], A method to complete generic character tables of groups of Lie type with focus on \(SL_4(q)\) and \(Spin_8^+(q)\)
Damiano Rossi [Bergische Universität Wuppertal], Character Triple Conjecture for psolvable groups
 Freitag, 18. September 2020:
Caroline Lassueur [TU Kaiserslautern], On the characters of trivial source modules
Birte Johansson [TU Kaiserslautern], On the inductive GaloisMcKay condition in defining characteristic \(p=2\)
Emily Norton [TU Kaiserslautern], Some decomposition matrices of classical groups
Gunter Malle [TU Kaiserslautern], Brauer's height zero conjecture for principal blocks
Wintersemester 2019/20
Organisatoren: Niamh Farrell, Caroline Lassueur, Lucas Ruhstorfer
 Dienstag, 15. Oktober 2019: A Day of Representation Theory at the Villa Denis 2019
Olivier Dudas [Paris VII], Unitriangular shape of decomposition matrices of unipotent blocks
Emily Norton [Guest @ TU Kaiserslautern], HarishChandra series of unipotent representations of finite unitary groups
Noelia Rizo Carrión [Università degli Studi di Firenze], Galois action on the principal block
Lucas Ruhstorfer [TU Kaiserslautern], Jordan decomposition for the AlperinMcKay conjecture
 Donnerstag, 31. Oktober 2018: Julian Brough [Bergische Universität Wuppertal]
Titel: A criterion for the inductive Alperin weight condition [Abstract]Many of the localglobal conjectures have been reduced to so called inductive conditions on simple groups. Moreover Späth has been successful in verifying the inductive McKay condition for most families of simple groups by considering an alternative criterion on a larger group with some additional information about the corresponding Clifford theory. In this talk I will discuss a joint work with Späth, in which we have provided an analogous criterion for the Alperin weight conjecture.  Donnerstag, 7. November 2019: Paul Wedrich [MPI Bonn]
Titel: Quivers for SL(2) tilting modules [Abstract]I will explain how diagrammatic algebra can be used to give an explicit generatorsandrelations presentation of all morphisms between indecomposable tilting modules for SL(2) over an algebraically closed field of positive characteristic. The result takes the form of a path algebra of an infinite, fractallike quiver with relations, which can be considered as the (semiinfinite) Ringel dual of SL(2). Joint work with Daniel Tubbenhauer.  Donnerstag, 14. November 2019: Jens Eberhardt [MPI Bonn]
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 algebrogeometric 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 KLRalgebras. We also will explain how more "exotic" versions of motivic sheaves provide exciting new opportunities in geometric representation theory.  Donnerstag, 28. November 2019: Ivo Dell'Ambrogio [Université de Lille, Frankreich]
Titel: The Green correspondence in algebra and topology [Abstract]The Green correspondence, a classic and basic result of modular representation theory, describes a bijection between indecomposable Hmodules and indecomposable Gmodules having the same vertex Q, provided H is a subgroup of G containing the normaliser of Q. Already in the 70's, Green himself noticed that the correspondence can be strengthened to an equivalence of categories between certain additive subquotients of Gmodules and Hmodules. I will explain how the latter Green equivalence can be proved and clarified in the general setting of "Mackey 2functors", and how one can make sense of it also in situations where the KrullSchmidt theorem does not hold (where indecomposable objects are not helpful). For example, and perhaps most notably, there follows a version of the Green equivalence for equivariant stable homotopy theory. This is joint work with Paul Balmer.  Donnerstag, 20. Dezember 2019: Ellen Henke [University of Aberdeen, Schottland]
Titel: A generalization of a theorem by Camina—Herzog [Abstract]An old theorem of Camina—Herzog states that a finite group has abelian Sylow 2subgroups if the centralizer of any 2element is of odd index. After an introduction to the theory of fusion systems, I will present a generalization of this theorem to fusion systems together with a very short and elementary proof. As a corollary, one obtains a theorem about 2blocks of finite groups which was conjectured by Külshammer, Sambale and Tiep.  Donnerstag, 9. Januar 2020: Henri Mühle [TU Dresden]
Titel: TALK CANCELLED [Abstract]  Donnerstag, 23. Januar 2020: Jenny August [MPI Bonn]
Titel: Contraction Algebras in the Homological Minimal Model Programme [Abstract]In this talk, I will give an introduction to a class of symmetric, finite dimensional algebras, known as contraction algebras, which are one of the main tools used in the Homological Minimal Model Programme. This program aims to use noncommutative algebra to answer questions in algebraic geometry and my goal for the talk is to not only convince you that the contraction algebras are very useful for this purpose, but to also show you that they have some very interesting algebraic properties.  Donnerstag, 30. Januar 2020: Vincent Beck [Université de Tours, Université d'Orléans, Frankreich]
Titel: Cohomological computations for reflection groups and their braid groups and application to the determination of the finite subgroups of B/[P,P] [Abstract]In 2015, Guaschi, Goncalves and Ocampos started to study the finite subgroups of B_n/[P_n,P_n] where B_n is the standard braid group on $n$ strands and $P_n$ the pure braid group. They exhibited cyclic and abelian subgroup and even a non abelian subgroup of B_7/[P_7,P_7]. Their constructions were generalized to complex reflection groups by Marin. Finally, using cohomological methods, in a joint work with Marin, we were able to give a complete description of the finite subgroups of $B/[P,P]$ for every complex reflection groups $W$ (where $B$ is the braid group of $W$ and $P$ its pure braid group). I will present these results and how the cohomological description of the exact sequence 1> P/[P,P] > B/[P,P] > W > 1 has led to our classification of finite subgroups of B/[P,P].  Donnerstag, 6. Februar 2020: Maike Gruchot [RWTH Aachen]
Titel: On a relative version of complete reducibility [Abstract]n this talk we discuss a relative version of Serre’s notion of complete reducibility introduced by Bate, Martin, Röhrle and Rudolf. In the first part of the talk I introduce Serre’s notion of complete reducibility. In the second part, we introduce the relative version and discuss a characterization of relative Gcomplete reducibility which directly generalizes equivalent formulations of complete reducibility in representation theory. In the last part of the talk, we consider the case of a subgroup of G which normalizes the identity component K^0 of K, it turns that such a subgroup is relatively Gcompletely reducible with respect to K if and only if its image in the automorphism group of K^0 is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting.
Sommersemester 2018
Organisatoren: Niamh Farrell, Alessandro Paolini
 Donnerstag, 3. Mai 2018: Keivan MallahiKarai [Jacobs University, Bremen]
Titel: Kirillov's orbit method and the polynomiality of the essential dimension of pgroups [Abstract]The faithful dimension of a finite group G is defined to be the smallest dimension of a faithful complex representation of G. Aside from its intrinsic interest, the problem of determining the faithful dimension of finite groups is intimately related to the notion of essential dimension, introduced by Buhler and Reichstein.
The problem of determining the faithful dimension of families of pgroups arising from F_ppoints of a nilpotent algebraic group defined over the field of rational numbers has been studied in some special cases, e.g. the Heisenberg and the full uppertriangular unipotent group.
In this paper, we will use Kirillov’s orbit method to address this problem for a large family of groups. It will be shown that this function is always a piecewise polynomial function along certain “numbertheoretically defined” sets, while, in some specific cases, it is given by a uniform polynomial in p.
This talk is based on a joint work with Mohammad Bardestani and Hadi Salmasian.  Montag, 7. Mai 2018: Thomas Gobet [University of Sydney]
Titel: Dual and classical generators of Artin groups of spherical type [Abstract]The nstrand braid group has two Garside structures, a classical and a dual one, leading to different solutions to the word problem in the group. The corresponding Garside monoids are the classical braid monoid and the BirmanKoLee braid monoid. These monoids exist for any Artin group of spherical type (i.e., attached to a finite Coxeter group).
The aim of the talk is to explain how to express the generators of an Artin group of spherical type coming from the dual braid monoid (and more generally the simples, that is, the divisors of the Garside element of the dual monoid) in terms of the classical generators: we give a closed formula to express a simple of the dual braid monoid, using combinatorial objects called csortable elements (introduced by Reading). It gives a word of shortest possible length in the classical generators representing a simple element of the dual braid monoid. It has as an immediate consequence that the simples of the dual braid monoids are Mikado braids. 
Donnerstag, 17. Mai 2018: Zhicheng Feng [TU Kaiserslautern]
Titel: On the inductive blockwise Alperin weight condition for special linear and unitary groups [Abstract]The inductive blockwise Alperin weight condition is a system of conditions whose verification for all nonabelian finite simple groups would imply the blockwise Alperin weight conjecture. This talk will explain how one can obtain the weights of a special linear or unitary group X from the general linear or unitary group and verify the inductive blockwise Alperin weight condition for any unipotent block of X when the prime does not divide the order of Z(X).  Donnerstag, 7. Juni 2018: two seminars.
16.3017.30: İpek Tuvay [Mimar Sinan Fine Arts University, Istanbul]
Titel: Brauer indecomposability of Scott modules for the quadratic group Qd(p) [Abstract]Brauer indecomposability of Scott modules is important for constructing stable equivalences of Morita type between the principal blocks of two different finite groups. For an algebraically closed field k of prime characteristic p, we compute the Scott kGmodule with vertex P when F is a constrained fusion system on P and G is Park’s group for F. In the case F is a fusion system of the quadratic group on a Sylow psubgroup P of Qd(p) and G is Park’s group for F, we prove that the Scott kGmodule with vertex P is Brauer indecomposable. In this talk these results will be presented. This is a joint work with Shigeo Koshitani.
17.3018.30: Kivanç Ersoy [Freie Universität Berlin]
Titel: Groups with certain conditions on fixed points of automorphisms [Abstract]Thompson proved that a finite group with a fixed point free automorphism of prime order is nilpotent. Later Higman, Kreknin and Kostrikin proved that in this case the nilpotency class is indeed bounded by the order of the automorphism. On the other hand, Brauer and Fowler proved that order of a finite simple group is indeed bounded in terms of the order of centralizers of involutions. Since then, there has been a lot of questions of the following type: Let G be a group with a particular type of centralizer (of fixed point set). How does the structure of the centralizer affect the structure of the group? In the first part of the talk we will give a survey of results obtained by imposing certain conditions on the centralizers. In the second part we will prove new results about finite and locally finite groups and their automorphisms.  Donnerstag, 14. Juni 2018: Iulian Simion [BabeşBolyai University, ClujNapoca]
Titel: Minimal realizations of finite groups of isometries [Abstract]Let G be a finite group. The minimal dimension in which G can be realized by isometries is known to be the dimension of a minimal faithful real representation. The minimal number of points which realize the group as their full group of isometries appears to be less wellunderstood. We discuss some known bounds (this is work in progress).  Donnerstag, 21. Juni 2018: Alex Malcolm [Heilbronn Institute for Mathematics, Bristol]
Titel: Strong reality in finite simple groups: on products of classes and characters [Abstract]In a finite simple group, the strongly real elements are those that are conjugate to their inverse by an involution. A related notion exists in the ordinary representation theory of the group, where we call an irreducible character orthogonal if it is not only realvalued, but has an underlying real representation. It was recently shown that any element of a finite simple group decomposes as a product of two strongly real elements, motivating the question as to whether an analogous result holds for irreducible characters? I.e. does every irreducible representation appear in the tensor product of two real reps? We will discuss the methods used in answering both of these questions, as they range from the analysis of finite simple group substructure, to a classical problem of computing Kronecker coefficients in the symmetric group.  Donnerstag, 28. Juni 2018: Kein Seminar. Norddeutsches GruppentheorieKolloquium in Braunschweig, 29.06.1830.06.18.
 Donnerstag, 5. Juli 2018: Kein Seminar. Computational Lie Theory in Stuttgart, 05.07.1806.07.18.
 Donnerstag, 12. Juli 2018: Madeleine Whybrow [Imperial College London, UK]
Titel: Constructing Majorana representations [Abstract]Majorana theory is an axiomatic framework to study objects related to the Monster group and its 196884 dimensional representation, the Griess algebra. The objects at the centre of the theory are known as Majorana algebras and can be studied either in their own right, or as Majorana representations of certain group. Inspired by a paper of A. Seress, and joint with M. Pfeiffer, I have developed an algorithm in GAP to construct the Majorana representations of a given group. I will present the methods of this work as well as some consequences and results. I will also discuss future theoretical and computational approache to the study of Majorana and axial algebras.
Wintersemester 2017/18
Organisatoren: Caroline Lassueur, Alessandro Paolini
 Donnerstag, 26. Oktober 2017: Andrew Mathas [University of Sydney]
Titel: Jantzen filtrations and graded Specht modules [Abstract]The Jantzen sum formula is a classical result in the representation theory of the symmetric and general linear groups that describes a natural filtration of the Specht modules over any field. Analogues of this result exist for many algebras including the cyclotomic Hecke algebras of type A. Quite remarkably, the cyclotomic Hecke algebras of type A are now know to admit a Zgrading because they are isomorphic to cyclotomic KLR algebras. I will explain how to give an easy proof, and stronger formulation, of Jantzen sum formula for the cyclotomic Hecke algebras of type A using the KLR grading. I will discuss some consequences and applications of this approach.  Donnerstag, 02. November 2017: Caroline Lassueur [TU Kaiserslautern]
Titel: Lifting Morita equivalences with an endopermutation source [Abstract]Motivated by current work of BoltjeKessarLinckelmann on the Picard group of Morita selfequivalences with endopermutation source, we consider the problem of whether Morita equivalences with an endopermutation source are liftable from positive characteristic to characteristic zero in general. The first part of the talk will give a gentle introduction to the concepts of vertices, sources and multiplicity modules in modular representation theory, while the second part of the talk will focus on the aforementioned lifting problem.  Donnerstag, 09. November 2017: Kein Seminar (Darstellungstheorietage 2017, Eichstätt 10.11.1711.11.17).
 Donnerstag, 16. November 2017: Olivier Dudas [Université Paris Diderot  Paris 7]
Titel: On the unitriangular shape of decomposition matrices for finite reductive groups [Abstract]In order to classify the unipotent characters of finite reductive groups, Kawanaka constructed a family of representations induced from 'almost' unipotent groups, and conjectured a multiplicity formula for the unipotent constituents of these representations. I will explain how one can prove this formula providing some mild assumptions on unipotent classes. This can be used to show that the decomposition matrices of unipotent blocks have unitriangular shape. This is a joint work with O. Brunat and J. Taylor.  Donnerstag, 23. November 2017: Reda Chaneb [Université Paris Diderot  Paris 7]
Titel: Basic sets for unipotent blocks of finite reductive groups [Abstract]Let G(q) be a finite group of Lie type and l be a prime number not dividing q. Geck and Hiss have proved that, if l is good, the restriction of unipotent characters to lregular elements is a basic set for unipotent blocks of G(q). When l is bad, there are usually not enough unipotent characters to form a basic set. An alternative strategy to get a basic set is to construct projective characters satisfying some properties. For classical groups with connected center and l=2, by using a family of projective representations constructed by Kawanaka, Geck proved the existence of a basic set such that the decomposition matrix of the unipotent block has a unitriangular shape for this basic set. I will introduce those results and present generalizations of the results of Geck for the case of groups with disconnected center.  Donnerstag, 7. December 2017, Oberseminar Representation Theory : Emil Rotilio
Titel: Lie superalgebras in Physics [Abstract]The current understanding of nature finds in the „Standard Model“ the most complete and verified theory (for now). The mathematics it involves heavily relies on Lie theory (Lie groups and Lie algebras). To better describe the universe, phisicists have come up with a „Supersymmetry“ theory (among others). This theory is described in terms of Lie superalgebras. The goal of this talk is to give an overview of which Lie algebras/superalgebras are used in Physics and why they help describing nature.  Donnerstag, 14. December 2017: Stefano Sannella [University of Birmingham]
Titel: Broué's conjecture and perverse equivalences [Abstract]The representation theory of a finite group G over a field F of positive characteristic carries many questions that have not been answered yet. Most of them can be stated as global/local conjectures: in various forms, they state that the representation theory of G is somehow controlled by its plocal subgroups. Here we will mostly focus on one of these conjectures, Broué's Abelian Defect Group Conjecture, which might be considered as an attempt to give a structural explanation of what is actually connecting G and its local psubgroups in the abelian defect case. In particular, we explain how the strategy of looking for a perverse equivalence (a specific type of derived equivalence) works successfully in some cases and how this procedure is related to some DeligneLusztig varieties.  Donnerstag, 21. December 2017: Patrick Wegener [TU Kaiserslautern]
Titel: Hurwitz action in elliptic Weyl groups and coherent sheaves on a weighted projective line [Abstract]Since 2000 the poset of noncrossing partitions attached to a Coxeter group (independently introduced by Bessis and BradyWatt) has gained a lot of attention from different areas of mathematics. In 2010 Igusa, Schiffler and Thomas showed that there exists an order preserving bijection between this poset and the set of thick subcategories in the derived category of mod(A) generated by an exceptional sequence, where A is a hereditary Artin algebra. Following Happel's classification of hereditary categories, it seems natural to ask if there is an analogous statement when replacing mod(A) by the category of coherent sheaves on a weighted projective line. I will give a short summary on hereditary categories, explain how elliptic Weyl groups show up in this context and then generalize the result of IgusaSchifflerThomas to tubular weighted projective lines. (This is joint work with B. Baumeister and S. Yahiatene.)  Donnerstag, 18. Januar 2018: Florian Eisele [City, University of London]
Titel: A counterexample to the first Zassenhaus conjecture [Abstract]There are many interesting problems surrounding the unit group U(RG) of the ring RG, where R is a commutative ring and G is a finite group. Of particular interest are the finite subgroups of U(RG). In the seventies, Zassenhaus conjectured that any u in U(ZG) is conjugate, in the group U(QG), to an element of the form +/g, where g is an element of the group G. This came to be known as the "Zassenhaus conjecture". In recent joint work with L. Margolis, we were able to construct a counterexample to this conjecture. In this talk I will give an introduction to the various conjectures surrounding finite subgroups of U(RG), and how they can be reinterpreted as questions on the (non)existence of certain R(GxH)modules, where H is another finite group. This establishes a link with the representation theory of finite groups, and I will explain how, plocally, our example is made up of certain ppermutation modules.  Donnerstag, 25. Januar 2018: Petra Schwer [Karlsruher Institut für Technologie]
Titel: Reflection length in affine Coxeter groups [Abstract]Affine Coxeter groups have a natural presentation as reflection groups on some affine space. Hence the set R of all its reflections, that is all conjugates of its standard generators, is a natural (infinite) set of generators. Computing the reflection length of an element in an affine Coxeter group means that one wants to determine the length of a minimal presentation of this element with respect to R. In joint work with Joel Brewster Lewis, Jon McCammond and T. Kyle Petersen we were able to provide a simple formula that computes the reflection length of any element in any affine Coxeter group. In this talk I would like to explain this formula, give its simple uniform proof and allude to the geometric intuition behind it.
Sommersemester 2017
Organisatoren: Alessandro Paolini
 Donnerstag, 27. April 2017: Kivanç Ersoy [Technische Universität Kaiserslautern]
Titel: Locally finite groups with small centralizers [Abstract]In this talk we will present recent results about fixed points of automorphisms in locally finite groups. Let $p$ be a prime and $G$ a locally finite group containing an elementary abelian $p$subgroup $A$ of rank at least $3$ such that $C_G(A)$ is Chernikov and $C_G(a)$ involves no infinite simple groups for any $a \in A^{\#}$. We show that $G$ is almost locally soluble. To prove this result, we first give a characterization of $PSL_p(k)$:
Theorem. [1] An infinite simple locally finite group $G$ admits an elementary abelian $p$group of automorphisms $A$ such that $C_G(A)$ is Chernikov and $C_G(a)$ involves no infinite simple groups for any $a \in A^{\#}$ if and only if $G$ is isomorphic to $PSL_p(k)$ for some locally finite field $k$ of characteristic different from $p$ and $A$ has order $p^2$. This is a joint work with Mahmut Kuzucuoglu and Pavel Shumyatsky.
[1] Ersoy, K., Kuzucuoglu, M., Shumyatsky, P., Locally finite groups and their subgroups with small centralizers, J. Algebra, Vol. 481, 1 July 2017, p.111.  Donnerstag, 4. Mai 2017: Emily Norton [Max Planck Institute Bonn]
Titel: The \mathfrak{sl}_\inftycrystal combinatorics of higher level Fock spaces [Abstract]A crystal is a type of directed graph encoding representation theoretic information. The crystal I will discuss is called the Heisenberg or $\mathfrak{sl}_\infty$ crystal, originally defined by Shan and Vasserot. Its vertices are multipartitions, and its arrows arise from a categorical action of a Heisenberg algebra on cyclotomic Cherednik category O. The representation theoretic meaning of this crystal is to keep track of one part of the support of simple modules. There are two crystals needed to determine supports; the other is the $\widehat{\mathfrak{sl}_e}$ crystal. Likewise, DudasVaragnoloVasserot recently constructed categorical actions of these two crystals on the unipotent category of finite classical groups in order to classify HarishChandra series. The problem of computing the arrows in the $\mathfrak{sl}_\infty$ crystal was reduced to a combinatorial problem by Thomas Gerber. I will explain the solution to this problem: the rule for the arrows, and the rule for determining depth of a multipartition in the crystal. This is joint work with Thomas Gerber.  Donnerstag, 11. Mai 2017: Andreas Bächle [Vrije Universiteit Brussel]
Titel: Rationality of groups and centers of integral group rings [Abstract]A finite group is called rational if all entries of its character table are rational integers. Being rational has significant implications for the structure of the group, e.g. it is a classical result of R. Gow that the only primes dividing the order of such a group are 2, 3 and 5, if the group is solvable. The concept of rationality was generalized in 2010 by D. Chillag and S. Dolfi by introducing the term (inverse) semirational group. It turned out that being an inverse semirational group has quite some impact in the study of integral group rings. We will discuss this connection and recent results.
 Donnerstag, 08. Juni 2017: William Wong [Technische Universität Kaiserslautern]
Titel: A phenomenon in the representation of SL(2,q) in defining characteristics [Abstract]I will talk about my PhD research, which uncovers some properties of modules of SL(2,q) in defining characteristics. It heavily depends on information from representations of its Borel subgroup, which is equivalent to the normaliser of the defect group in this case. In this talk I will present the results using combinatorial properties in the local representation.  Dienstag, 13. Juni 2017: Wolfgang Willems [OttovonGuerickeUniversität Magdeburg]
Titel: On quasiprojective Brauer characters [Abstract]We study pBrauer characters of a finite group G which are restrictions of generalized characters vanishing on psingular elements for a fixed prime p dividing the order of G. Such Brauer characters are called quasiprojective. We show that for each irreducible Brauer character there exists a minimal ppower, say $p^{a(\varphi)}$, such that $p^{a(\varphi)}\varphi$ is quasiprojective. The exponent $a(\varphi)$ only depends on the Cartan matrix of the block to which $\varphi$ belongs. Moreover $p^{a(\varphi)}$ is bounded by the vertex of the module afforded by $\varphi$, and equality holds in case that G is psolvable. We give some evidence for the conjecture that $a(\varphi)=0$ occurs if and only if $\varphi$ belongs to a block of defect 0. Finally, we study indecomposable quasiprojective Brauer characters of a block B. This set is finite and corresponds to a minimal Hilbert basis of the rational cone defined by the Cartan matrix of B.  Donnerstag, 22. Juni 2017: Baptiste Rognerud [IRMA, Straßburg]
Titel: A Morita theory for permutation modules [Abstract]It is known that the trivial source modules (or ppermutation modules) over a block of group algebra share a lot of similarities with the projective modules. For example, there are only finitely many of them, and working over a pmodular system, any trivial source module over the field of positive characteristic lifts uniquely to a trivial source module over the valuation ring. It has also been shown by Arnold that it is possible to do homological algebra with this family of modules.
The aim of the talk is to explain what happens when you replace projective modules by trivial source modules in the classical Morita theory between blocks of group algebras. This is a joint work with Markus Linckelmann.  Donnerstag, 29. Juni 2017: Julian Külshammer [Universität Stuttgart]
Titel: Ringel duality as a special case of Koszul duality [Abstract]Quasihereditary algebras are a class of finite dimensional algebras which occur frequently in representation theory. Prominent examples are blocks of BGG category O as well as Schur algebras of reductive algebraic groups. In 1991, Ringel constructed a duality on the class of quasihereditary algebras using a characteristic tilting module. Koszul duality dates back to a 1978 paper by Bernstein, Gelfand, and Gelfand which provides an equivalence between the bounded derived categories of a symmetric and an exterior algebra. In this talk, we show how to interpret Ringel duality as a special case of Koszul duality for differential graded algebras. This is joint work with Agnieszka Bodzenta which is based on ideas of Sergiy Ovsienko.
 Mittwoch, 05. Juli 2017: Tung Le [University of Pretoria]
Titel: On the automorphisms of designs constructed from finite simple groups [Abstract](Joint work with J. Moori) We study the automorphism groups of 1designs constructed from finite nonabelian simple groups by using two methods presented in Moori (Information Security, Coding Theory and Related Combinatorics, 2011). We obtain some general results on the automorphism groups from both methods, and improve one of these methods.  Donnerstag, 13. Juli 2017: Kein Seminar ("Representation Theory in Kaiserslautern" Konferenz).
 Donnerstag, 20. Juli 2017: Yujiao Sun [Universität Stuttgart]
Titel: Supercharacter theories for Sylow psubgroups of finite exceptional groups of Lie type [Abstract]Classifying the conjugacy classes of the full unitriangular groups U_n(q) for all n and q is known to be a "wild" problem, where q is a power of some prime p. C.A.M. Andre using Kirillov's orbit method, and later N. Yan using a more elementary method constructed supercharacters and superclasses for U_n(q) as approximations of irreducible characters and conjugacy classes. One may conjecture that similar results can be obtained for Sylow psubgroups of all finite groups of Lie type. My PhD thesis was concerned with testing this for some exceptional groups of Lie type. I will present some of the results in this talk.  *Freitag, 25. August 2017, 14:0015:00*: Shigeo Koshitani [University of Chiba]
Titel: Brauer indecomposability of AlperinScott modules for finite nonabelian 2groups [Abstract]We will be discussing the socalled AlperinScott module for the group algebra kG of a finite group G over a field k of characteristic p>0. It is a ppermutation indecomposable kGmodule such that it has the trivial module k in the socle with multiplicity one. Brauer indecomposabitily is considered in the 2011 paper by KessarKunugiMitsuhashi, and it is useful to get for instance a stable equivalence of Morita type in order to prove Broue's abelian defect group conjecture. In the above paper they prove the Brauer indecomposability for abelian pgroups. So as next step we will consider nonabelian 2groups cases.
(*) This is a part of joint work with Caroline Lassueur.
Wintersemester 2016/17
Organisatoren: Julian Brough, Caroline Lassueur
 Donnerstag, 3. November 2016: Yanjun Liu [TU Kaiserslautern]
Titel: Principal and defectzero blocks of finite groups [Abstract]This talk will consist of two parts. The first one is about the intersection of principal blocks of a finite group. Recently, the block distributions of complex irreducible characters across distinct primes were investigated by several experts. For instance, C. Bessenrodt and J. Zhang showed that the property that any two principal blocks intersect trivially is equivalent to the nilpotency of a finite group. Based on this result, we will talk about a criterion of solvability in terms of small intersections of any two principal blocks of a finite group. The second part is about the existence of blocks of defect zero. We will pay attention to the situation that all subgroups of a finite groups with a given prime order are conjugate.
 Donnerstag, 24. November 2016: Niamh Farrell [TU Kaiserslautern/City, University of London]
Titel: The rationality of blocks of quasisimple finite groups [Abstract]The Morita Frobenius number of an algebra is the number of Morita equivalence classes of its Frobenius twists. They were introduced by Kessar in 2004 the context of Donovan’s Conjecture. We aim to calculate the Morita Frobenius numbers of all blocks of group algebras of quasisimple finite groups. I will present the latest results and discuss how they relate to Donovan’s Conjecture.
 Freitag, 25. November 2016: Jesper Grodal [Københavns Universitet]
Titel: Endotrivial modules via homotopy theory [Abstract]In the modular representation theory of finite groups, a special class of interesting modules are those indecomposable kGmodules that upon restriction to a Sylow psubgroup split as the trivial module plus a projective module. Isomorphism classes of these form a group under tensor product, a key subgroup inside the group of all socalled endotrivial modules, whose classification has been a longterm goal in modular representation theory. In my talk I'll show how this subgroup can be calculated via homotopy theory  the talk will be a gentle introduction to the preprint arXiv:1608.00499.
 Donnerstag, 1. Dezember 2016: Emilio Pierro [Universität Bielefeld]
Titel: Quantities associated to the subgroup lattice [Abstract]We begin with the Möbius function of a finite group $G$ and how this can be used to determine Eulerian functions of $G$. These are natural generalisations of the Euler totient function $\phi(n)$ when viewed as counting generators in the cyclic group of size $n$. In the case of the small Ree groups we use this to give new results and new proofs of old results on their generation and asymptotic generation. We also discuss the interpretation of the Möbius function and Eulerian functions from the perspective of topology.  Donnerstag, 8. Dezember 2016: Kein Seminar: 9.10. Dez. Nikolauskonferenz
 Donnerstag, 15. Dezember 2016: Kein Seminar: Konferenz "Finite simple groups, their fusion systems and representations", EPFL
 Donnerstag, 19. Januar 2017: Iris Köster [Universität Stuttgart]
Titel: Sylow Numbers in Character Tables and Integral Group Rings [Abstract]The Sylow pnumber of a finite group is defined as the number of Sylow psubgroups of the given group. In this talk we want to consider a question of G. Navarro whether character tables determine the Sylow numbers of the underlying group. In the second part of the talk we analyze the question whether the integral group rings determines the Sylow numbers.  Donnerstag, 26. Januar 2017: Kein Seminar: 27.28. Jan. Darstellungstheorietage 2016
 Montag, 6. Februar 2017: Magdalena Boos [RUHR Universtität Bochum]
Titel: Using quiver representations to prove finiteness criteria [Abstract]It is a natural question to ask if a given group action is finite (has only finitely many orbits) or not. In this session, we show how the theory of quiver representations of certain finitedimensional algebras can be used to prove finiteness or infiniteness of some particular actions.  Donnerstag, 9. Februar 2017: Charles Eaton [University of Manchester]
Titel: Loewy lengths of blocks [Abstract]I will give a brief survey on results and conjectures about the Loewy length of a block of a finite group. This includes work with Michael Livesey giving precise upper and lower bounds for 2blocks with abelian defect groups  Mittwoch, 8. März 2017: Michael Geline [Northern Illinois University]
Titel: Some quantitative and qualitative aspects of Knörr lattices [Abstract]These lattices are the $p$adic representations of finite groups whose invertible endomorphisms can be distinguished by the trace function. They include the absolutely irreducible lattices as well as the indecomposable lattices of rank not divisible by $p$. Knörr introduced them in a 1988 paper in connection with Brauer's height zero conjecture. They are now known also to be connected with Alperin's weight conjecture. I will explain this in the talk, and, time permitting, also give a quantitative result: Knörr lattices for the elementary abelian group of order $8$, over an dvr unramified over the $2$adics, do not exist.
Sommersemester 2016
Organisatoren: Thomas Gobet, Caroline Lassueur
 Donnerstag, 21. April 2016, 15:3016:30, Raum 48582: Paul Balmer [UCLA, Bielefeld]
Titel: A quick tour of tensor triangular geometry. [Abstract]We shall recall the motivation to study tensor triangulated categories through the geometric invariant called the "spectrum". We shall then see how some of the wellknown classification results due to HopkinsSmith, Neeman, Thomason, BensonCarlsonRickard, and others, can be elegantly expressed using that spectrum. Finally, we shall see how new techniques of separable extensions of tensortriangulated categories allow to approach new computations of such spectra, thus obtaining new classification results, for instance in equivariant stable homotopy theory.  Mittwoch, 11. Mai 2016, 17:0018:00, Raum 48436: Jay Taylor [Padova]
Titel: Sums of Skew Characters of Symmetric Groups. [Abstract]In previous work Regev considered the character of the symmetric group which is the sum of all irreducible characters labelled by hook partitions. He showed that the values of were particularly simple. Namely, we have is unless is a product of nontrivial oddlength cycles, in which case . To prove this result Regev considered a character , which is defined for any integers , and proved a surprising identity for its values. The character turns out to be so one easily obtains the previous result from this more general case. Regev's proof of the formula for the values of the character relies, in a crucial way, on the representation theory of Lie superalgebras. In this talk we will present a proof of Regev's formula for the values of the character which relies only on techniques from the representation theory of the symmetric group.  Donnerstag, 12. May 2016, 17:0018:00, Raum 48436: Christopher Voll [Bielefeld]
6 Titel: Representation growth of finitely generated nilpotent groups. [Abstract]A finitely generated nilpotent group has only finitely many irreducible complex representations of each finite dimension up to twists by onedimensional representations. The study of the representation growth of such groups aims to understand the asymptotic and arithmetic features of the resulting arithmetic functions.  Donnerstag, 2. Juni 2016, 17:0018:00, Raum 48436: René Reichenbach [Jena]
6 Titel: The BrouéPuig \(\ast\)construction and some applications to blocks. [Abstract]The \(\ast\)construction, as introduced by Broué and Puig, gives a method to construct generalized characters of a block of a finite group using generalized characters of the defect group of which are stable relative to the fusion system. Given only the defect group and the fusion system of , in many cases this method yields strong restrictions for the block invariants , , as well as the Cartan matrix of . In this talk we describe some of these cases if . We will focus on defect groups which are groups with exactly three involutions and nontrivial automorphisms of odd order or particular sections of such groups.  Donnerstag, 16. Juni 2016, 17:0018:00, Raum 48436: AnneLaure Thiel [Dijon]
6 Titel: Categorical action of the braid group of the cylinder. [Abstract]In their seminal work, Khovanov and Seidel have used the polymorphous nature of the usual braid group (ie of finite type A) and hence its various definitions (diagrammatic presentation, mapping class group...) to construct a categorical action of this group which categorifies its famous Burau linear representation. The important fact is that this action detects subtle topological properties which ensures its faithfulness unlike the Burau linear representation. The aim of this talk is to describe a generalization of their approach to another Artin group, the one of type B  aka braid group of the cylinder or extended braid group of affine type A. Joint work with Agnès Gadbled and Emmanuel Wagner.  Donnerstags, 23. Juni 2016, 17:0018:00, Raum 48436: Christian Stump [FU Berlin]
6 Titel: Free reflection multiarrangements and their surprising exponents. [Abstract]The module of derivations of a multiarrangements of hyperplanes was introduced by Ziegler in the 1980's. Maybe the most important example is the situation of Coxeter arrangements with constant multiplicities resolved by Terao in 2002 by showing that these are free with exponents given uniformly in terms of Coxeter numbers and invariant degrees. In this talk, I plan to present a (partially conjectured) generalization of this result to all complex reflection group, using the most general notion of Coxeter numbers and an operator on irreducible components introduced by Malle in the context of the associated cyclotomic Hecke algebra. If time permits, I will try to also give a brief science fiction detour through the combinatorics of Catalan and Shi arrangements for finite Weyl groups and their resistance of being generalized to more general reflection groups. This talk is based on results in an ongoing collaboration with Torsten Hoge and Gerhard Röhrle.  Donnerstag, 7. Juli 2016, 16:0017:00, Raum 48519: Alastair Litterick [Bielefeld]
Titel: Rigidity and triangle generation of simple groups [Abstract]A group is called an group if it is generated by a pair of elements of orders and , whose product has order . A 2010 conjecture of Claude Marion roughly states that, if is a simple algebraic group whose elements of orders and satisfy a certain dimension condition, then only finitely many of the corresponding groups are groups. This was proved 'generically' by Larsen, Lubotzky and Marion in 2014 using deformation theory, but the remaining cases are quite interesting, and have required a variety of different approaches. We will discuss a few of these.
Wintersemester 2015/16
Organisatoren: Thomas Gobet, Caroline Lassueur
 Donnerstag, 15. Oktober 2015, 16:0017:00, Raum 48436: Mikaël Cavallin [EPFL, Ecublens, CH]
Titel: Restriction of irreducibles and structure of Weyl modules [Abstract]Let $K$ be an algebraically closed field of positive characteristic $p>0$ and consider the usual embedding of $X=\mbox{SO}_{2n}(K)$ in $Y=\mbox{SL}_{2n}(K).$
In this talk, we study certain families of irreducible $KY$modules, on which $X$ acts with at most three composition factors. We then see how this help to better understand the structure of those composition factors and the corresponding Weyl modules, as well as other consequences.
(⚠ Termin in der Vorlesungsfreie Zeit!)  Donnerstag, 29. Oktober 2015: Philipp Perepelitsky [TU Kaiserslautern]
Titel: \(p\)permutation equivalences between blocks of finite groups [Abstract]  Donnerstag, 5. November 2015: Benjamin Sambale [TU Kaiserslautern]
Titel: Broué's Conjecture for sporadic groups  Donnerstag, 12. November 2015: Neil Saunders [EPFL, Ecublens, CH]
Titel: Irreducible Components of Springer Fibres [Abstract]The celebrated correspondence by Springer gives a bijection between the irreducible rep resentations of the Weyl group of a (connected, reductive) algebraic group and a certain subset of perverse sheaves on the nilpotent cone. Looking at the Springer resolution of the nilpotent cone, one sees the irreducible representations of the Weyl group in the top degree cohomology of these socalled “Springer Fibres”. The geometry of these Springer Fibres can be quite subtle, but often admit nice combinatorial descriptions. In this talk, I will review the work of Steinberg and Spaltenstein in Type A and report on joint work with Vinoth Nandakumar and Daniele Rosso in Type C.  Donnerstag, 19. November 2015: Thomas Gobet [TU Kaiserslautern]
Titel: On twisted filtrations on Soergel bimodules and linear Rouquier complexes [Abstract]The IwahoriHecke algebra of a Coxeter group has a standard and a costandard basis, as well as two canonical bases. If the Coxeter group is finite, it was shown by Dyer that the product of an element of the canonical basis with an element of the standard basis has positive coefficients when expressed in the standard basis. Using Dyer’s notion of biclosed sets of reflections, we consider a family of bases containing both the standard and costandard bases and show that an element of the canonical basis has a positive expansion in any basis from this family. The key tool for this is to consider twisted filtrations on Soergel bimodules (these bimodules categorify the canonical basis of the Hecke algebra) and interpret the coefficients as multiplicities in these filtrations. This generalizes Dyer’s result to a more general family of bases as well as to arbitrary Coxeter groups. Elements of these bases turn out to be images of "mikado braids" as introduced in a joint work with F. Digne. It time allows, we will mention a conjecture on the Rouquier complexes of these braids, which would imply a generalized "inverse" KazhdanLusztig positivity, and prove it in case the biclosed set comes from an inversion set of an element or its complement. This generalizes another positivity result of Dyer and Lehrer to arbitrary Coxeter groups.  Mittwoch, 25. November 2015: Olivier Dudas [University Paris Diderot  Paris 7, FR]
Titel: On the unitriangular shape of decomposition matrices for finite reductive groups [Abstract]In a work in progress with O. Brunat and J. Taylor, we study a family of characters of finite groups of Lie type defined by Kawanaka. These are projective characters which vanish on many conjugacy classes. We will explain how this property can be used to determine the multiplicity of some of their unipotent constituents, and lead to a proof of the unitriangular shape of the decomposition matrices of unipotent blocks.  Donnerstag, 26. November 2015: Chris Bowman [City University, London, UK]
Titel: The coPieri rule for Kronecker coefficients [Abstract]A central problem in algebraic combinatorics is to provide an algorithm for calculating the coefficients arising in the decomposition of a tensor product of two simple representations of the symmetric group. The coefficients in such a decomposition are known as the “Kronecker coefficients”; these coefficients include the Littlewood—Richardson coefficients as a special case. In this subcase, the solution to the problem takes the form of a tableaux counting algorithm known as the Littlewood—Richardson rule.
The ultimate goal in this area is to generalise the Littlewood—Richardson rule to the general case. We shall discuss recent work with Maud De Visscher and John Enyang in which this problem is solved for Kronecker coefficients labelled by “coPieri triples" of partitions.  Donnerstag, 3. Dezember 2015: Benedetta Lancellotti [Università degli Studi di MilanoBicocca, IT]
Titel: Trivial source lattices and a strong form of McKay conjecture [Abstract, Abstract.pdf]In a preprint, Shigeo Koshitani and Thomas Weigel tried to link McKay's conjecture for a finite group $G$ and a prime number $p$ with indecomposable linear source \(\mathcal{O}G\)modules $L$ of maximal vertex, where $\mathcal{O}$ is a suitable discrete valuation ring of characteristic $0$ with residue field of characteristic $p$ and quotient field $\mathbb{K}$. Although it is known that the associated $\mathbb{K}G$module $L_{\mathbb{K}}=L\otimes_{\mathcal{O}}\mathbb{K}$ is not irreducible in general, they investigated some examples showing that, in some sense, it is not too far away from being irreducible. One may expect that there exists a sign $\varepsilon_L\in\{\pm1\}$ and an irreducible $\mathbb{K}G$module $M_L$ such that $\varepsilon_L\cdot[L]_{\mathbb{K}}+[R_L]_\mathbb{K}=M_L$, where $R_L$ is an element in the ideal of the Grothendieck ring of linear source modules spanned by the indecomposable linear source modules which do not have maximal vertex. By choosing the sign $\varepsilon_L$, one can define an injective map from the set of isomorphism types of irreducible $\mathbb{K} N_G(P)/P^\prime$modules to $\mathrm{Irr}(G)$, where $P\in\mathrm{Syl}_p(G)$. In this talk I will present some details about this link and we will discuss this map in case that $G$ has a Sylow $p$subgroup of order $p$.  Mittwoch, 9. Dezember 2015, 16:0017:00, Raum 48582: Patrick Wegener [Universität Bielefeld]
Titel: Hurwitz action in Coxeter Groups [Abstract]  Donnerstag, 17. Dezember 2015: Oberseminar Complex Reflection Groups
 Donnerstag, 14. Januar 2016: Alessandro Paolini [University of Birmingham, UK]
Titel: A reduction of characters of Sylow \(p\)subgroups of finite groups of Lie type [Abstract]Let \(G\) be a split finite group of Lie type over \(\mathbb{F}_q\), where \(q\) is a power of a prime \(p\). The problem of determining the irreducible complex characters of a Sylow \(p\)subgroup \(U\) of \(G\) has attracted a lot of interest for many years, with motivation going back to the work of Higman in the 1960's. This has applications to the representation theory of \(G\), in particular to determine its decomposition numbers.\par The aim of this talk is to discuss a reduction procedure of irreducible complex characters of \(U\), applicable when \(p\) is not a very bad prime for \(G\). This process, first introduced by Himstedt, Le and Magaard to parametrize the irreducible characters of \(U\) in type \(\mathrm{D}_4\), has recently been developed in a joint work with Goodwin, Le and Magaard, to construct all irreducible characters of \(U\) in any type and rank at most \(4\). In particular, the parameterization in type \(\mathrm{F}_4\) is “uniform” over all primes \(p > 3\), where all character degrees are of the form \(q^d\) for some integer \(d\), but this does not happen for the bad prime \(p = 3\).  Donnerstag, 28. Januar 2016: Oliver Goodbourn [RuhrUniversität Bochum]
Titel: Reductive pairs from representations of algebraic groups [Abstract]Reductive pairs are a class of nice embeddings of algebraic groups. They have been used to salvage some good behaviour observed in characteristic 0 in the positive characteristic case, for instance relating to Serre's notion of Gcomplete reducibility, and also in providing uniform proofs of otherwise technical results. I will discuss research 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.  Donnerstag, 4. Februar 2016: Julian Brough [TU Kaiserslautern]
Titel: Determining finite group structure from the size of the conjugacy classes [Abstract]The conjugacy class sizes form a natural set of integers that can be associated to a finite group and much research has been carried out since the work of Sylow about how this set can be used to determine finite group structure. In this talk I will present two different types of result. First I will discuss finite groups in which, for a prime $p$, all $p$elements have conjugacy class size not divisible by $p$. In particular, I will highlight when this condition implies the group has an abelian Sylow $p$subgroup; this has ties to Brauer's problem of using the character table to determine if a group has an abelian Sylow $p$subgroup. Second I shall introduce the notion of vanishing conjugacy classes, which is a property that is defined using the character table. I will then present some new results which show that certain structural properties deduced from conjugacy class sizes can in fact be obtained by only considering the vanishing conjugacy class sizes.  Donnerstag, 11. Februar 2016: John Murray [Maynooth University, IR]
Titel: Strong and weak reality and principal indecomposable modules in characteristic 2 [Abstract](joint with R. Gow, University College Dublin) Brauer proved that the number of principal indecomposable modules (pims) of a finite group $G$ over a field of characteristic $p$ equals the number of $p$regular conjugacy classes of $G$. Moreover the number of selfdual pims equals the number of real $p$regular conjugacy classes. Suppose that $p\ne2$. Then each selfdual pim affords either a quadratic or a symplectic geometry. W. Willems and J. Thompson independently showed that the type is detected by the FrobeniusSchur (FS)indicator of a real irreducible character which occurs with odd multiplicity in the principal indecomposable character. Now suppose that $p=2$. Then each selfdual pim has a quadratic geometry or no discernible geometry. R. Gow and W. Willems showed how to determine the type using the action of involutions on primitive idempotents. Recently R. Gow and I have developed a charactertheoretic criterion for the type of a pim. Recall that a real class is strongly real if its elements are inverted by involutions and is weakly real otherwise. We detect the type using character values on strong and weak 2regular classes. As a consequence we can show that $\#$ quadratic type pims $=\#$ strongly real 2regular classes. Brauer's problem 14 asks for a determination of the numbers of irreducible characters with a given FSindicator in grouptheoretic terms. We conjecture that $\#$ weakly real 2regular classes $\leq\#$ FS indicator $1$ irreducible characters.
Sommersemester 2015
Organisatoren: Caroline Lassueur, Britta Späth
 Donnerstag, 23. April, 15:3016:30, Raum 48582: Ulrike Faltings [TU Kaiserslautern]
Titel: On the decomposition numbers of \(F_4(q)\)for \(q\) even [Abstract]For odd prime powers \(q\), a lot is already known about the decomposition matrix of the Chevalley group \(F_4(q)\). However, for \(q\) even, this is not the case. I wish to find out something about the shape of the decomposition matrix for \(q\) a power of \(2\).  Donnerstag, 30. April, 15:3016:30, Raum 48582: Carolina Vallejo [Universitat de València]
Titel: Coprime action and Brauer characters [Abstract]Let \(A\) and \(G\) be finite groups. If \(A\) acts coprimely on \(G\), it is wellknown that there is a natural correspondence between the irreducible \(A\)invariant characters of \(G\) and the irreducible characters of \(C_G(A)\), the fixed point subgroup of the action. In the modular case, no natural correspondence is known in general. Indeed, it is an open conjecture that the number of irreducible \(A\)invariant Brauer characters of \(G\) coincides with the number irreducible Brauer characters of \(C_G(A)\). We reduce this conjecture to a question on finite simple groups. (Joint work with B. Spaeth).  Dienstag, 12. Mai, 15:3016:30, Raum 48436: Danny Neftin [University of Michigan, Ann Arbor]
Titel: Monodromy groups of rational functions [Abstract]A fundamental invariant associated to every rational function is its monodromy group. We shall discuss the accumulating group theoretic work towards determining the complex rational functions with a specified monodromy group, and its applications to problems concerning values of rational functions over finite fields, Hilbert irreducibility, Nevanlinna theory, and Mazur's torsion theorem.  Donnerstag, 21. Mai, 15:3016:30, Raum 48519: Vivien Ripoll [Universität Wien]
Titel: Coxeter elements in wellgenerated complex reflection groups [Abstract]Coxeter elements are specific elements in a reflection group W, that play a key role in CoxeterCatalan combinatorics, in particular in the construction of the Wnoncrossing partition lattice. Several nonequivalent definitions coexist in the literature. I will explain and motivate the following: for W an irreducible real reflection group, or a wellgenerated complex reflection group, a Coxeter element in W is an element having an eigenvalue of order h (where h is the highest invariant degree of W, also known as Coxeter number). The set of these elements does not always form a unique conjugacy class, but it forms a unique orbit under the action of reflection automorphisms of W. Using a Galois action on these elements, we obtain that the number of conjugacy classes is equal to the degree of the field of definition of W. When W is real, c is a Coxeter element if and only if there exists a simple system S constituted of reflections and such that c is the product of the elements in S. Finally I will explain how some of the properties of these Coxeter elements can be extended to Springerregular elements. (Joint work with Vic Reiner and Christian Stump)  Donnerstag, 15:301630, 11. Juni: Amanda SchaefferFry [Metropolitan State University Denver]
Titel: Finite groups with an irreducible character of large degree [Abstract](This is joint work with Hung Nguyen (Univ. Akron) and Mark Lewis (Kent State Univ).) Let $G$ be a finite group, $d$ an irreducible character degree of $G$, and write $G=d(d+e).$ N. Snyder observed that $G$ is bounded in terms of e whenever $e>1$, which consequently led to a number of papers by various authors working to improve Snyder's original bound. We prove the (best possible) bound $G\leq e^4e^3 .$ Moreover, when $G$ is nonabelian simple, we improve this bound to $G \leq 2e^2 .$  Donnerstag, 18. Juni, 17:0018:00, Raum 48436: Thomas Gerber [RWTH Aachen]
Titel: Weak HarishChandra theory for finite unitary groups [Abstract]HarishChandra theory is a classical tool to label the simple modules of a finite group of Lie type $G$, in nondefining characteristic. In this talk, we focus on the case where $G$ is a finite unitary group of index $n$, and we aim at determining the HarishChandra series, by slightly modifying the classical theory.
After introducing the HarishChandra branching graph for unipotent $G$modules on the one hand, and the crystal graph of the combinatorial Fock space on the other hand, we explain how these two graphs are related, and what new information this brings. This is part of joint works with Gerhard Hiss and Nicolas Jacon.  Donnerstag, 25. Juni, 15:3016:15: Matthias Klupsch [RWTH Aachen]
Titel: Supercuspidal modules of finite reductive groups [Abstract]HarishChandra induction is a very useful tool in the representation theory of finite reductive groups (finite groups of Lie type) as this refined form of induction allows the construction of representations of a finite reductive group from its Levi subgroups.
The cuspidal (supercuspidal) modules form the part of the representation theory which remains undetected when looking only at the submodules (composition factors) of HarishChandra induced modules. As the complementary part can essentially be understood using the representation theory of the proper Levi subgroups, the classification of cuspidal and supercuspidal modules is of considerable interest.
In my talk I will show how to reduce the problem of classifying the supercuspidal modules to the case where we only have to consider the finite redutive groups coming from simple simply connected algebraic groups.  Donnerstag, 16. Juli, 17:0018:00, Seminarraum 48436: Oksana Yakimova [FriedrichSchillerUniversität Jena]
Titel: On symmetric invariants of semidirect products [Abstract]Let $\mathfrak g$ be a complex reductive Lie algebra. By the Chevalley restriction theorem, the subalgebra of symmetric invariants $S(\mathfrak g)^{\mathfrak g}$ is a polynomial ring in rank $\mathfrak g$ variables. A quest for nonreductive Lie algebras with a similar property has recently become a trend in invariant theory. Several classes have been suggested, centralisers of nilpotent elements (Premet's conjecture), truncated biparabolic subalgebras (Joseph's conjecture), $\mathbb Z_2$contractions (Panyushev's conjecture). We will discuss results on symmetric invariants of semidirect products and their relevance to these conjectures.  Donnerstag, 23. Juli: Kein Seminar
Konferenz Blocks of Finite Groups and Beyond (Jena)
Wintersemester 2014/15
Organisatoren: Caroline Lassueur
 Donnerstag, 2. Oktober: Shigeo Koshitani [University of Chiba]
Titel: Remarks on the Loewy length of a block algebra [Abstract]We will be discussing a problem on Loewy lengths of blocks of group algebras of finite groups over a field of characteristic \(p>0\). This is sort of a survey talk and part of this contains joint work with B. Kuelshammer and B. Sambale.
 Donnerstag, 30. Oktober: Attila Maróti [TU Kaiserslautern]
Titel: Groups equal to a product of three conjugate proper subgroups [Abstract]Martino Garonzi and Dan Levy proved that every nonsolvable finite group is equal to the product of no more than 36 conjugate proper subgroups. We outline a proof for the fact that 36 can be replaced by 3. One ingredient of this result is that any (not necessarily finite) group with a BNpair and a finite Weyl group is equal to the product of three conjugates of the subgroup B. This is work in preparation with John Cannon, Martino Garonzi, Dan Levy, and Iulian Simion.
 Donnerstag, 6. November: Kein Seminar
November 7.8.: DarstellungstheorieTage 2014  Donnerstag, 27. November: Imke Toborg [Universität KoblenzLandau]
Titel: Local methods need some representation theory to prove $Z_3^*\!$type results [Abstract]In my PhDthesis I reproved a theorem of Rowley which is a weaker version of a generalisation of Glauberman's Z*theorem for the prime 3 using local group theoretical methods. In view of a generalisation of the Z*theorem for the prime 3 this new proof can give an idea of range of local methods. In this talk I will give an overview of the strategy of my proof. Then I will disclose parts of my thesis where also representation theory is needed.
 Donnerstag, 4. Dezember: Kein Seminar
Dezember 56.: Nikolauskonferenz Aachen  Donnerstag, 18. Dezember: Sebastian Herpel [RuhrUniversität Bochum]
Titel: Maximal subalgebras of Cartan Type in the exceptional Lie algebras [Abstract]Consider a simple exceptional Lie algebra over an algebraically closed field of good but positive characteristic. In joint work with David Stewart we determine the simple Cartantype maximal subalgebras in this setup. It turns out that only the first Witt algebra occurs, and in the talk we will describe how its embeddings may be controled with the use of representation theory and the theory of nilpotent orbits.  Donnerstag, 8. Januar: Tommy Hofmann [TU Kaiserslautern]
Titel: Integrality of Representations of Finite Groups [Abstract]It is well known that a complex character of a finite group \(G\) can be obtained from a representation \(G \to \mathrm{GL}_n(K)\!\), where \(K\) is a finite extension of \(\mathbf Q\). In this talk I will discuss the question, whether we can adjust this representation such that all matrices are elements of \(\mathrm{GL}_n(\mathcal O)\), where \(\mathcal O\) denotes the ring of integers of \(K\).  Donnerstag, 5. Februar, 17:00 Raum 48436: Eirini Chavli [Université Paris Diderot  Paris 7]
Titel: The BMR freeness conjecture: the exceptional groups of rank 2 [Abstract]Between 1994 and 1998, the work of M. Broue, G. Malle, and R. Rouquier generalized in a natural way the definition of the Hecke algebra beyond a Coxeter group, for any finite arbitrary complex reflection group. Attempting to also generalize the properties of the Coxeter case, they stated a number of conjectures concerning the Hecke algebras. One specific example of importance regarding those yet unsolved conjectures is the socalled BMR freeness conjecture. This conjecture is known to be true apart from 19 cases, the exceptional groups of rank 2, and the case of a large complex reflection group, that hasn’t been checked yet. In this talk, we will explain some methods we used for proving the conjecture for some of the exceptional groups of rank 2, methods we are optimistic that can work for the rest of the cases, as well (work in progress).  Donnerstag, 19. Februar, 17:00 Raum 48436: Barbara Baumeister [Universität Bielefeld]
Titel: Moufang twin buildings and KacMoody groups [Abstract]Every reductive algebraic group acts nicely on a geometric object, the spherical building, and the simple algebraic and classical groups are characterized by that property as Tits showed.
Ronan and Tits generalized the concept of spherical buildings to that of twin buildings in order to obtain geometric objects for the KacMoody groups. Twin buildings are defined in such a way that they possess similar properties to those of spherical buildings; for instance the Moufang property  named after Ruth Moufang  can also be defined for twin buildings.
We will introduce the relevant terminology and discuss whether a KacMoody group is characterized by its action on a Moufang twin building. We will then focus on the case of trivalent Moufang twin trees. In that case we are able to completely determine the structure of the "unipotent" subgroups which is also of independent interest.
Sommersemester 2014
Organisatoren: Britta Späth, Jay Taylor, Caroline Lassueur
24.4.2014 17:00  18:00 (Room 48436) Simon Schmider: On AuslanderReiten components for Hecke algebras of type A
For a finitedimensional algebra $\Lambda$, the stable AuslanderReiten quiver $\Gamma_{s}(\Lambda)$ is an important homological invariant. In this talk, we will consider the case where $\Lambda$ is the Hecke algebra of a symmetric group over some field of characteristic zero, and try to understand the shape of the components of $\Gamma_{s}(\Lambda)$.
8.5.2014 17:00  18:00 (Room 48436) Frank Himstedt (TU München): On the decomposition numbers of the groups $\rm {SO}_7(q)$ and $\rm {Sp}_6(q)$
In my talk I describe the completion of the lmodular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups $\rm {SO}_7(q)$ and the symplectic groups $\rm {Sp}_6(q)$ for all prime powers q and all odd primes $\ell$ dividing $q+1$. This is joint work with Felix Noeske.
22.5.2014 17:00  18:00 (Room 48436) Erwan Biland (Sherbrooke University, Kanada): Stable equivalences related to the odd Z*ptheorem
Let $G$ be a finite group, $p$ be a prime number, and $P$ be a $p$subgroup of $G$ such that the centraliser $C_G(P)$ "controls the pfusion" in $G$. Then the famous $Z^*_p$thorem asserts that the group $G$ may be factorised as $G = O_{p'}(G).C_G(P)$. This theorem has been proven for $p=2$ by Glauberman (1962), using modular representation theory. It is also known to be true for odd p, thanks to the classification of finite simple groups, but no "modular proof" of it has been found yet. In this talk, I will explain how the $Z^*_p$theorem can be restated in terms of Morita equivalences, and how I have been able to prove a partial result that points towards a modular proof of that theorem, namely a stable equivalence. This, the main result of my PhD thesis, is obtained by gluing methods which may have an interest for themselves, and which rely on a natural generalisation of the widely used ppermutation modules.
28.5.2014 17:00  18:00 (Room 48436) Pablo Luka : Vertexbounded defects
Let $K$ be an algebraically closed field of positive characteristic $p$, and let $Q$ be a finite $p$group. A set $X$ of finite groups satisfies the vertexboundeddefect property with respect to $Q$ iff there exists an integer $n_Q$ such that whenever $G$ is an element of $X$ and $M$ is a simple left $KG$module having vertex $Q$, then the defect groups of the block containing $M$ have order at most $n_Q$.
I will motivate the talk with a conjecture of Feit and then speak about this vertexboundeddefect property and results for classical groups.
5.6.2014 15:45  16:45 (Room 48582) Eugenio Gianelli (Royal Holloway University of London): On permutation modules and decomposition numbers of the symmetric group
The determination of the decomposition numbers of symmetric groups is a famous longstanding open problem.
In this talk I will firstly introduce my recent joint work with Dr. Wildon on the topic. We have been studying some structural properties of the family of $p$permutation modules known as Foulkes modules; I will show how we used this information to completely determine a number of columns of the decomposition matrix of the symmetric group.
In the second part of the talk I will present a new, more general result on the modular structure of the Foulkes modules. In particular I will give a classication of the vertices of all their indecomposable summands. I will conclude by summarising some of the open questions in the area.
5.6.2014 17:00  18:00 (Room 48436) Hung P. TongViet (Universität Bielefeld): Some results on Brauer character degrees of finite groups
For a finite group $G$ and a prime $p,$ denote by $\mathrm {IBr}_p(G)$ the set of irreducible $p$Brauer characters of $G$ and by $\mathrm {cd}_p(G)$ the set of their degrees. In this talk, I will present recent results on the structure of groups under some arithmetic conditions on the set $\mathrm {cd}_p(G).$ We are interested in groups in which the degrees of nonlinear irreducible Brauer characters are prime powers or have the same $p$part.
12.6.2014 17:00  18:00 (Room 48436) Martino Garonzi (Università di Padova): Factorizing a group with conjugate subgroups
Let $G$ be a ﬁnite group. We study the minimal number $n$ such that there exist $n$ pairwise conjugated proper subgroups of $G$ whose product, in some order, equals $G$. We call cp$(G)$ this number. I will present the following results.
Theorem 1. If $G$ is nonsolvable then $G$ is the product of at most $36$ conjugates of a proper subgroup. In other words cp$(G)$ is at most $36$.
Theorem 2. For all integer $n > 2$ there exists a solvable group $G$ with cp$(G) = n$.
3.7.2014 16:00  17:00 (Room 48582) Giovanna Carnovale (Università di Padova): Conjugacy classes in finite simple groups and pointed Hopf algebras
The classification problem of finite dimensional Hopf algebras is very hard in general but there are good strategies to classify special families of interest. I will explain the strategy, due to Andruskiewitsch and Schneider, for classifying finitedimensional pointed Hopf algebras and show how one step in the procedure can be obtained by analyzing combinatorial properties of conjugacy classes in finite (simple) groups. Next, I will describe our results in the case of simple groups of Lie type. The talk is based on joint work with N. Andruskiewitsch and G. A. Garcia.
17.7.2014 17:00  18:00 (Room 48436) Martina Lanini (FAU ErlangenNürnberg): Finite dimensional representations of rational Cherednik algebras.
Given a highest weight irreducible representation for a simple Lie algebra, the question whether it is finite dimensional has an affirmative answer if and only if its highest weight is dominant. In the case of a lowest weight simple module for a rational Cherednik algebra at a given parameter, the above, natural, question does not have an answer in general yet. In this talk, I will discuss joint work with S.Griffeth, A.Gusenbauer and D.Juteau providing a necessary condition for finite dimensionality of such mudules.
Wintersemester 2013/14
Organisatoren: Britta Späth
7.11.2013, 15:3016:30 Philipp Perepelitsky: $p$permutation equivalences between blocks of finite groups
In this talk we describe joint work with Robert Boltje. Let F be an algebraically closed eld of positive characteristic p: Let $G$ and $H$ be nite groups. Let A be a block of $FG$ and let B be a block of $FH$: A $p$permutation equivalence between $A$ and $B$ is an element in the group of $(A,B)$$p$permutation bimodules with twisted diagonal vertices such that $\gamma \cdot_H\gamma^\circ = [A]$ and $\gamma^\circ \cdot_G \gamma = [B]$. A $p$permutation equivalence lies between a splendid Rickard equivalence and an isotypy.
We introduce the notion of a $\gamma$Brauer pair, which generalizes the notion of a Brauer pair for a $p$block of a finite group. The $\gamma$Brauer pairs satisfy an appropriate Sylow theorem. Furthermore, each maximal $\gamma$Brauer pair identies the defect groups, fusion systems and KülshammerPuig classes of $A$ and $B$: Additionally, the Brauer construction applied to $\gamma$ induces a $p$permutation equivalence at the local level, and a splendid Morita equivalence between the Brauer correspondents of $A$ and $B$.
14.11.2013, 15:3016:30 Iulian Simion: Refining the Bruhat decomposition
Following Shimura we will describe Hecke rings. We then specialize to the case of IwahoriHecke algebras and consider their structure constants for standard basis elements. A result of Kawanaka describes these constants by means of certain subexpression for elements of the Weyl group. Curtis reinterpreted this result and generalized results of Deodhar related to a refinement of the Bruhat decomposition. We show how, in some particular cases, this point of view allows for a simple description of the structure constants.
21.11.2013 no talk (BaerKolloquium, 23.11.2013)
28.11.2013, 15:3016:30 Fuat Erdem: On the generating graph of a finite group
The generating graph of a finite group $G$, denoted by $\Gamma(G)$, is the graph on the nonidentity elements of $G$ in which two distinct vertices are joined by an edge if and only if they generate $G$. An important notion in the study of graphs is that of a Hamiltonian cycle. A Hamiltonian cycle in an undirected simple graph is a cycle that visits every vertex exactly once. We will first mention several aspects of the generating graph and then raise the subject to a more recent question, namely that, for which finite groups $G$ does there exist a Hamiltonian cycle in $\Gamma(G)$? We will mainly discuss the case of the symmetric group.
5.12.2013, no talk (Darstellungstheorietage und Nikolauskonferenz 2013)
12.12.2013 16:0017:00 Jürgen Müller: The Abelian Defect Group Conjecture for sporadic simple groups
Let G be a finite group, let A be a prime block of G having an abelian defect group P, let N be the normaliser in G of P, and let B be the Brauer correspondent of A. Then the Abelian Defect Group Donjecture (in various forms due to M. Broue and J. Rickard) says that the bounded derived categories of the module categories of A and B are equivalent as triangulated categories. Although being in the focus of intensive study since two decades now, and considerable progress has been made, a general proof seems to be out of sight.
A possible strategy is a reduction to the simple (and closely related) groups. Hence it seems worthwhile to tackle these groups first. In view of this, in this talk we show how a combination of theoretical strategies and techniques from computational representation theory can be pursued to prove the Abelian Defect Group Conjecture for some of the sporadic simple groups. This is joint work with Shigeo Koshitani and Felix Noeske.
19.12.2013 15:3016:30 Olivier Dudas: Ordering conjugacy classes in Weyl groups
Conjugacy classes in a Weyl group or more generally a Coxeter group do not behave well with respect to the Bruhat order. This difficulty can be overcome by looking at elements of minimal length in a given conjugacy class. I will explain how the order that is induced on these elements should reflect the closure relations on unipotent classes of the corresponding reductive group.
9.1.2014 15:3016:30 Caroline Lassueur: The position of endo$p$permutation modules and relatives in the AuslanderReiten quiver
16.1.2014 17:0018:00 (Room 48436)Serge Bouc: The poset of posets
In this joint work with Jacques Thévenaz, we study the structure of the algebra $E(X)$ of essential relations on a finite set $X$: this algebra (which doesn't seem to have been considered so far) is the quotient of the algebra of the monoid of all relations on $X$ by the submodule generated by relations which factor through a (strictly) smaller set. In particular, we determine all the simple $E(X)$modules. These are parametrized in particular by order relations on $X$. This motivates further study of combinatorial and topological properties of the "poset of posets" on $X$.
23.1.2014 15:3016:30 Elisabeth Schulte: Alperin's Weight Conjecture  On Inductive Conditions for $G_2(q)$
In my talk I give a brief introduction to the Weight Conjecture of J. L. Alperin (1986) and its reduction to finite simple groups, which leads to the socalled inductive Alperin Weight Condition. We will consider this condition for the Chevalley groups $G_2(q)$.
30.1.2014 15:3016:30 Burkhard Külshammer: Some questions in block theory
In my talk I will address some results in block theory which were recently obtained with various coauthors. These results are concerned with defect groups, fusion systems and Loewy lengths of blocks. One of these results is also related to one of R. Brauer's problems.
10.2.2014 15:3016:30 (Room 48436) Alessandro Paolini (University of Birmingham): A Basis of the Gelfand–Graev Algebra of a Chevalley Group
6.3.2014 10:30  11:30 (Room 48436) Michel Enguehard (Institut de Mathématique de Jussieu): Towards a Jordan decomposition of blocks of finite reductive groups
Let $G$ be a connected algebraic reductive group over an algebraically closed field of non zero characteristic $p$, let $F$ be an endomorphism of $G$ defining $G$ over a finite field ${\Bbb F}_q$, let $G^F$ be the subgroup of rational points. Let $\ell$ be a prime number different from $p$. Let $(G^*,F)$ be `` in duality with $(G,F)$ ". By a theorem of BrouéMichel, for any $\ell$block $B$ of $G^F$ there exist a unique $G^{*\, F}$conjugacy class $(s)$ of semisimple $\ell'$elements in $G^{*\, F}$ such that at least one element of the Lusztig's series of irreducible representations of $G^F$ defined by $(s)$ is a representation of $B$. If $s=1$, $B$ is said to be unipotent.
For a large class of data $((G,F),\ell)$, we associate to $((s),B)$ as above a datum $((G(s),F),B(s))$ where
 $(G(s),F)$ is an algebraic reductive group defined over ${\Bbb F}_q$, with connected component $G(s)^\circ$, and
 $B(s)$ is an $\ell$block of $G(s)^F$ that dominates a unipotent $\ell$block of $G(s)^{\circ \, F}$,
in such a way that
 The defect groups of $B$ and of $B(s)$ are isomorphic.
 There is a onetoone map preserving height between the sets of irreducible representations of $B$ and of $B(s)$.
 The Frobenius categories associated to $B$ and $B(s)$ are equivalent.
Sommersemester 2013
Organisatoren: Britta Späth
Thursday, 15:30  16:30, room 48436.
25.4.2013, 17:00  18.00 and 30.4.2013, 15:3016:30 Olivier Dudas: On DeligneLusztig varieties and their cohomology
In this talk I will explain how Soergel bimodules provide a good framework to study the cohomology of DeligneLusztig varieties. I will derive some results from (as yet conjectural) properties of the action of a Braid group on these bimodules and their homotopy category.
8.5.2013, 15:30  16:30 Michael Livesey: Rouquier Blocks of Finite Classical Groups and Other Groups
Rouquier blocks were important for solving Broué's Abelian defect group conjecture for symmetric groups and finite general linear groups. We attempt to generalize the notion of a Rouquier block for some finite classical groups and show how this will (hopefully!) fit into a proof of Broué's conjecture for these groups. We will also briefly look at what progress has been made in this area for the double covers of the symmetric group.
16.5.2013, 15:3016:30 Radu Stancu: Fusion systems and the double Burnside ring
Saturated fusion systems were introduced by Puig as a generalization of the $p$local structure of a finite group or of a block algebra of a finite group. Broto, Levi and Oliver introduced the notion of characteristic biset associated to a saturated fusion system. This biset is not unique but Ragnarsson proved that there is a unique characteristic idempotent in the pcompleted double Burnside ring associated to a saturated fusion system. In this talk, based on a joint work with Kari Ragnarsson, I will give a characterization of saturated fusion systems on a $p$group $S$ in terms of idempotents in the $p$local double Burnside ring of $S$ that satisfy a Frobenius reciprocity relation. This helps us to reformulate fusiontheoretic phenomena in the language of idempotents and give some applications in stable homotopy.
23.5.2013, 15:3016:30 Michael Pleger: Decomposition numbers for HeckeAlgebras of complex reflection groups
6.6.2013, 15:3016:30 Gunter Malle: On Ore's conjecture
Ore's conjecture asserts that in a nonabelian finite simple group, every element is a commutator. The proof of this statement was recently completed by Liebeck, O'Brien, Shalev and Tiep. We report on the various ingredients used in that proof, reaching from DeligneLusztig character theory to explicit computations. We also mention several related, still open problems.
13.6.2013, 15:3016:00 Jay Taylor: On Lusztig's Conjecture for Character Sheaves of Classical Type Groups
Let $G$ be a connected reductive algebraic group defined over a field of good characteristic and $F : G \to G$ a Frobenius endomorphism of $G$; we denote by $G^F$ the corresponding finite reductive group. We will assume that the centre $Z(G)$ is connected and $G/Z(G)$ is simple of type $B$, $C$ or $D$. In addition to the irreducible characters of $G^F$ Lusztig has defined two additional bases of the space of all class functions of $G^F$. The first is the basis of almost characters and the second is the basis of characteristic functions of character sheaves. Lusztig has conjectured that these bases coincide up to multiplication by roots of unity. In our talk we will discuss new results which explicitly compute these scalars for those character sheaves whose support contains unipotent elements. If time permits we will also discuss applications of this to Kawanaka's Generalized GelfandGraev Representations.
18.6.2013, 15:3016:30 Meinolf Geck: Triangular shape of decomposition matrices
20.6.2013, 15:3016:30 Nicolas Jacon: Lusztig symbols and Representation Theory of Weyl groups of type B
It is wellknown that the representation theory of the symmetric group is related to the combinatorics of partitions in both the ordinar and the modular case. The aim of this talk is to show that, in the context of the Weyl group of type B and its Hecke algebra, the notion of Lusztig symbols play a similar role. We present several results around this notion and we show how they can be possibly generalized to a class of complex reflection groups.
4.7.2013. 15:30  16:30 Michael Cuntz: Weyl groupoids and arrangements
The Weyl groupoid is a symmetry structure which was originally introduced as an invariant of Nichols algebras. The classification of finite Weyl groupoids revealed further applications in geometry and combinatorics. In this talk we will see this connection for more general Cartan schemes, and discuss the recently initiated classification of affine Weyl groupoids.
11.7.2013. 15:30  16:30 Caroline Lassueur: On simple endotrivial modules for quasisimple groups
I will report on our joint work with G. Malle and E. Schulte. First I will give an overview of our results. Second I will explain, with proof, how, in the cyclic defect case, one can read from the Brauer tree whether or not a simple module (in characteristic p>0) is endotrivial. (I will also explain why I hyphenate the word endotrivial!)
16.7.2013, 15:30  16:30 (48516) Benjamin Sambale: Finite groups with two conjugacy classes of pelements
I report on a joint paper with Külshammer, Navarro and Tiep about the classification of finite groups in which every two nontrivial pelements (for a fixed prime p) are conjugate. This leads to consequences for blocks of finite groups. We also raise an interesting question about fusion systems.
18.7.2013, 17:00  18:00 Frieder Ladisch: Clifford theory and rationality questions
Let $G$ be a finite group, $N$ a normal subgroup of $G$ and $\theta\in { \mathrm Irr}(N)$ an irreducible character. When investigating the character theory of $G$ "above $\theta$'', one can often replace the triple $(\theta, N, G)$ by another triple $(\theta_1, N_1,G_1)$, where $N_1\subset {\mathrm Z}(G_1)$. This is a consequence of classical Clifford theory and the theory of projective representations. This result is no longer true when rationality questions of the characters involved are considered. In the talk we plan to outline a proof that it is possible to reduce to a triple $(\theta_1, N_1,G_1)$ where $N_1$ contains a cyclic subgroup $C$ which is normal in $G_1$, where the factor group $N_1/C$ is abelian and where $\theta_1\in { \mathrm Irr} (N_1)$ is induced from $C$. An important tool in the proof is a certain subgroup (the "SchurClifford subgroup'') of the BrauerClifford group introduced by Alexandre Turull.
Oberseminar Darstellungstheorie
 Wintersemester 2020/21: Equivalences of Blocks of Finite Groups [Organisiert von Caroline Lassueur  Programme]
 Sommersemester 2020: Abgesagt
 Wintersemester 2019/20: Block theory of finite reductive groups [Organisiert von Gunter Malle  Programme]
 Sommersemester 2019: Topics in the block theory of finite groups [Organisiert von Niamh Farrell  Programme]
 Wintersemester 2018/19: Perverse equivalences [Organisiert von Olivier Dudas/Alessandro Paolini  Programme]
 Sommersemester 2018: Cluster algebras [Organisiert von Patrick Wegener Programme]
 Wintersemester 2017/18: Lie superalgebras [Organisiert von Caroline Lassueur  Programme]
 Sommersemester 2017: Representation theory – A homological algebra point of view [Organisiert von Benjamin Sambale  Programme]
 Wintersemester 2016/17: Quivers and reprerentations of algebras [Organisiert von Mikaël Cavallin  Programme]
 Sommersemester 2016: Buildings [Organisiert von Thomas Gobet  Programme]
 Wintersemester 2015/16: Complex Reflection Groups [Organisiert von Caroline Lassueur  Programme]
 Sommersemester 2015: KLRalgebras [Organisiert von Gunter Malle  Programme]
 Wintersemester 2014/15: Der AuslanderReitenKöcher von Gruppenalgebren [Organisiert von Caroline Lassueur  Programme]
 Sommersemester 2014: The Alperin weight conjecture [Organisiert von Gunter Malle und Britta Späth  Programme]
 Wintersemester 2013/14: Diagram algebras [Organisiert von Susanne Danz and Jay Taylor  Programme]
 Sommersemester 2013: Darstellungen endlicher reduktiver Gruppen [Organisiert von Gunter Malle  Programme]
 Wintersemester 2012/13: Representations of finite reductive groups [Organisiert von Gunter Malle  Programme]
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