### Seminar Gruppen und Darstellungen

(UNDER CONSTRUCTION) This page contains an archive of the talks given in the Seminar Groups and Representations (TU Kaiserlautern) since the Wintersemester 2012/13.

#### Sommersemester 2018

**Organisation:**Niamh Farrell, Alessandro Paolini

**Donnerstag, 3. Mai 2018**: Keivan Mallahi-Karai [Jacobs University, Bremen]

Titel:**Kirillov's orbit method and the polynomiality of the essential dimension of p-groups**[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 p-groups arising from F_p-points 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 upper-triangular 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 “number-theoretically 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 n-strand 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 Birman-Ko-Lee 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 c-sortable 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 non-abelian 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.

**16.30-17.30**: 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 kG-module 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 p-subgroup P of Qd(p) and G is Park’s group for F, we prove that the Scott kG-module with vertex P is Brauer indecomposable. In this talk these results will be presented. This is a joint work with Shigeo Koshitani.

**17.30-18.30**: 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 [Babeş-Bolyai University, Cluj-Napoca]

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 well-understood. 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 real-valued, 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 sub-structure, to a classical problem of computing Kronecker coefficients in the symmetric group.**Donnerstag, 28. Juni 2018**: Kein Seminar. Norddeutsches Gruppentheorie-Kolloquium in Braunschweig, 29.06.18-30.06.18.

**Donnerstag, 5. Juli 2018**: Kein Seminar. Computational Lie Theory in Stuttgart, 05.07.18-06.07.18.

**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

**Organisation:**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 Z-grading 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 endo-permutation source**[Abstract]Motivated by current work of Boltje-Kessar-Linckelmann on the Picard group of Morita self-equivalences with endo-permutation source, we consider the problem of whether Morita equivalences with an endo-permutation 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 10.11.17-11.11.17).**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 l-regular 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 p-local 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 p-subgroups 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 Deligne-Lusztig 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 Brady-Watt) 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 Igusa-Schiffler-Thomas 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, p-locally, our example is made up of certain p-permutation 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

**Organisation:**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.1-11.**Donnerstag, 4. Mai 2017**: Emily Norton [Max Planck Institute Bonn]

Titel:**The \mathfrak{sl}_\infty-crystal 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, Dudas-Varagnolo-Vasserot recently constructed categorical actions of these two crystals on the unipotent category of finite classical groups in order to classify Harish-Chandra 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) semi-rational group. It turned out that being an inverse semi-rational 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 [Otto-von-Guericke-Universität Magdeburg]

Titel:**On quasi-projective Brauer characters**[Abstract]We study p-Brauer characters of a finite group G which are restrictions of generalized characters vanishing on p-singular elements for a fixed prime p dividing the order of G. Such Brauer characters are called quasi-projective. We show that for each irreducible Brauer character there exists a minimal p-power, say $p^{a(\varphi)}$, such that $p^{a(\varphi)}\varphi$ is quasi-projective. 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 p-solvable. 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 quasi-projective 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 p-permutation 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 p-modular 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]Quasi-hereditary 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 quasi-hereditary 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 1-designs 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 p-subgroups 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 p-subgroups 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, 14:00-15:00***: Shigeo Koshitani [University of Chiba]

Titel:**Brauer indecomposability of Alperin-Scott modules for finite non-abelian 2-groups**[Abstract]We will be discussing the so-called Alperin-Scott module for the group algebra kG of a finite group G over a field k of characteristic p>0. It is a p-permutation indecomposable kG-module such that it has the trivial module k in the socle with multiplicity one. Brauer indecomposabitily is considered in the 2011 paper by Kessar-Kunugi-Mitsuhashi, 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 p-groups. So as next step we will consider non-abelian 2-groups cases.

(*) This is a part of joint work with Caroline Lassueur.

#### Wintersemester 2016/17

**Organisation:**Julian Brough, Caroline Lassueur

**Donnerstag, 3. November 2016**: Yanjun Liu [TU Kaiserslautern]

Titel:**Principal and defect-zero 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 quasi-simple 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 quasi-simple 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 kG-modules that upon restriction to a Sylow p-subgroup 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 so-called endotrivial modules, whose classification has been a long-term 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 p-number of a finite group is defined as the number of Sylow p-subgroups 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 finite-dimensional 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 2-blocks 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

**Organisation:**Thomas Gobet, Caroline Lassueur

#### Wintersemester 2015/16

**Organisation:**Thomas Gobet, Caroline Lassueur

**Donnerstag, 15. Oktober 2015, 16:00-17:00, Raum 48-436**: 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 so-called “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 Iwahori-Hecke 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" Kazhdan-Lusztig 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 co-Pieri 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 “co-Pieri triples" of partitions.**Donnerstag, 3. Dezember 2015**: Benedetta Lancellotti [Università degli Studi di Milano-Bicocca, 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, 16:00-17:00, Raum 48-582**: 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 [Ruhr-Universität 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 G-complete 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 self-dual pims equals the number of real $p$-regular conjugacy classes. Suppose that $p\ne2$. Then each self-dual pim affords either a quadratic or a symplectic geometry. W. Willems and J. Thompson independently showed that the type is detected by the Frobenius-Schur (FS)-indicator of a real irreducible character which occurs with odd multiplicity in the principal indecomposable character. Now suppose that $p=2$. Then each self-dual 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 character-theoretic 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 2-regular classes. As a consequence we can show that $\#$ quadratic type pims $=\#$ strongly real 2-regular classes. Brauer's problem 14 asks for a determination of the numbers of irreducible characters with a given FS-indicator in group-theoretic terms. We conjecture that $\#$ weakly real 2-regular classes $\leq\#$ FS indicator $-1$ irreducible characters.

#### Sommersemester 2015

**Organisation:**Caroline Lassueur, Britta Späth

**Donnerstag, 23. April, 15:30-16:30, Raum 48-582**: 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, 15:30-16:30, Raum 48-582**: 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 well-known 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, 15:30-16:30, Raum 48-436**: 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, 15:30-16:30, Raum 48-519**: Vivien Ripoll [Universität Wien]

Titel:**Coxeter elements in well-generated complex reflection groups**[Abstract]Coxeter elements are specific elements in a reflection group W, that play a key role in Coxeter-Catalan combinatorics, in particular in the construction of the W-noncrossing partition lattice. Several non-equivalent definitions coexist in the literature. I will explain and motivate the following: for W an irreducible real reflection group, or a well-generated 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 Springer-regular elements. (Joint work with Vic Reiner and Christian Stump)**Donnerstag, 15:30-16-30, 11. Juni**: Amanda Schaeffer-Fry [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 $|G|\leq e^4-e^3 .$ Moreover, when $G$ is nonabelian simple, we improve this bound to $|G| \leq 2e^2 .$**Donnerstag, 18. Juni, 17:00-18:00, Raum 48-436**: Thomas Gerber [RWTH Aachen]

Titel:**Weak Harish-Chandra theory for finite unitary groups**[Abstract]Harish-Chandra theory is a classical tool to label the simple modules of a finite group of Lie type $G$, in non-defining 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 Harish-Chandra series, by slightly modifying the classical theory.

After introducing the Harish-Chandra 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, 15:30-16:15**: Matthias Klupsch [RWTH Aachen]

Titel:**Supercuspidal modules of finite reductive groups**[Abstract]Harish-Chandra 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 Harish-Chandra 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, 17:00-18:00, Seminarraum 48-436**: Oksana Yakimova [Friedrich-Schiller-Universität Jena]

Titel:**On symmetric invariants of semi-direct 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 non-reductive 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 bi-parabolic subalgebras (Joseph's conjecture), $\mathbb Z_2$-contractions (Panyushev's conjecture). We will discuss results on symmetric invariants of semi-direct products and their relevance to these conjectures.**Donnerstag, 23. Juli**: Kein Seminar

Konferenz Blocks of Finite Groups and Beyond (Jena)

#### Wintersemester 2014/15

**Organisation:**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 non-solvable 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 BN-pair 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.: Darstellungstheorie-Tage 2014**Donnerstag, 27. November**: Imke Toborg [Universität Koblenz-Landau]

Titel:**Local methods need some representation theory to prove $Z_3^*\!$-type results**[Abstract]In my PhD-thesis 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 5-6.: Nikolauskonferenz Aachen**Donnerstag, 18. Dezember**: Sebastian Herpel [Ruhr-Universität 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 Cartan-type 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 48-436**: 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 so-called 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 48-436**: Barbara Baumeister [Universität Bielefeld]

Titel:**Moufang twin buildings and Kac-Moody 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 Kac-Moody 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 Kac-Moody 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

**Organisation:**Britta Späth, Jay Taylor, Caroline Lassueur

#### 24.4.2014 17:00 - 18:00 (Room 48-436) Simon Schmider: On Auslander--Reiten components for Hecke algebras of type A

For a finite-dimensional algebra $\Lambda$, the stable Auslander--Reiten 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 48-436) 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 l-modular 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 48-436) Erwan Biland (Sherbrooke University, Kanada): Stable equivalences related to the odd Z*p-theorem

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 p-fusion" 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 p-permutation modules.

#### 28.5.2014 17:00 - 18:00 (Room 48-436) Pablo Luka : Vertex-bounded 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 vertex-bounded-defect 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 vertex-bounded-defect property and results for classical groups.

#### 5.6.2014 15:45 - 16:45 (Room 48-582) 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 long-standing 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 48-436) ** Hung P. Tong-Viet** (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 48-436) 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 48-582) 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 finite-dimensional 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 48-436) Martina Lanini (FAU Erlangen-Nürnberg): 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

**Organisation:**Britta Späth

#### 7.11.2013, 15:30-16:30 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 Külshammer-Puig 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, 15:30-16:30 Iulian Simion: Refining the Bruhat decomposition

Following Shimura we will describe Hecke rings. We then specialize to the case of Iwahori-Hecke 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 (Baer-Kolloquium, 23.11.2013)

#### 28.11.2013, 15:30-16:30 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 non-identity 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 16:00-17:00 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 worth-while 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 15:30-16:30 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 15:30-16:30 Caroline Lassueur: The position of endo-$p$-permutation modules and relatives in the Auslander-Reiten quiver

#### 16.1.2014 17:00-18:00 (Room 48-436)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 15:30-16:30 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 so-called inductive Alperin Weight Condition. We will consider this condition for the Chevalley groups $G_2(q)$.

#### 30.1.2014 15:30-16:30 Burkhard Külshammer: Some questions in block theory

In my talk I will address some results in block theory which were recently obtained with various co-authors. 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 15:30-16:30 (Room 48-436) Alessandro Paolini (University of Birmingham): A Basis of the Gelfand–Graev Algebra of a Chevalley Group

#### 6.3.2014 10:30 - 11:30 (Room 48-436) Michel Enguehard (Institut de Mathematique de Jussiue): 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 Broué-Michel, for any $\ell$-block $B$ of $G^F$ there exist a unique $G^{*\, F}$-conjugacy class $(s)$ of semi-simple $\ell'$-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 one-to-one 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

**Organisation:**Britta Späth

Thursday, 15:30 - 16:30, room 48-436.

#### 25.4.2013, 17:00 - 18.00 and 30.4.2013, 15:30-16:30 Olivier Dudas: On Deligne-Lusztig varieties and their cohomology

In this talk I will explain how Soergel bimodules provide a good framework to study the cohomology of Deligne-Lusztig 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, 15:30-16:30 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 p-completed 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 fusion-theoretic phenomena in the language of idempotents and give some applications in stable homotopy.

#### 23.5.2013, 15:30-16:30 Michael Pleger: Decomposition numbers for Hecke-Algebras of complex reflection groups

#### 6.6.2013, 15:30-16:30 Gunter Malle: On Ore's conjecture

Ore's conjecture asserts that in a non-abelian 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 Deligne-Lusztig character theory to explicit computations. We also mention several related, still open problems.

#### 13.6.2013, 15:30-16:00 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 Gelfand-Graev Representations.

#### 18.6.2013, 15:30-16:30 Meinolf Geck: Triangular shape of decomposition matrices

#### 20.6.2013, 15:30-16:30 Nicolas Jacon: Lusztig symbols and Representation Theory of Weyl groups of type B

It is well-known 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 endo-trivial modules for quasi-simple 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 endo-trivial. (I will also explain why I hyphenate the word endo-trivial!)

16.7.2013, 15:30 - 16:30 (48-516) Benjamin Sambale: Finite groups with two conjugacy classes of p-elements

I report on a joint paper with Külshammer, Navarro and Tiep about the classification of finite groups in which every two non-trivial p-elements (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 "Schur-Clifford subgroup'') of the Brauer-Clifford group introduced by Alexandre Turull.

#### Wintersemester 2012/13

**Organisation:**Britta Späth

6.12.2012, 15:00 - 16:00: Susanne Danz: A ghost algebra of the double Burnside algebra

6.12.2012, 16:00 - 17:00: Elisabeth Schulte: Simple endotrivial modules for finite simple groups

TU Kaiserslautern • FB Mathematik • AG Algebra & Geometrie • Arbeitsgruppe Last updated: 11.08.2018. Icons designed by Freepik.