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 Φ(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.
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14 entries found
 December

01. December17:00  18:00
Location: 48436AG Algebra, Geometrie and Computer AlgebraEmilio Pierro, Universtität Bielefeld; Quantities associated to the subgroup lattice
 November

25. November11:00  12:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraJesper Grodal, University of Copenhagen; Endotrivial modules via homotopy theory
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.

24. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraNiamh Farrell , TU KL/University of London; The rationality of blocks of quasisimple finite groups
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.

03. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraYanjun Liu, TU Kaiserslautern; Principal and defectzero blocks of finite groups
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.
 June

16. June17:00  18:00
Location: 48436Algebra, Geometrie und ComputeralgebraAnneLaure Thiel, Dijon: Categorical action of the braid group of the cylinder
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.  May

12. May17:00  18:00
Location: 48436AG Algebra, Geometrie, ComputeralgebraChristopher Voll, Bielefeld: 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.
An important tool in this area are socalled representation zeta functions, viz. Dirichlettype generating functions encoding the arithmetic functions' values. These zeta functions share a number of features with their numbertheoretic and algebrogeometric predecessors, e.g. Euler products with rational factors, (local) functional equations, (some) meromorphic continuation etc. It is of particular interest to understand how representation zeta functions vary with the group, e.g. in families of groups obtained by base extensions.
I will discuss some arithmetic features (uniform rationality of and functional equations for Euler factors) as well as uniform analytic properties of representation zeta functions of finitely generated nilpotent groups. I will also say something about the general tools available to compute representation zeta functions of nilpotent (and other!) groups, such as the Kirillov Orbit Method and padic integration, without assuming any familiarity with any of them.

11. May17:00  18:00
Location: 48436AG Algebra, Geometrie, ComputeralgebraJay Taylor, Padova: 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.
 April

21. April15:30  16:30
Location: 48582AG Algebra, Geometrie und ComputeralgebraBalmer, Paul; UCLA Bielefeld: A quick tour of tensor triangular geometry
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.  February

11. February15:30  16:30
Location: 48582AG Algebra, Geometrie und ComputeralgebraJohn Murray, Maynooth University IR: Strong and weak reality and principal indecomposable modules in characteristic 2
(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 pequals the number of pregular conjugacy classes of G. Moreover the number of selfdual pims equals the number of real pregular conjugacy classes. Suppose that p≠2. 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 ≤ # indicator 1 irreducible characters.

08. February15:30  17:00
Location: 48438AG Algebra, Geometrie und ComputeralgebraJean Michel, Paris 7: The $K(\pi, 1)$ property for complex braid groups and the dual braid monoid