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.
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Vorträge 2017
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 Juni

29. Juni17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraJulian Külshammer, Stuttgart: Ringel duality as a special case of Koszul duality
Kategorie: AG Algebra, Geometrie und Computeralgebra 
22. Juni17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraBaptiste Rognerud, IRMA Strasbourg: A Morita theory for permutation modules
Kategorie: AG Algebra, Geometrie und ComputeralgebraIt 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. 
13. Juni17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraWolfgang Willems, Magdeburg: On quasiprojective Brauer characters
Kategorie: AG Algebra, Geometrie und ComputeralgebraWe 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(φ)}, such that p^{a(φ)} φ is quasiprojective. The exponent a(φ) only depends on the Cartan matrix of the block to which φ belongs. Moreover p^{a}(φ) is bounded by the vertex of the module afforded by φ, and equality holds in case that G is psolvable. We give some evidence for the conjecture that a(φ) occurs if and only if φ 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.

08. Juni17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraWilliam Wong,TU Kaiserslautern: A phenomenon in the representation of SL(2,q) in defining
Kategorie: AG Algebra, Geometrie und ComputeralgebraI 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.
 Mai

11. Mai17:00  18:00
Ort: 48519Algebra, Geometrie und Computer AlgebraAndreas Bächle, University Brussels: Rationality of groups and centers of integral group rings
Kategorie: AG Algebra, Geometrie und ComputeralgebraA 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.

04. Mai17:00  18:00
Ort: 48436Algebra, Geometrie und Computer AlgebraEmil Norton, MPI Bonn: The sl_∞crystal combinatorics of higher level Fock spaces
Kategorie: AG Algebra, Geometrie und ComputeralgebraA crystal is a type of directed graph encoding representation theoretic information. The crystal I will discuss is called the Heisenberg or <nobr></nobr> 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 <nobr></nobr> 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 <nobr></nobr> 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.
 April

27. April17:00  18:00
Ort: 48436Algebra, Geometry and Computer AlgebraKivanv Ersoy, TU Kaiserslautern: Locally finite groups with small centralizers
Kategorie: AG Algebra, Geometrie und ComputeralgebraIn this talk we will present recent results about fixed points of automorphisms in locally finite groups. Let pbe a prime and G a locally finite group containing an elementary abelian psubgroup 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 ∈ 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 pgroup of automorphisms A such that C_{G }(A) is Chernikov and C_{G }(a) involves no infinite simple groups for any a ∈ 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 <nobr> 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. ρ  März

08. März17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraMichael Geline, Northern Illinois University: Some quantitative and qualitative aspects of Knörr lattices
Kategorie: AG Algebra, Geometrie und ComputeralgebraThese lattices are the <nobr></nobr> padic 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 2adics, do not exist.
 Februar

23. Februar16:00  17:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraProf. Dr. Eugenii Shustin, Tel Aviv University; Real morsifications of plane curve singularities.
Kategorie: AG Algebra, Geometrie und ComputeralgebraA real morsification of a real plane curve singularity is a real deformation with the maximal possible number of hyperbolic nodes (i.e., equivalent to x^{2}y^{2}=0 over R). We prove that any real plane curve singularity admits a real morsification. This was known before only in the case of all local branches being real (A'Campo, GusseinZade). We also discuss a relation between real morsifications and the topology of singularities and extend to arbitrary real morsifications the BalkeKaenders theorem stating that the A'CampoGusseinZade diagram associated to the morsification uniquely determines the real topological type of the initial curve singularity. Joint work with Piter Leviant.

09. Februar17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraCharles Eaton, University of Manchester: Loewy length of blocks
Kategorie: AG Algebra, Geometrie und ComputeralgebraI 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.