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.
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26 entries found
 July

20. July17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraYujiao Sun, Universität Stuttgart: Supercharacter theories for Sylow psubgroups of finite exceptional groups of Lie type

12. July11:30  12:30
Location: 48436AG Algebra, Geometrie und Computer AlgebraRémi Imbach, TU KL: Certified numerical tools for computing the topology of projected curves
We are interested in computing the topology of the projection of an algebraic or analytic space curve in the plane. Such a projection is not a smooth curve and has singularities. State of the art approaches to compute the topology of algebraic plane curves use symbolic calculus but in the case of a projection, latter approaches suffer from the size of the implicit representation of the curve as a resultant polynomial.
Using numerical tools to compute the projected curve or its singularities is a challenging problem since the projected curve is not smooth and stateoftheart characterizations of the singularities use overdetermined systems of equations. We will first propose a new characterization of the singularities of the projection of an algebraic curve using a square system polynomials; its solutions are regular and it can be solved numerically.
However its equations are coefficients of resultant polynomials and are still very large polynomials. The demand in arithmetic precision to carry out numerical calculus with such polynomials makes classical solvers either incomplete or dramatically time consuming. We will present a multiprecision solver using interval subdivision specially designed to handle large polynomials to compute the solutions of this system. We will then consider the more general case of projections of analytic space curves, and propose a geometric approach to describe its singularities. It results in a square system of equations with only regular solutions, that do not involve resultant theory, and that can be solved with our certified numerical solver. Finally we will present a new approach to compute the topology of the projected curve, i.e. find a graph that has the same topology. We use a certified numerical path tracker to enclose the space curve in a sequence of boxes, allowing both to simplify the research of singularities and to compute smooth branches linking singularities. 
05. July17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraTung Le, University of Pretoria: On the automorphisms of designs constructed from finite simple groups
(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.
 June

29. June17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraJulian Külshammer, Stuttgart: Ringel duality as a special case of Koszul duality
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.

22. June17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraBaptiste Rognerud, IRMA Strasbourg: A Morita theory for permutation modules
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. 
13. June17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraWolfgang Willems, Magdeburg: On quasiprojective Brauer characters
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(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. June17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraWilliam Wong,TU Kaiserslautern: A phenomenon in the representation of SL(2,q) in defining
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.
 May

11. May17:00  18:00
Location: 48519Algebra, Geometrie und Computer AlgebraAndreas Bächle, University Brussels: Rationality of groups and centers of integral group rings
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.

04. May17:00  18:00
Location: 48436Algebra, Geometrie und Computer AlgebraEmil Norton, MPI Bonn: The sl_∞crystal combinatorics of higher level Fock spaces
A 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
Location: 48436Algebra, Geometry and Computer AlgebraKivanv Ersoy, TU Kaiserslautern: Locally finite groups with small centralizers
In 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