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.
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Talks 2017
<< 201619 entries found
- August
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25. August14:00 - 15:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraShigeo Koshitani, University of Chiba: Brauer indecomposability of Alperin-Scott modules for finite non-abelian 2-groups
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09. August11:30 - 13:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraDaniel Schultz, TU Kaiserslautern: Modular Equations in Two Variables
By adding certain equianharmonic elliptic sigma functions to the
coefficients of the Borwein cubic theta functions, an interesting set of six
two-variable theta functions may be derived. These theta functions
invert a special case of Appell’s hypergeometric function and
satisfy several identities akin to those satisfied by the Borwein
cubic theta functions. In this talk I will discuss the modular
properties of these functions as well as the computation of their
modular equations, which turn out to be algebraic surfaces. An
application of these results is a new two-parameter family X^9 −
3*X^8 + 4*t*X^6 − 6*s*X^5 − 6*s*X^4 + 4*s*t*X^3 − 3*s^2*X + s^2 = 0
of solvable nonic equations. - July
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25. July11:00 - 12:30
Location: 48-210AG Algebra. Geometrie und Computer AlgebraMichael Monagan/Simon Fraser University: Toward High Performance Factorization
To factor a multivariate polynomial Wang's incremental
design must solve many multivariate polynomial diophantine equations
of the form sigma A + τ B = C for sigma and τ in Zp[x1,...,xn].
We present a new efficient sparse approach.
We also modify the main Hensel lift may always be done modulo a machine prime.
Our goal is a high performance algorithm.
This is joint work with Baris Tuncer. -
20. July17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraYujiao Sun, Universität Stuttgart: Supercharacter theories for Sylow p-subgroups of finite exceptional groups of Lie type
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.
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12. July11:30 - 12:30
Location: 48-436AG 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 state-of-the-art characterizations of the singularities use over-determined 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 multi-precision 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: 48-436AG 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 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.
- June
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29. June17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraJulian Külshammer, Stuttgart: Ringel duality as a special case of Koszul duality
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.
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22. June17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraBaptiste Rognerud, IRMA Strasbourg: A Morita theory for permutation modules
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. -
13. June17:00 - 18:00
Location: 48-436AG Algebra, Geometrie und Computer AlgebraWolfgang Willems, Magdeburg: On quasi-projective Brauer characters
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 pa(φ), such that pa(φ) φ is quasi-projective. The exponent a(φ) only depends on the Cartan matrix of the block to which φ belongs. Moreover pa(φ) is bounded by the vertex of the module afforded by φ, and equality holds in case that G is p-solvable. 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 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.
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08. June17:00 - 18:00
Location: 48-436AG 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.
