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.
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22 entries found
 November

16. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraOlivier Dudas, Université Paris Diderot: On the unitriangular shape of decomposition matrices for finite reductive group

02. November17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraCaroline Lassueur: Lifting Morita equivalences with an endopermutation source
Motivated by current work of BoltjeKessarLinckelmann on the Picard group of Morita selfequivalences with endopermutation source, we consider the problem of whether Morita equivalences with an endopermutation 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.
 October

26. October17:00  18:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraAndrew Mathas (University of Sydney): Jantzen filtrations and graded Specht modules
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 Zgrading 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.
 August

25. August14:00  15:00
Location: 48436AG Algebra, Geometrie und Computer AlgebraShigeo Koshitani, University of Chiba: Brauer indecomposability of AlperinScott modules for finite nonabelian 2groups
We will be discussing the socalled AlperinScott module for the group algebra kG of a finite group G over a field k of characteristic p>0. It is a ppermutation indecomposable kGmodule such that it has the trivial module k in the socle with multiplicity one. Brauer indecomposabitily is considered in the 2011 paper by KessarKunugiMitsuhashi, 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 pgroups. So as next step we will consider nonabelian 2groups cases.
(*) This is a part of joint work with Caroline Lassueur. 
09. August11:30  13:00
Location: 48436AG 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
twovariable 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 twoparameter 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

25. July11:00  12:30
Location: 48210AG 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[x_{1},...,x_{n}].
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: 48436AG Algebra, Geometrie und Computer AlgebraYujiao Sun, Universität Stuttgart: Supercharacter theories for Sylow psubgroups 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 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.

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.