Group representation | EPFL Graph Search

In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. M

Invariant integrals on topological groups

Vasco Schiavo

We generalize the fixed-point property for discrete groups acting on convex cones given by Monod in [23] to topological groups. At first, we focus on describing this fixed-point property from a functional point of view, and then we look at the class of groups that have it. Finally, we go through some applications of this fixed-point property. To accomplish these tasks, we introduce a new class of normed Riesz spaces that depend on group representation. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

ACADEMIC PRESS INC ELSEVIER SCIENCE2022

Sur quelques foncteurs de bi-ensembles

Rosalie Françoise Chevalley

This thesis is in the context of representation theory of finite groups. More specifically, it studies biset functors. In this thesis, I focus on two biset functors: the Burnside functor and the functor of p-permutation modules. For the Burnside functor we first give a result that characterize some B-groups; B-groups being the essential ingredient in the classification of composition factors of the Burnside functor. The second result compares the Burnside functor and the functor of free modules. Note that the functor of free modules is not a biset functor since the inflation of a free module is not necessarily free. To compare those functors we will work on an adjunction between the category of biset functors and the category of functors that do not have inflation. An aspect of the work done on the functor of p-permutation module is to compare the functor of p-permutation modules and the functor of ordinary representations. On the other hand, because of the classification of p-permutation modules, we try to express the functor o p-permutation modules in terms of the functor of projective modules (which is not a biset functor). We will use an adjunction between the category of biset functors and a category that contains the functor of projective modules.

EPFL2015

Computational Aspects of Jacobians of Hyperelliptic Curves

Alina Dudeanu

Nowadays, one area of research in cryptanalysis is solving the Discrete Logarithm Problem (DLP) in finite groups whose group representation is not yet exploited. For such groups, the best one can do is using a generic method to attack the DLP, the fastest of which remains the Pollard rho algorithm with

r

-adding walks. For the first time, we rigorously analyze the Pollard rho method with

r

-adding walks and prove a complexity bound that differs from the birthday bound observed in practice by a relatively small factor. There exist a multitude of open questions in genus

2

cryptography. In this case, the DLP is defined in large prime order subgroups of rational points that are situated on the Jacobian of a genus~

2

curve defined over a large characteristic finite field. We focus on one main topic, namely we present a new efficient algorithm for computing cyclic isogenies between Jacobians. Comparing to previous work that computes non cyclic isogenies in genus~

2

, we need to restrict to certain cases of polarized abelian varieties with specific complex multiplication and real multiplication. The algorithm has multiple applications related to the structure of the isogeny graph in genus~

2

, including random self-reducibility of DLP. It helps support the widespread intuition of choosing \emph{any} curve in a class of curves that satisfy certain public and well studied security parameters. Another topic of interest is generating hyperelliptic curves for cryptographic applications via the CM method that is based on the numerical estimation of the rational Igusa class polynomials. A recent development relates the denominators of the Igusa class polynomials to counting ideal classes in non maximal real quadratic orders whose norm is not prime to the conductor. Besides counting, our new algorithm provides precise representations of such ideal classes for all real quadratic fields and is part of an implementation in Magma of the recent theoretic work in the literature on the topic of denominators.

EPFL2016