ICMS 2016 Session: Computational aspects of homological algebra, group, and representation theory

ICMS 2016 (July 11 to July 14 held at ZIB Berlin): Home, Sessions

Organizers

Mohamed Barakat (University of Siegen, Germany)
Max Horn (Justus Liebig University Giessen, Germany)

Aim and Scope

Homological algebra is a universal language with an established presence in many fields of mathematics. The session will focus on modern applications of computational homological methods to the representation theory of groups and algebras. These methods range from the computability of group, Lie algebra, and Hochschild cohomology to the constructivity of Morita, tilting, and various other equivalences of derived and differentially enriched categories. Such equivalences also connect the representation theory to equivariant coherent sheaves on varieties admitting a tilting object and hence to algebraic geometry.

Topics (including, but not limited to)

group cohomology
Lie algebra cohomology
modules over path algebras
constructive category theory
coherent sheaf cohomology

Publications

A short abstract will appear on the permanent conference web page (see below) as soon as accepted.
An extended abstract will appear on the permanent conference web page (see below) as soon as accepted.
It will also appear on the proceedings that will be distributed during the meeting.
A journal special issue consisting of full papers will be organized immediately after the meeting.

Submission Guidelines

If you would like to give a talk at ICMS, you need to submit first a short abstract and then later an extended abstract. See the guideline for the details.
After the meeting, the submission guideline for a journal special issue will be communicated to you by the session organizers.

Talks

Group cohomology and efficient methods for group algebras of large p-groups.
David Green (Friedrich-Schiller-Universität Jena, Germany) (Slides)
Abstract: How could one perform efficient calculations in the modular group algebra of a large p-group? For the p-group itself one would want to use a power-commutator presentation; I shall report on adapting this approach to the case of the group algebra. One intended application is the computation of the low-dimensional mod-2 cohomology groups of the sporadic finite simple group J₄. Parts are joint work with Robert Müller.
Endo-p-permutation modules: a computational approach via character theory
Caroline Lassueur (Technische Universität Kaiserslautern, Germany)
Abstract: The endo-p-permutation and endo-trivial modules over the group algebra of a finite group over a field of prime characteristic are classes of modules with the nice property to be liftable to characteristic zero. Thus the problem of their classification leads naturally to consider properties of the corresponding ordinary characters over the field of complex numbers. Through this approach many groups can be investigated using computational methods.
Constructing morphisms by diagram chases
Sebastian Posur (University of Siegen, Germany) (Slides)
Abstract: In homological algebra, a method called diagram chasing is used to define important homomorphisms between modules over a ring. Examples of such morphisms are the connecting homomorphism in the snake lemma, and the differentials on the pages of a spectral sequence. In this talk, we discuss data structures that can be used to perform diagram chases constructively in the context of an arbitrary abelian category. These data structures are already implemented in our GAP-package CAP (Categories, Algorithms, and Programming). This is joint work with Mohamed Barakat and Sebastian Gutsche.
Test for infinite projective dimension
Øyvind Solberg (NTNU Trondheim, Norway)
Abstract: Let Λ be a finite dimensional algebra over an algebraically closed field (or an admissible quotient of path algebra). An open problem for such algebras is the Finitistic Dimension Conjecture: The supremum of the projective dimension of all the finitely generated modules which have finite projective dimension is finite.
If one would know that this is true and one could compute the finitistic dimension of an algebra, one would have a finite test for showing that a module have infinite projective dimension. As this is still open, such a finite test is to my knowledge unknown.
We introduce the notion of a matrix factorization associated to all finitely generated Λ-modules, a dimension vector invariant of any finitely generated module over Λ and an abelian group G(Λ) in which all dimension vector invariants give rise to an element. We show that if the dimension vector invariant of a module M gives rise to a non-zero element in G(Λ), then the projective dimension of M is infinite. This is a finite test as it only involves computing a projective presentation of the module.