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.
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.
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.
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.