Computer Algebra in Representation Theory of Algebras

Abstracts

David Green: Computing cohomology rings of finite groups (joint work with Simon King and Graham Ellis)
Abstract: The mod-$p$ cohomology ring of a finite group $G$ is a graded commutative ring whose Krull dimension is the $p$-rank of $G$. Topologists are particularly interested in the Cohen-Macaulay case, where the depth is as large as the dimension. Using the cohomology package Simon wrote for Sage, we computed the mod-2 cohomology of the third Conway sporadic finite simple group, and demonstrated that it is Cohen-Macaulay.

Edward Green: Applications of computer algebra in representation theory
Abstract: In this talk I will begin with a brief discussion of some basic tools of computer algebra useful in the study of representations of finite dimensional algebras, namely, Groebner bases and projective resolutions. I will follow this in a survey of some applications.

Lutz Hille: Tilting Modules and Dense Orbits, Computational Aspects
Abstract: Given a finite dimensional algebra it is desirable to classify all tilting modules. There are many examples where the number of isomorphism classes of tilting modules is finite. In such a case we have algorithms that compute all tilting modules. We discuss the principal ideas and several computational problems.
This problem is closely related to a problem for actions of algebraic groups. For those actions we also can use probabilistic methods. Finally the combination of both approaches seems to be very powerful to solve the classification with a computer.

Torsten Hoge: Computation of the ring of invariants of triples of 3x3 matrices
Abstract: The ring of invariants of triples of 3x3 matrices is the coordinate ring of the affine variety which points correspond to the three-dimensional semisimple representations of the free associative algebra in three variables. In this talk I will present a minimal presentation of this invariant ring and discuss the main problems of the computations.

Helmut Lenzing: Remarks on the K-theory of triangulated categories
Abstract: There will be a few comments on a computer-implementation of K-theoretic properties of triangulated categories with a full exceptional sequence. The comments concern a) the fractional Calabi-Yau property, b) perpendicular calculus, c) one-point extensions. We will also address open mathematical problems connected to the topics above.

Phillip Linke: Computations in generic representation theory
Abstract: In this talk I would give a quick introduction on the usage of GAP in generic representation theory. The first part will consist of an overview of the posed problem, the Artinian Conjecture by L. Schwartz. Afterwards I will introduce my algorithmical approach and give some examples.

Grischa Studzinski: Computations over the free associative algebra
Abstract: In this talk I will present the recent developments in Singular for computations in the free associative algebra, especially for computations of Groebner bases of two-sided ideals and K-bases of factor algebras. Moreover, I will give a preview for the current plans for the (near and medium term) future, as well as applications for our methods.