The investigator studies computational representations of a finite group over finite fields. Groups can be thought of as the mathematical abstraction of the common notion of symmetries. As such they arise in mathematics and natural sciences such as for example physics, chemistry, and biology. The symmetry group of an object usually also acts on function spaces that are related to the given object. This is a so-called linear action and in the case where the function space is finite dimensional the group acts as invertible matrices on the space. Such a realization is called a (matrix-) representation of the group. In this context one distinguishes two cases, the one where the matrices are written over a field of characteristic 0 such as the complex numbers, or the case where the entries are in a finite field. This case is particularly interesting since here the representation theory is tightly connected to the structure of the group. The investigator develops a share package of programs in the computer algebra system GAP. This system is free of charge, well documented, and widely used. The theory of representations in the finite characteristic case is far from being fully understood. For example, up to now not even all the irreducible representations for all finite simple groups are known. The share package implements algorithms to help solve several questions concerning the representations of a finite group. First of all it enables the user to compute the projective indecomposable representations of a given group or alternatively access them via a data base that is provided by the package. These representations are of fundamental importance when one is interested in getting an overview of all representations of the group. From the projective indecomposable representations one can derive an algebra called the basic algebra that has the same representation theory as the group. This algebra is much smaller than the group algebra itself and hence also has important applications in computing with representations. The share package therefore also contains functions to compute the basic algebra. Even though the basic algebra is a very important invariant, barely anything is known about it even for specific groups. The share package increases the knowledge about specific groups tremendously. Moreover, there is an explicit algorithmic connection between representations of a group and that of its basic algebra. The share package incorporates functions that allow the user to analyze and construct representations using this connection. This is of particular importance in computations of the cohomology ring or the Ext-algebra of a group. A group is a mathematical object that captures notions of arrangement and symmetry -- for example, the different orientations of a square when turned ninety degrees at a time comprise a group. Groups arise in mathematics and natural sciences such as physics, chemistry, and biology. Exploiting the group structure often leads to deep insights into problems in these areas. The investigator develops ways to represent and computationally study finite groups. He implements these methods in computer software, building on the freely available software system GAP. The package he develops contains data bases of computed results. These are accessible to the public via the GAP system and alternatively on the world wide web. GAP itself is used in the educational environment and as part of GAP one possible application of the software package is in the class room. It gives students hands-on experience with representation theory free of charge.
