Adalbert Kerber

Representations of permutation groups

Characters of wreath products.- An application to representation theory: Symmetrization of inner tensor products of representations.- An application to combinatorics: The theory of enumeration under group action.

MOLGEN+, a generator of connectivity isomers and stereoisomers for molecular structure elucidation

The structure generator MOLGEN produces all the molecular graphs that correspond to a given chemical formula and (optionally) prescribed and forbidden substructures as well as further optional conditions like intervals for possible ring sizes, hybridization of carbon atoms (not completely implemented), etc. It does this in a fast and redundancy free way, which seems, to our knowledge, to be more efficient than all the other molecular generators known [see R. Grund, A. Kerber and R. Laue, MATCH, 27 (1992) 87, for introductions to various other generators]. Afterwards these molecular graphs can be displayed on the computer screen, 3D-placements of them (using an energy model) can be computed, and the stereoisomers can be calculated and visualized basing on this placement. Therefore, this generator serves very well as the mathematical core of computer packages for molecular structure elucidation. Moreover, it can easily be used for demonstrations in chemical education. MOLGEN developed from MOLGRAPH [introduced in A. Kerber, R. Laue and D. Moser, Anal. Chim. Acta, 235 (1990) 221]. In contrast to MOLGRAPH it has a much wider range of application since it does not need a catalog of precomputed cyclic substructures.

SYMMETRICA, an object oriented computer-algebra system for the symmetric group

Abstract This is a review of an object oriented computer algebra system which is devoted to epresentation theory, invariant theory and combinatorics of the symmetric group. Moreover, it can be used for classical multivariate polynomials via the different actions of the symmetric group on the algebra of polynomials. The review contains a brief introduction to the basic methods used. Schubert polynomials are introduced, examples are given, and some applications are described. In particular, they provide a new algorithm for the evaluation of Littlewood-Richardson coefficients via symbolic computations using integer sequences instead of partitions, tableaux or lattice permutations.

Error-correcting linear codes : classification by isometry and applications

Linear Codes.- Bounds and Modifications.- Finite Fields.- Cyclic Codes.- Mathematics and Audio Compact Discs.- Enumeration of Isometry Classes.- Solving Systems of Diophantine Linear Equations.- Linear Codes with a Prescribed Minimum Distance.- The General Case.

MOLGEN-CID: A canonizer for molecules and graphs accessible through the internet

The MOLGEN Chemical Identifier MOLGEN-CID is a software module freely accessible via the Internet. For a molecule or graph entered in molfile format (2D) it produces, by a canonical renumbering procedure, a canonical molfile and a unique character string that is easily compared by computer to a similar string. The mode of operation of MOLGEN-CID is detailed and visualized with examples.

Simple 8-Designs with Small Parameters

We show the existence of simple 8-(31,10,93) and 8-(31,10,100) designs. For each value of λ we show 3 designs in full detail. The designs are constructed with a prescribed group of automorphisms PSL(3,5) using the method of Kramer and Mesner KramerMesner76. They are the first 8-designs with small parameters which are known explicitly. We do not yet know if PSL(3,5) is the full group of automorphisms of the given designs. There are altogether 138 designs with λ = 93 and 1658 designs with λ = 100 and PSL(3,5) as a group of automorphisms. We prove that they are all pairwise non-isomorphic. For this purpose, a brief account on the intersection numbers of these designs is given. The proof is done in two different ways. At first, a quite general group theoretic observation shows that there are no isomorphisms. In a second approach we use the block intersection types as invariants, they classify the designs completely.

The Discovery of Simple 7-Designs with Automorphism Group PTL (2, 32)

A computer package is being developed at Bayreuth for the generation and investigation of discrete structures. The package is a C and C++ class library of powerful algorithms endowed with graphical interface modules. Standard applications can be run automatically whereas research projects mostly require small C or C++ programs. The basic philosophy behind the system is to transform problems into standard problems of e.g. group theory, graph theory, linear algebra, graphics, or databases and then to use highly specialized routines from that field to tackle the problems. The transformations required often follow the same principles especially in the case of generation and isomorphism testing.

QSPR Using MOLGEN-QSPR: The Example of Haloalkane Boiling Points

MOLGEN-QSPR is a software newly developed for use in quantitative structure property relationships (QSPR) work. It allows to import, to manually edit, or to generate chemical structures, to detect duplicate structures, to import or to manually input property values, to calculate the values of a broad pool of molecular descriptors, to establish QSPR equations (models), and using such models to predict unknown property values. In connection with the molecule generator MOLGEN, MOLGEN-QSPR is able to predict property values for all compounds in a predetermined structure space (inverse QSPR). Some of the features of MOLGEN-QSPR are demonstrated on the example of haloalkane boiling points. The data basis used here is broader than in previous studies, and the models established are both more precise and simpler than those previously reported.

Isometry Classes of Indecomposable Linear Codes

In the constructive theory of linear codes, we can restrict attention to the isometry classes of indecomposable codes, as it was shown by Slepian. We describe these classes as orbits and we demonstrate how they can be enumerated using cycle index polynomials. The necessary tools are already incorporated in SYMMETRICA, a (public domain) computer algebra package devoted to representation theory and combinatorics of symmetric groups and of related classes of groups. Moreover, we describe how systems of representatives of these classes can be evaluated using double coset methods.

Ein strukturgenerator für molekulare graphen

Abstract ( A structure generator for molecular graphics. ) The structure generator presented is suitable for molecular structure elucidation. The generator is self-standing and can (in suitable cases) construct a nonredundant exhaustive list of connected multigraphs with appropriate ranking and prescribed or forbidden subgraphs.