Preface

# Preface

SYMMETRICA is a computer algebra package devoted to representation theory, invariant theory and combinatorics of finite symmetric groups and related classes of groups (like the alternating groups, wreath products of symmetric or alternating groups, but also finite and infinite linear groups and other classes of groups). It is meant as a tool for the scientist working on representation theory or on its applications in mathematics, physics or chemistry.

The aim was to develop a system that can be run on any computer which contains a C-compiler, whatever the operating system is.

The development started with a program devoted to the Littlewood-Richardson Rule, for which the method of Schubert polynomials was used. Procedures for the evaluation and decomposition of characters were added, and in particular programs for the evaluation of irreducible matrix representations were implemented, both for the ordinary and for the modular case.

Symmetric group representation theory is an interesting playground for all kinds of symbolic calculations since its results are mostly formulated in terms of sequences of natural numbers or of tableaux. The programs are written in an object oriented way that allows to avoid the implementation of an extra language on top of the procedures. We wanted to keep the usability of programming tools like optimizer, debugger, profiler and so on.

SYMMETRICA provides routines for handling in particular the following mathematical structures:

• Ordinary irreducible and Brauer characters as well as decomposition numbers of symmetric groups,
• ordinary irreducible characters of alternating groups,
• ordinary irreducible characters of wreath products of symmetric groups,
• ordinary and modular irreducible matrix representations of symmetric groups,
• ordinary irreducible polynomial representations of general linear groups GLm(C),
• multivariate polynomials and in particular Schubert polynomials, also zonal polynomials,
• Schur polynomials as well as several other series of symmetric polynomials together with base change matrices,
• cycle indicator polynomials for combinatorial enumeration,
• the ordinary group algebra of the symmetric groups.
Using these structures and appropriate procedures, you can evaluate irreducible characters, and decompose reducible ones. You can do combinatorial enumeration to some extent, and you can also apply symmetry adapted bases by an application of irreducible matrix representations, which can be evaluated explicitly. For these procedures you can use
• integer arithmetic, including long integers which are used automatically and if necessary,
• cyclotomic fields, which are necessary, for example, if you want to evaluate characters of alternating groups.
At present for modular purposes only primes fields of prime characteristics are used and necessary. Later on this will be extended.

The following pages contain a brief description how SYMMETRICA can be installed and how the inexperienced but interested beginner who has not much knowledge in programming, in computer systems like UNIX or DOS, or in programming languages like C, can make his first steps towards a systematic use of SYMMETRICA . This is the content of the first two chapters. After that we give more details. In the third chapter we describe how object oriented programming works so that the interested reader can start writing his own programs in this style. The third chapter also contains a review of the main topics that can be treated using SYMMETRICA .

The definitions of the most important objects can be found in the fourth chapter together with the corresponding main routines for handling these objects.

The fifth chapter contains an index of the available functions listed lexicographically and seperated into sections that contain their definition.

The experienced reader may immediately start using SYMMETRICA by installing it from the diskette, reading the file install.doc first and then the documentation files which are the files with names ending by .doc. A good part of this documentation is also contained in the present manual.

Our aim was to put the reader of these pages in a position that she or he should be able to continue playing and more seriously working while reading from time to time in the documentation files which are available in SYMMETRICA . It is our hope that she or he then might like this tool and find it useful in his or her own research on finite groups and their applications. Moreover we expect a feed-back or even an extension of SYMMETRICA which then can be added to a further release of this computeralgebra system.

At present, SYMMETRICA contains programs written by M. Bauch, C. Carré, U. Eidt, P. Frank, Th. Fürbringer, A. Golembiowski, R. Hager, M. Hain, A. Kerber, I. Klein, A. Kohnert, A. Lascoux, Th. Leitner, T. McDonough, C. Precetti, N. Schüler, F. Stötzer, W. Weber.

Bayreuth, November 19, 2001        A. Kerber, A. Kohnert

harald.fripertinger@kfunigraz.ac.at,
last changed: November 19, 2001

 Preface