SYMMETRICA manual

Harald Fripertinger1

November 19, 2001

Copyright © by

Lehrstuhl II für Mathematik

Department of Mathematics

University of Bayreuth

Postfach 10 12 51

8580 Bayreuth, Germany

This program system is devoted to the linear representation theory, the invariant theory and the combinatorics of finite symmetric groups and related classes of groups. It was developed jointly at the Lehrstühle II/IV für Mathematik, University of Bayreuth, at the LITP, Université Paris VII, and at the University of Wales, Aberystwyth. Financial support by the Deutsche Forschungsgemeinschaft and the European exchange programs PROCOPE and ARC is gratefully acknowledged.

The permission to redistribute this software remains at the Lehrstuhl II für Mathematik. Copies are distributed freely for non-commercial purposes only. You will get the program source together with a LaTeX-source for this introductory user's guide.

SYMMETRICA is distributed without any warranty. Please inform us about bugs, modifications, wishes, improvements and applications.

  • Preface
  • The installation
  • The installation
  • First steps in a UNIX environment
  • First steps in a DOS environment
  • The documentation files
  • The example files
  • The use of SYMMETRICA
  • The file test.c
  • More details
  • Factorial numbers
  • The first object
  • How SYMMETRICA does it
  • Characters
  • Mixed examples
  • Combinatorics
  • Symmetric polynomials
  • Multivariate polynomials
  • Finite group actions
  • Orbit-enumeration in SYMMETRICA
  • Some basic cycle index formulae
  • Multi-dimensional cycle indices
  • The cycle indices of the symmetry groups of platonic solids
  • Linear, affine and projective groups
  • Induced cycle indices
  • Products of cycle indices
  • The Redfield operators
  • Substitutions into cycle indices
  • Some further cycle index routines
  • Some example programs
  • Orbit-construction in SYMMETRICA
  • Generators of permutation groups
  • Generators of induced actions
  • Special routines for wreath products
  • Some further generators
  • Actions of the form GX
  • Action on the domain of functions
  • Applications
  • Enumeration of isometry classes of linear codes
  • Endofunctions of given cycle type
  • Combinatorics of the fullerenes
  • Characters
  • Plethysms
  • Matrix representations
  • Symmetry adapted bases
  • The arithmetic
  • The Objects
  • The empty object
  • How to select the object kind
  • How to change the object kind
  • Integers
  • How to make an INTEGERobject
  • How to select its INTvalue
  • How to change an INTEGERobject
  • How to build an INTEGERobject
  • Particular routines for INTEGERobjects
  • General routines
  • Vectors
  • How to build a VECTORobject
  • Selection of and access to parts
  • Initialization
  • Numeric routines
  • Further routines
  • Partitions
  • Changes
  • Constructions
  • Testing and sorting
  • Building
  • Advanced routines
  • Comparing partitions
  • Operations on partitions
  • General routines
  • Permutations
  • How to change a PERMUTATIONobject
  • How to build a PERMUTATIONobject
  • How to handle PERMUTATIONobjects
  • Barred Permutations
  • Lists
  • Selection
  • Construction
  • Empty LISTobject
  • Insertion
  • Transformation
  • General routines
  • Polynomials
  • Selections
  • Making and building
  • Further routines
  • General routines
  • Schur polynomials
  • Access to and change of parts
  • Building
  • Matrices
  • Selections and changes
  • Special routines
  • General routines
  • Characters of symmetric groups
  • Selections
  • Building or making such objects
  • Evaluation of characters
  • General routines
  • Skewpartitions
  • Selections
  • Buildung or making
  • Tableaux
  • Selections
  • Building and making
  • Words, rows, columns, content, shape
  • Jeu de taquin, Lehmer code
  • General routines
  • Index
  • References
  • Footnotes

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