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Manuscript Title: Generation of the Clebsch-Gordan coefficients for Sn.
Authors: S. Schindler, R. Mirman
Program title: SYMCGM
Catalogue identifier: ACXV_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 15(1978)131
Programming language: Fortran.
Computer: IBM 370/168.
Operating system: OS/MVT/ASP.
RAM: 252K words
Word size: 8
Peripherals: magnetic tape.
Keywords: General purpose, Symmetric group, Clebsch-Gordan coefficient, Tensor product, Decomposition, Irreducible Representation, Algebras.
Classification: 4.2.

Subprograms used:
Cat Id Title Reference
ACXW_v1_0 SYMFUNC CPC 15(1978)147

Nature of problem:
To find the decomposition of the tensor product of two irreducible representations of the symmetric group Sn (for any n) into a direct sum of irreducible representations, and to compute the matrices for the similarity transformations.

Solution method:
An iterative formula has been derived which gives the tensor coupling coefficients for Sn in terms of the Clesbsch-Gordan coefficients for Sn-1 and the matrix elements for the transposition (n-1n). The computation is then essentially by direct substitution. Symmetry relations are used to reduce the number of coefficients which need to be calculated. The Clebsch-Gordan coefficients are then found from the tensor coupling coefficients by methods previously developed.

Restrictions:
These are determined by the amount of storage space and running time allotted. For large n imprecision of the coefficients can be greater than their magnitude. The values of n for which the program works can be increased by using greater precision. If n is greater than 12 the integer format of the function FAC will have to be changed. With the values presently specified in the dimension statements the program will work for all n through 5 and in addition for all triplets for n=6. However, the larger triplets will have to be done individually.

Unusual features:
Since this is an iterative procedure the coefficients for Sn-1 must be stored for use in calculating those for Sn. This requires a large amount of storage. The program is designed to store only the non-zero coefficients. Symmetry properties allow the calculation and storage of only a subset of these. Thus, a large part of the program consists of search routines for finding coefficients.