Programs in Physics & Physical Chemistry
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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.|
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.
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.
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.
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.
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