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Manuscript Title: Construction of symmetric group representation matrices and states.
Authors: M.F. Soto Jr., R. Mirman
Program title: ORTHNRM
Catalogue identifier: AAMG_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 23(1981)95
Programming language: PL/1.
Computer: IBM 3033.
Operating system: OS/MVT/ASP.
RAM: 224K words
Word size: 8
Peripherals: magnetic tape.
Keywords: General purpose, Algebras, Symmetric groups, Basis states, Orthonormality, Verification.
Classification: 4.2.

Subprograms used:
Cat Id Title Reference
AAME_v1_0 SYMSTATS CPC 23(1981)95

Nature of problem:
To check that the states produced by SYMSTATS form an orthonormal set.

Solution method:
An acceptable basis for the representations of the symmetric group consists of a set of multinomials, over which the action of the generators of the group is defined, which are complete and orthonormal. Orthonormality requires the definition of the product of terms. We define the product of any single product of boson terms with any other single product of terms as 1 in all the indices are in the same order in both terms, zero otherwise. From this product of sums can be found. This program verifies that the computed states are orthonormal. For each value of NMAX, the product of every state of every representation with every state of every representation is found. This product should be zero, except when the two states are identical, when it should be 1. In both states of a product, the terms corresponding to the different permutations are in the same order. Thus to find the product, what is done is to take the product of each of the two coefficients of each of the corresponding terms and sum them. Then the result is divided by the product of the normalization coefficients. These coefficients are given by the vector TH. The output, after NF, is the value of NMAX, with the normalization coefficients, the TH matrix, then followed, for each pair of states by the indices then the product, which is PS. This should be 1 if the first two indices are the same, the third and fourth are the same, and the fifth and sixth are also the same, zero otherwise. After PS, on the same line, is PSC, which is the correct value. These two numbers should agree (within rounding errors). One disagreement indicates an error. If there is a disagreement, the program outputs 4 blank lines (for emphasis) then "ERROR".

Any data produced by SYMSTATS can be checked. However, for S(10) and above, there must be some modifications because the numbers then consist of two digits.

Running time:
Through S(4): 16.78 s.