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Manuscript Title: Construction of symmetric group representation matrices and states.
Authors: M.F. Soto Jr., R. Mirman
Program title: SYMRPMAT
Catalogue identifier: AAMF_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: 174K words
Word size: 8
Peripherals: magnetic tape.
Keywords: General purpose, Algebras, Symmetric group, Representation matrix, Permutation, Explicit numerical Values.
Classification: 4.2.

Subprograms used:
Cat Id Title Reference
AAMC_v1_0 SYMGRPTB CPC 23(1981)81
AAME_v1_0 SYMSTATS CPC 23(1981)95

Nature of problem:
To find the explicit representation matrices for every permutation, for every representation, for any symmetric group.

Solution method:
The representation matrices for the neighboring transpositions are found from the formulas. Every permutation can be written as a product of neighboring transpositions. Thus, the representation matrix for permutation is found by simply multiplying the matrices of the neighboring transpositions into which the permutation decomposes. The main program is similar to SYMSTATS, in setting variables and calling the subroutines which are standard to all the programs in this set. The subroutine packages used are ROWLEN, GENPERM, FAC and CONSMAT. It is in the subroutine RM of the latter package that the matrices are actually found. In the main program there are DO-loops over the representations, and over the permutations, inside of which is a call to RM to compute the matrix for the current permutation. It is stored in UTR, indexed by the permutation index and by the matrix row and column indices. There are two structures found in the program, in order to provide a convenient way of placing on tape parameters needed by the programs which reformat and check (and in other ways may use) the matrices. One is NUMTRNS, which contains the variable LH, the largest dimension of any of the representations of the current symmetric group (plus 1 to avoid problems when DO-loop bounds are checked), and NRP, the estimate of the number of representations of the current group. The other is DATTRNS, which contains IG, the set of dimensions and NU, the number of dimensions for each symmetric group up to, and including, the present one. The variables in the structures all end in T, and are used to transfer the corresponding variables to the other programs (the variables themselves could not be in the structures since these are DECLARED inside the DO-loop giving the symmetric group, and thus would be reinitialized). The output of the program is NF, which gives the largest symmetric group considered, then for each NMAX, the value of NMAX and the two structures, followed for each representation by NMAX and KA, the representation index, and then by the matrices, UTR, for that representation. The results of this program are checked by the accompanying program MATTAB. That it is running correctly can be seen by checking that all representations for each NMAX are printed out, for all permutations, and that the neighboring transpositions are correct.

The program can be used for any symmetric group, for which machine time and storage are available. However, for S(10) and above, there must be some modifications because the numbers then consist of two digits.

Running time:
Through S(4): 23.18 s.