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Manuscript Title: Programs for symmetry adaptation coefficients for semisimple symmetry chains: the general case.
Authors: M. Ramek, B. Gruber
Program title: LIE_A0, LIE_A1, LIE_A2
Catalogue identifier: ACHL_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 70(1992)371
Programming language: Pascal.
Computer: VAX 11/750.
Operating system: VAX/VMS, CMS.
RAM: 170K words
Word size: 32
Keywords: General purpose, Semisimple symmetry Chains, Lie algebra embedding, Symmetrization.
Classification: 4.2.

Nature of problem:
1. Calculation of orthonormal bases for irreducible unitary representations of the special unitary algebras (groups) SU(l+1) of any symmetry, and of orthonormal bases for direct products of such representations.

2. Calculation of orthonormal bases of irreducible unitary representations of the Lie algebras L=SU(l'+1), SO(2l'), SO(2l'+1), Sp(2l'), l' <=l, and direct products of these algebras, considered as subalgebras of an algebra SU(l+1) or a direct product of SU(l+1)'s. The bases for the irreducible unitary representations of the subalgebras L are obtained in terms of the bases of the representations of the algebra SU(l+1) by following a symmetry chain of algebras SU(l+1)-> L (symmetrization of states according to a symmetry chain).

3. The matrix elements for the generators of the Lie algebras are obtained together with the symmetrized wavefunctions.

4. The special cases SU(l+1) times SU(l+1) -> SU(l+1), SU(l+1) times SU(l+1) times SU(l+1) -> SU(l+1), etc. for the decomposition of a direct product of representations into its irreducible constituents are included.


Solution method:
Starting from the state vector corresponding to the highest weight of an irreducible representation of SU(l+1) of given symmetry, repeated application of shift operators generates all states within this irreducible representation. The initial states of all other irreducible representations of the given symmetry chain, which are contained in the representation of SU(l+1), are automatically generated using a precomputed list of all dominant subalgebra weights.

Restrictions:
The number of particles is limited to the number of bits in a word; no other restrictions except machine dependent storage limitations apply.

Unusual features:
To avoid rounding errors, only integer arithmetic is used throughout the programs. Linear combination coefficients, and all quantities related with these, are treated in the explicit form +/- sqrt (p/q), p and q being integers which are stored and manipulated in portions of a few digits in several variables; the programs are therefore not restricted to any machine dependent integer arithmetic limitations. Final output may be obtained by pipelining output files generated by the programs to TEX.

Running time:
Calculations of the totally antisymmetric representations of the embedding SU(6) -> SU(3) required the following CPU-times on a VAX station 3200:
 for 2 particles:                                  5 seconds             
 for 3 particles:                                 13 seconds             
 for 4 particles:                                 28 seconds             
 for 5 particles:                                 42 seconds             
 for 6 particles:                           1 min 18 seconds