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Manuscript Title: Programs for symmetry adaption coefficients for semisimple symmetry chains: the completely symmetric representations.
Authors: T. Nomura, M. Ramek, B. Gruber
Program title: LIE_S1,LIE_S2
Catalogue identifier: ABTO_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 61(1990)410
Programming language: Pascal.
Computer: VAX 11/750, VAX SERVER 3600, IBM 3090.
RAM: 120K words
Word size: 8
Keywords: General purpose, Semisimple symmetry chains, Lie algebra embedding, Symmetrization, Quantum states.
Classification: 4.2.

Nature of problem:
1. Calculation of orthonormal bases for the completely symmetric irreducible unitary representations [[N]] of the special unitary algebras (groups) SU(l+1), and of orthonormal bases for direct products of representations [[N1]] otimes [[N2]] otimes [[N3]]... of the algebras SU(l1+1) otimes SU(l2+1) otimes SU(l3+1)... . (N denotes the number of particles and [[N]] denotes the completely symmetric representations of the pair (SU(l+1), SN), SN being the symmetric group generated by N particles.)
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. That is, the bases for the irreducible unitary representations of the subalgebras L are obtained in terms of the bases of the completely symmetric representations [[N]] of an 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) otimes SU(l+1) -> SU(l+1), SU(l+1) otimes SU(l+1) otimes SU(l+1) -> SU(l+1), etc. for the decomposition of a direct product of representations [[N1]] otimes [[N2]], [[N1]] otimes [[N2]], otimes [[N3]], etc. 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), 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 a representation [[N]] of SU(l+1), are automatically generated using a precomputed list of all dominant subalgebra weights.

None except machine dependent storage limitations.

Unusual features:
To avoid rounding errors, only integer arithmetic is used throughout the the programs. Linear combination coefficients, and all quantities related with these, are treated in the explicit form +- square root (p/q), p and q being integers 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 embedding SU(6) -> SU(3) required the following CPU- times on a VAXserver 3600:
 representation [[2]]:                     6 sec;                        
 representation [[3]]:                    15 sec;                        
 representation [[4]]:                    44 sec;                        
 representation [[5]]:              2 min 24 sec;                        
 representation [[6]]:              7 min 14 sec;                        
 representation [[7]]:             20 min 53 sec;                        
 representation [[8]]:         1 h 05 min 53 sec;                        
 representation [[9]]:         1 h 59 min 17 sec.