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Manuscript Title: A Macsyma program for computing analytic Clebsch-Gordan coefficients of U(3) contained in SO(3).
Authors: T. Draeger
Program title: exactU3clebsch
Catalogue identifier: ADOL_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 139(2001)246
Programming language: Macsyma 2.4 (pro).
Computer: PC 300 MHz.
Operating system: MS Windows 95/98/NT.
Keywords: U(3), SU(3), SO(3), Clebsch-Gordan coefficients, Wigner coefficients, Isoscalar factors, Matrix elements reduced, Nuclear physics, Fractional parentage.
Classification: 17.18.

Nature of problem:
Elliott's nuclear SU(3) model [1,2] describes collective nuclear motion by considering the leading SU(3) irrep within the totally antisymmetric tensor product space of the valence nucleons. A rotational spectrum is expected, hence the basis for the irrep space should be symmetry adapted to the rotation group SO(3). The reduced matrix elements of the spherical tensor operator Q are needed to compute the electromagnetic transition rates. When coupling two SU(3) contained in SO(3) irreps to good SU(3) symmetry, the isoscalar factors together with the known SO(3) Clebsch-Gordan coefficients provide a unitary basis transformation from the uncoupled to the coupled basis. In the interacting boson model [3,4], one of the subgroup chains containing the rotation group is U(6) contained in SU(3) contained in SO(3). When writing the U(6) generators as SU(3) tensor operators, their matrix elements can be expressed in terms of the triple-barred matrix elements and the isoscalar factors [5].

Solution method:
It follows from the Borel-Weil theorem [6], that every irreducible representation of the compact group U(n) can be realized on the space of polynomials in the coordinates of GL(n,C). This space can be endowed with a 'differentiation' inner product [7], which is equivalent to the usual 'integration' inner product but more time-efficient for computer algebra programs like Macsyma. Applying the lowering operators to the highest weight and all subsequently generated polynomials while doing Gram-Schmidt orthogonalization, an orthonormal polynomial basis is generated. Then the representation when restricted to the generators of the desired subgroup (in this case SO(3) contains U(3)) is decomposed by diagonalizing the Cartan subalgebra of that subgroup, identifying the weight spaces and 'peeling off' the irreps of the subgroup. This is the standard mathematical procedure according to the Cartan-Weyl theory. Macsyma's sparse matrix routines allow for large matrix manipulation by way of keeping track only of the nonzero elements.

When working with a noncanonical basis, the reduction of an irrep to a subgroup is usually not multiplicity free, that is, there are missing labels. Racah has shown that there is no operator with integer eigenvalues that could be used as the missing label. However operators in the enveloping algebra with noninteger eigenvalues can be used. Since it is in general mathematically impossible to find the roots of a polynomial of degree greater than four using analytical methods (i.e. exactly), the computer algebra routines can fail if the degeneracy of the missing label is greater than four. Another limitation is the amount of memory of the PC. The larger the dimension of the U(3) irreps the more space is allocated by Macsyma for the different kinds of memory (cons, binary, ...). See Section 1.5 for details.

Unusual features:
When Macsyma computes in batch mode, it dynamically allocates space to the different types of memory while computing, but it cannot decrease the spaces already allocated unless one starts a new math engine. When the allocation levels get to high, more and more time is spent on garbage collecting up the the point where no more progress is made. That is why the program periodically writes intermediate results to temporary files, so that the computation can be interrupted at any time (except when saving). Then the computation can be resumed at the point of the most recent saving in a new notebook.

Running time:
A 300 MHz processor with 64 MB memory needs for the computation of all reduced matrix elements of Q about 30 minutes for the 125-dimensional (8,4,0) and about 2 days for the 315-dimensional (12,4,0). To compute all isoscalars, it needs about 20 minutes for the coupling of (3,1,0) with (2,1,0) and about 2 days for the coupling of (4,2,0) with (4,2,0).

[1] J.P. Elliott, Collective motion in the nuclear shell model, I, Proc. Roy. Soc. (London) A 245 (1958) 128-145.
[2] M. Harvey, The nuclear SU(3) model, Adv. in Nuclear Phys. 1 (1968) 67-182.
[3] F.Iachello, A. Arima, The Interacting Boson Model, Cambridge University Press, 1987.
[4] F. Alejandro, P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics, Wiley-Interscience, New York, 1994.
[5] G. Rosensteel, Analytic formulae for interacting boson model matrix elements in the SU(3) basis, Phys. Rev. C 41 (2) (1990) 730-735.
[6] A. Knapp, Representation Theory of Semisimple Groups, and Overview Based on Examples, Princeton University Press, 1986.
[7] W.H. Klink, T. Ton-That, Holomorphic induction and the tensor product decomposition of irreducible representations of compact groups. I. SU(n) groups, Ann. Inst. H. Poincare, Section A XXXI (2) (1979) 77-97.