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Manuscript Title: LieART - A Mathematica Application for Lie Algebras and Representation Theory
Authors: Robert Feger, Thomas W. Kephart
Program title: LieART
Catalogue identifier: AEVL_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 192(2015)166
Programming language: Mathematica.
Computer: x86, x86_64, PowerPC.
Operating system: cross-platform.
RAM: ≥ 1 GB recommended. Memory usage depends strongly on the Lie algebra's rank and type, as well as the dimensionality of the representations in the computation.
Keywords: Lie algebra, Lie group, Representation theory, Irreducible representation, Tensor product, Branching rule, GUT, Model building.
PACS: 02.20.Hj, 02.20.Qs, 02.20.Sv, 11.30.-j.
Classification: 4.2, 11.1.

External routines: Wolfram Mathematica 8-10

Nature of problem:
The use of Lie algebras and their representations is widespread in physics, especially in particle physics. The description of nature in terms of gauge theories requires the assignment of fields to representations of compact Lie groups and their Lie algebas. Mass and interaction terms in the Lagrangian give rise to the need for computing tensor products of representations of Lie algebras. The mechanism of spontaneous symmetry breaking leads to the application of subalgebra decomposition. This computer code was designed for the purpose of Grand Unified Theory (GUT) Model building, where compact Lie groups beyond the U(1), SU(2) and SU(3) of the Standard Model of particle physics are needed. Tensor product decomposition and subalgebra decomposition have been implemented for all classical Lie groups SU(N), SO(N) and Sp(2N) and the exceptionals E6, E7, E8, F4 and G2.

Solution method:
LieART generates the weight system of an irreducible representation (irrep) of a Lie algebra by exploiting the Weyl reflection groups, which is inherent in all simple Lie algebras. Tensor products are computed by the application of Klimyk's formula, except for SU(N)'s, where the Young-tableaux algorithm is used. Subalgebra decomposition of SU(N)'s are performed by projection matrices, which are generated from an algorithm to determine maximal subalgebras as originally developed by Dynkin [1, 2].

Restrictions:
Internally irreps are represented by their unique Dynkin label. LieART's default behavior in TraditionalForm is to print the dimensional name, which is the labeling preferred by physicists. Most Lie algebras can have more than one irrep of the same dimension and different irreps with the same dimension are usually distinguished by one or more primes (e.g. 175 and 175′ of A4). To determine the need for one or more primes of an irrep a brute-force loop over other irreps must be performed to search for irreps with the same dimensionality. Since Lie algebras have an infinite number of irreps, this loop must be cut off, which is done by limiting the maximum Dynkin digit in the loop. In rare cases for irreps of high dimensionality in high-rank algebras, if the cutoff used is too low, then the assignment of primes will be incorrect, but the problem can be avoided by raising the cutoff. However, in either case, this can only affect the display of the irrep because all computations involving this irrep are correct, since the internal unique representation of Dynkin labels is used.

Running time:
From less than a second to hours depending on the Lie algebra's rank and type and/or the dimensionality of the representations in the computation.

References:
[1] E. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Trans.Am.Math.Soc. 6 (1957) 111.
[2] E. Dynkin, Maximal subgroups of the classical groups, Trans.Am.Math.Soc. 6 (1957) 245.