Computation of character decompositions of class functions on compact semisimple Lie groups

Authors:
R. V. Moody and J. Patera

Journal:
Math. Comp. **48** (1987), 799-827

MSC:
Primary 22E46

DOI:
https://doi.org/10.1090/S0025-5718-1987-0878707-3

MathSciNet review:
878707

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Abstract: A new algorithm is described for splitting class functions of an arbitrary semisimple compact Lie group *K* into sums of irreducible characters. The method is based on the use of elements of finite order (EFO) in *K* and is applicable to a number of problems, including decompositions of tensor products and various symmetry classes of tensors, as well as branching rules in group-subgroup reductions. The main feature is the construction of a decomposition matrix *D*, computed once and for all for a given range of problems and for a given *K*, which then reduces any particular splitting to a simple matrix multiplication. Determination of *D* requires selection of a suitable set *S* of conjugacy classes of EFO representing a finite subgroup of a maximal torus *T* of *K* and the evaluation of (Weyl group) orbit sums on *S*. In fact, the evaluation of *D* can be coupled with the evaluation of the orbit sums in such a way as to greatly enhance the efficiency of the latter. The use of the method is illustrated by some extensive examples of tensor product decompositions in ${E_6}$. Modular arithmetic allows all computations to be performed exactly.

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Article copyright:
© Copyright 1987
American Mathematical Society