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

Moody, R. V.; Patera, J.

doi:10.1090/S0025-5718-1987-0878707-3

Computation of character decompositions of class functions on compact semisimple Lie groups
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by R. V. Moody and J. Patera PDF

Math. Comp. 48 (1987), 799-827 Request permission

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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