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

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**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.

## References

- J. Frank Adams,
*Lectures on Lie groups*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0252560** - N. Bourbaki,
*Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR**0240238** - Murray R. Bremner,
*Fast computation of weight multiplicities*, J. Symbolic Comput.**2**(1986), no. 4, 357–362. MR**872785**, DOI 10.1016/S0747-7171(86)80003-7 - M. R. Bremner, R. V. Moody, and J. Patera,
*Tables of dominant weight multiplicities for representations of simple Lie algebras*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 90, Marcel Dekker, Inc., New York, 1985. MR**779462**
J. Conway & L. Queen, - John D. Dixon,
*High speed computation of group characters*, Numer. Math.**10**(1967), 446–450. MR**224726**, DOI 10.1007/BF02162877 - Dragomir Ž. Djoković,
*On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups*, Proc. Amer. Math. Soc.**80**(1980), no. 1, 181–184. MR**574532**, DOI 10.1090/S0002-9939-1980-0574532-7 - Dragomir Ž. Djoković,
*On conjugacy classes of elements of finite order in complex semisimple Lie groups*, J. Pure Appl. Algebra**35**(1985), no. 1, 1–13. MR**772157**, DOI 10.1016/0022-4049(85)90026-X
E. B. Dynkin, "Semisimple subalgebras of semisimple Lie algebras," - V. G. Kac,
*Automorphisms of finite order of semisimple Lie algebras*, Funkcional. Anal. i Priložen.**3**(1969), no. 3, 94–96 (Russian). MR**0251091** - W. G. McKay, R. V. Moody, and J. Patera,
*Tables of $E_8$ characters and decompositions of plethysms*, Lie algebras and related topics (Windsor, Ont., 1984) CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 227–263. MR**832202** - W. G. McKay, R. V. Moody, and J. Patera,
*Decomposition of tensor products of $E_8$ representations*, Algebras Groups Geom.**3**(1986), no. 3, 286–328. MR**900487** - W. G. McKay and J. Patera,
*Tables of dimensions, indices, and branching rules for representations of simple Lie algebras*, Lecture Notes in Pure and Applied Mathematics, vol. 69, Marcel Dekker, Inc., New York, 1981. MR**604363** - Robert V. Moody,
*Root systems of hyperbolic type*, Adv. in Math.**33**(1979), no. 2, 144–160. MR**544847**, DOI 10.1016/S0001-8708(79)80003-1 - R. V. Moody and J. Patera,
*Fast recursion formula for weight multiplicities*, Bull. Amer. Math. Soc. (N.S.)**7**(1982), no. 1, 237–242. MR**656202**, DOI 10.1090/S0273-0979-1982-15021-2 - R. V. Moody and J. Patera,
*Characters of elements of finite order in Lie groups*, SIAM J. Algebraic Discrete Methods**5**(1984), no. 3, 359–383. MR**752042**, DOI 10.1137/0605037 - R. V. Moody, J. Patera, and R. T. Sharp,
*Character generators for elements of finite order in simple Lie groups $A_{1},$ $A_{2},$ $A_{3},$ $B_{2},$ and $G_{2}$*, J. Math. Phys.**24**(1983), no. 10, 2387–2396. MR**718223**, DOI 10.1063/1.525618
R. V. Moody, J. Patera & R. T. Sharp, "Elements of finite order and symmetry classes of tensors of simple Lie groups." (In preparation.)
- Henri J. Nussbaumer,
*Fast Fourier transform and convolution algorithms*, Springer Series in Information Sciences, vol. 2, Springer-Verlag, Berlin-New York, 1981. MR**606376** - K. R. Parthasarathy, R. Ranga Rao, and V. S. Varadarajan,
*Representations of complex semi-simple Lie groups and Lie algebras*, Ann. of Math. (2)**85**(1967), 383–429. MR**225936**, DOI 10.2307/1970351 - A. Pianzola,
*On elements of finite order and cyclotomic fields*, Lie algebras and related topics (Windsor, Ont., 1984) CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 351–355. MR**832209**
A. J. Pianzola, "On the arithmetic of the representation ring and elements of finite order in Lie groups," - R. Slansky,
*Group theory for unified model building*, Phys. Rep.**79**(1981), no. 1, 1–128. MR**639396**, DOI 10.1016/0370-1573(81)90092-2 - T. A. Springer,
*Regular elements of finite reflection groups*, Invent. Math.**25**(1974), 159–198. MR**354894**, DOI 10.1007/BF01390173 - B. G. Wybourne and M. J. Bowick,
*Basic properties of the exceptional Lie groups*, Austral. J. Phys.**30**(1977), no. 3, 259–286. MR**462278**

*Computing the Character Table of a Lie Group*, Proc. Conf. on Finite Groups, Montreal, 1982.

*Amer. Math. Soc. Transl.*(2), 1957, pp. 111-244. E. B. Dynkin, "Maximal subgroups of the classical groups," Suppl. 23,

*Amer. Math. Soc. Transl.*(2), v. 6, 1957, pp. 245-378. M. J. Englefield,

*Tabulation of Kronecker products of representations of F*4,

*E*6,

*and E*7, Preprint, Univ. of Southampton, Math. N57, 1981.

*J. Algebra*. (To appear.) I. Schur,

*Über die Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen*, Dissertation, Berlin, 1901. Collected Works, Vol. I, Springer-Verlag, New York, 1973.

## Additional Information

- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp.
**48**(1987), 799-827 - MSC: Primary 22E46
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878707-3
- MathSciNet review: 878707