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Internal multiplicities in the Cartan classes
Physica 114A (1982) 341-344 North-Holland
INTERNAL
Publishing Co.
MULTIPLICITIES
IN THE CARTAN
CLASSES
V. AMAR, U. DOZZIO and C. OLEARI Istitulo di Fisica dell’Uniuersitic di Parma, Sezione di Fisica Teorica, via Massimo d’Azeglio 8.5, l-43100 Parma, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy
It is shown how the evaluation of the internal multiplicities of a simple Lie group G(n) is possible by means of the reduction G(n)+G(n - 1) @ U(1). An algorithm is then given for evaluating the branching multiplicity G(n)-+ G(n - 1) @J U(1) when G(n) = SO(2n) and G(n) = SO(2n + 1).
1. Introduction
The computation of the internal multiplicity for simple Lie groups is a classical problem, which has two general and elegant methods of solution: one is the enumerative Kostant formula’) and the other is the recursive Freudenthal formula*). Recently more effective methods have been obtained for SU(n + 1) groups’) and for Sp(2n) groups4). In this paper we propose algorithms for the groups SO(2n) and SO(2n + l), which are more effective than the usual recursive method and are very similar to the previously mentioned SU(n + 1) and Sp(2n) methods. The validity of these algorithms has been supported by a very large number of numerical tests, as will be seen below.
2. General theory Let G(n) denote a simple Lie group of rank n, the irreducible representations of G(n) be identified by the highest weight M = (Ml,. . . , M,,) and an arbitrary weight denoted by m = (ml,. . . , m,). We denote by rg, the internal multiplicity of weight m in the irreducible representation denoted by M. If G’ is a Lie group such that G’ C G, we denote by T$qG,the branching multiplicity of the irreducible representation of G’ defined by M’ in the irreducible representation of G defined by M (hereafter, fixed M and M’, the indices G and G’ are suppressed if the groups G and G’ appear unambiguously known from the text). 0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
342
V. AMAR et al.
If G’ is naturally embedded in G, i.e. the root system subsystem of the roots of G, the following formula’) holds
of G’ forms
a
(1) where L is the map of the weights of G on the weights of G’. On the left-hand side the summation is over weights m such that Lm = m’, while on the right-hand side the summation is over the irreducible representations of G’ which contain the weight m’ and are contained in the given irreducible representation of G. Now if we assume G’(n) = G(n - 1) @I U(l), where G(n - 1) belongs to the same Cartan class of G(n), then map L is the identity map, i.e. Lm = m, and formula (1) becomes -M
M'
(2)
YM’Y(mh
where
in agreement with the structure of G’. From eq. (2), if we find a method for evaluating -& by recursion it would also be possible to evaluate yfA)*. A method for evaluating 9% is known for G(n) = SU(n + 1) (cf. branching law of Weyl”)) and recently an algorithm for G(n) = Sp(2n) was given4). Rather simple procedures are proposed below for the groups G(n) = SO(2n) and G(n) = SO(2n + 1). So far, we have not been able to prove these algorithms; however, we have tried a very large set of numerical tests (vide infra) and we have no doubt that these algorithms are correct.
3. The SO(2n) case For evaluating ~~~z~!,,, U(I)the following algorithm (MI,. . . , MA) with MA-15 0 let us consider the relation
then evaluate the number of sets of n - 1 non-negative
holds:
given M’ =
numbers qi such that
* In (1) and (2) the condition M’ C M can be satisfied by using Gel’fand patterns6”) related to the different Cartan classes, while the condition M’ > m follows from the constraints given in refs. 8 and 9 for the groups SU(n + 1) and SO(2n). Moreover, analogous constraints can be easily obtained in a similar way also for the groups Sp(2n) and SO(2n + 1).
INTERNAL
MULTIPLICITIES
IN THE CARTAN CLASSES
343
eq. (3) is satisfied and such that M, 2 412 max(Mz, MI), miIl(Mi,
M:-1)
2
qi
S
IllZlX(Mi+~,
M:);
Vi
=
2,. . . , n - 1.
(4)
The number of possible different sets of non-negative numbers qi is $$. The condition ML_, 2 0 is not a limiting constraint and it was in fact proved? that
4. The SO(2n + 1) case Defining A = XyZI(Mi - MI), the multiplicity ?$, is the number of different sets of non-negative numbers qi satisfying (4) and
(5) where i) E = 0 and min(M,, ML,) 3 q,, 2 0 if A is an even integer; ii) E = 1 and min(M,,, MA-I) > qn 2 0 if A is an odd integer and the Mi’s are integral ;
iii) E = -1 and min(M,, MC,) 2 qn > 0 if A is an odd integer and the Mi’s are half an odd integer. Finally, we would like to remark that the existence of these algorithms for the groups SO(2n) and SO(2n + 1) is very surprising because the Gel’fand basis vectors are not eigenvectors of the n commuting elements of the Cartan subalgebra.
5. Numerical
tests and conclusions
For the proposed algorithms we have consistency tests which follow from the properties of the Gel’fand patterns, i.e. if +$::;$_, is the branching multiplicity of the reduction SO(2n + l)+ SO(2n - 1) or S0(2n)+ SO(2n - 2), the following must hold
&
-MI....,M,
TM;,..
.,MA_:_I,M;
-MI,...,M* =
YMi,.
,hf;_l
A
where the summation gives a non-zero contribution M’ = (M;, . . . , M@C M = (M,, . . . , M,).
provided that
344
V. AMAR et al.
The agreement between the proposed algorithm and formula (6) was verified for a large number of irreducible representations of SO(7), SO@), SO(9) and SO(10). For SO(5) and SO(6) we also have a global tes$q. Since there is an analogy between the proposed algorithms and between these and the algorithms valid for the other two Cartan classes, we think that it is possible to construct a unique computer program for evaluating the inner multiplicity for any Cartan class. This program, like our previous program’) for unitary groups, would be much more effective than other programs based on Freudenthal’s formula,
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
N. Jacobson, Lie Algebras (Dekker, New York, 1965) p. 261. N. Jacobson, Lie Algebras (Dekker, New York, 1965) p. 247. V.Amar, U. Dozzio and C. Oleari, Comput. Phys. Commun. 14 (1978) 413. J. Mickelson, Rep. Math. Phys. 3 (1972) 193. R.M. Delaney and B. Gruber, J. Math. Phys. 10 (1969) 252. I.M. Gel’fand, R.A. Minlos and ZYa. Shapiro, Representations of Rotations and Lorentz Groups (Pergamon Press, London, 1963). G.C. Hegerfeldt, J. Math. Phys. 8 (1967) 1195. V. Amar, U. Dozzio and C. Oleari, Lett. Nuovo Cimento 18 (1977) 13. V. Amar, U. Dozzio and C. Oleari, to be published. H. Weyl, The Theory of Groups and Quantum Mechanics (Methuen, London, 1931) p. 390.
INTERNAL
Publishing Co.
MULTIPLICITIES
IN THE CARTAN
CLASSES
V. AMAR, U. DOZZIO and C. OLEARI Istitulo di Fisica dell’Uniuersitic di Parma, Sezione di Fisica Teorica, via Massimo d’Azeglio 8.5, l-43100 Parma, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy
It is shown how the evaluation of the internal multiplicities of a simple Lie group G(n) is possible by means of the reduction G(n)+G(n - 1) @ U(1). An algorithm is then given for evaluating the branching multiplicity G(n)-+ G(n - 1) @J U(1) when G(n) = SO(2n) and G(n) = SO(2n + 1).
1. Introduction
The computation of the internal multiplicity for simple Lie groups is a classical problem, which has two general and elegant methods of solution: one is the enumerative Kostant formula’) and the other is the recursive Freudenthal formula*). Recently more effective methods have been obtained for SU(n + 1) groups’) and for Sp(2n) groups4). In this paper we propose algorithms for the groups SO(2n) and SO(2n + l), which are more effective than the usual recursive method and are very similar to the previously mentioned SU(n + 1) and Sp(2n) methods. The validity of these algorithms has been supported by a very large number of numerical tests, as will be seen below.
2. General theory Let G(n) denote a simple Lie group of rank n, the irreducible representations of G(n) be identified by the highest weight M = (Ml,. . . , M,,) and an arbitrary weight denoted by m = (ml,. . . , m,). We denote by rg, the internal multiplicity of weight m in the irreducible representation denoted by M. If G’ is a Lie group such that G’ C G, we denote by T$qG,the branching multiplicity of the irreducible representation of G’ defined by M’ in the irreducible representation of G defined by M (hereafter, fixed M and M’, the indices G and G’ are suppressed if the groups G and G’ appear unambiguously known from the text). 0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
342
V. AMAR et al.
If G’ is naturally embedded in G, i.e. the root system subsystem of the roots of G, the following formula’) holds
of G’ forms
a
(1) where L is the map of the weights of G on the weights of G’. On the left-hand side the summation is over weights m such that Lm = m’, while on the right-hand side the summation is over the irreducible representations of G’ which contain the weight m’ and are contained in the given irreducible representation of G. Now if we assume G’(n) = G(n - 1) @I U(l), where G(n - 1) belongs to the same Cartan class of G(n), then map L is the identity map, i.e. Lm = m, and formula (1) becomes -M
M'
(2)
YM’Y(mh
where
in agreement with the structure of G’. From eq. (2), if we find a method for evaluating -& by recursion it would also be possible to evaluate yfA)*. A method for evaluating 9% is known for G(n) = SU(n + 1) (cf. branching law of Weyl”)) and recently an algorithm for G(n) = Sp(2n) was given4). Rather simple procedures are proposed below for the groups G(n) = SO(2n) and G(n) = SO(2n + 1). So far, we have not been able to prove these algorithms; however, we have tried a very large set of numerical tests (vide infra) and we have no doubt that these algorithms are correct.
3. The SO(2n) case For evaluating ~~~z~!,,, U(I)the following algorithm (MI,. . . , MA) with MA-15 0 let us consider the relation
then evaluate the number of sets of n - 1 non-negative
holds:
given M’ =
numbers qi such that
* In (1) and (2) the condition M’ C M can be satisfied by using Gel’fand patterns6”) related to the different Cartan classes, while the condition M’ > m follows from the constraints given in refs. 8 and 9 for the groups SU(n + 1) and SO(2n). Moreover, analogous constraints can be easily obtained in a similar way also for the groups Sp(2n) and SO(2n + 1).
INTERNAL
MULTIPLICITIES
IN THE CARTAN CLASSES
343
eq. (3) is satisfied and such that M, 2 412 max(Mz, MI), miIl(Mi,
M:-1)
2
qi
S
IllZlX(Mi+~,
M:);
Vi
=
2,. . . , n - 1.
(4)
The number of possible different sets of non-negative numbers qi is $$. The condition ML_, 2 0 is not a limiting constraint and it was in fact proved? that
4. The SO(2n + 1) case Defining A = XyZI(Mi - MI), the multiplicity ?$, is the number of different sets of non-negative numbers qi satisfying (4) and
(5) where i) E = 0 and min(M,, ML,) 3 q,, 2 0 if A is an even integer; ii) E = 1 and min(M,,, MA-I) > qn 2 0 if A is an odd integer and the Mi’s are integral ;
iii) E = -1 and min(M,, MC,) 2 qn > 0 if A is an odd integer and the Mi’s are half an odd integer. Finally, we would like to remark that the existence of these algorithms for the groups SO(2n) and SO(2n + 1) is very surprising because the Gel’fand basis vectors are not eigenvectors of the n commuting elements of the Cartan subalgebra.
5. Numerical
tests and conclusions
For the proposed algorithms we have consistency tests which follow from the properties of the Gel’fand patterns, i.e. if +$::;$_, is the branching multiplicity of the reduction SO(2n + l)+ SO(2n - 1) or S0(2n)+ SO(2n - 2), the following must hold
&
-MI....,M,
TM;,..
.,MA_:_I,M;
-MI,...,M* =
YMi,.
,hf;_l
A
where the summation gives a non-zero contribution M’ = (M;, . . . , M@C M = (M,, . . . , M,).
provided that
344
V. AMAR et al.
The agreement between the proposed algorithm and formula (6) was verified for a large number of irreducible representations of SO(7), SO@), SO(9) and SO(10). For SO(5) and SO(6) we also have a global tes$q. Since there is an analogy between the proposed algorithms and between these and the algorithms valid for the other two Cartan classes, we think that it is possible to construct a unique computer program for evaluating the inner multiplicity for any Cartan class. This program, like our previous program’) for unitary groups, would be much more effective than other programs based on Freudenthal’s formula,
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
N. Jacobson, Lie Algebras (Dekker, New York, 1965) p. 261. N. Jacobson, Lie Algebras (Dekker, New York, 1965) p. 247. V.Amar, U. Dozzio and C. Oleari, Comput. Phys. Commun. 14 (1978) 413. J. Mickelson, Rep. Math. Phys. 3 (1972) 193. R.M. Delaney and B. Gruber, J. Math. Phys. 10 (1969) 252. I.M. Gel’fand, R.A. Minlos and ZYa. Shapiro, Representations of Rotations and Lorentz Groups (Pergamon Press, London, 1963). G.C. Hegerfeldt, J. Math. Phys. 8 (1967) 1195. V. Amar, U. Dozzio and C. Oleari, Lett. Nuovo Cimento 18 (1977) 13. V. Amar, U. Dozzio and C. Oleari, to be published. H. Weyl, The Theory of Groups and Quantum Mechanics (Methuen, London, 1931) p. 390.