Reprinted from FOUNDATIONS OF PHYSICS Vol. 27, No.7, July 1997Printed in Belgium

The Biedenharn-Louck-Hecht Resolution of the Outer

Multiplicity Problem for the U(3) and Ui3) Groups

R. M. Asherova,1 J. P. Draayer,2 Yu. I. Kharitonov,3and Yu. F. Smirnov4,5

Received March 28, 1997; revised April 3, 1997

The solution of the outer multiplicity problem in the tensor product of U(3)irreducible representations (irreps) developed by Biedenharn et al.(I·7) and realizedthrough the well-known Draayer-Akiyama (DA) computer code(S) is extended to

the quantum algebra Ui3). An analytic formula for special stretched U.(3)Wigner coefficients,

is derived using a projection operator method.(9-1O)In this expression Hi denotesthe highest weight vector of the (A.,p,) irrep; the subscript "max" means coef­ficients corresponding to a unit tensor operator with a maximal characteristic nullspace; and q is the usual quantum label so the standard U( 3) Wigner coefficient,which is required in the DA code, can be obtained in the q -> 1 limit of the theory.To illustrate the theory, some Uq(3) Wigner coefficients for the tensor product(22) x (22) are calculated. The procedure for evaluating nonhighest weight WignerUq(3) coefficients follow the DA prescription.

1. INTRODUCTION

The problem of the decomposition of a tensor product of U( 3} irreps intoirreducible components and a canonical definition for the outer multi­plicity, which distinguishes multiple occurrences of an irrep in product of

I Institute of Physics and Power Engineering, 249020 Obninsk, Russia.2 Department of Physics and Astronomy, Louisiana State University, Baton Rouge,

Louisiana.

3 Petersburg Institute or-Nuclear Physics, Gatchina, Russia.4 Institute of Nuclear Physics, Moscow State University, 119899 Moscow, Russia.5 Instituto de Ciencias Nucleares, UNAM, AP 70-543, 04510 Mexico, D.F. Mexico.

1035

OO15-9018/97/0700-1035SI2.50/0 © 1997 Plenum Publishing Corporation

1036 Asherova, Draayer, Kharitonov, and Smirnov

U( 3) irreps, has been thoroughly investigated by Biedenharn, Louck,Hecht (BLH) and co - workers. 1-7)Their mathematically canonical defini­tion puts the outer multiplicity on a sound group theoretical basis throughthe use of an upper Gelfand pattern for a Wigner operator with the U( 3)irreducible tensor character under consideration. Practical aspects of thischoice are the vanishing of certain Wigner (Clebsch-Gordan) and Racahcoefficients, very simple symmetry relations under conjugation, and attrac­tive limiting forms for the SU(3) =:J SU(2) x U(l) Wigner coefficients.

Using the essential features of the BLH definition, Draayer andAkiyama (DA)18) developed a practical algorithm for evaluating SU(3) =:J

SU( 2) x U( 1) Wigner coefficients. The starting point of the DA prescriptionis a set of special Wigner coefficients,

(1)

which are generated using an empirically deduced equation, Eq. (20) ofRef. 8, that up to now has escaped analytic verification.

This contribution gives an alternative analytic result for the startingcoefficients that are used in the DA algorithm. The derivation employs theprojection operator (PO) method developed by the Moscow group. 19-10)Inthis regard, it should be noted that the Moscow group also considered,early on, the decomposition of the tensor product of U( 3) irreps intoirreducible components.(9) Their approach employed the PO method11O)inconjunction with the introduction of an additional integral of motion for aresolution of the multiplicity, in accordance with a prescription recommendedby Racah and Moshinsky. (11)Some final results can be found in Ref. 12.

Based on these successes, a unification of the BLH tensor approachand the PO method appears to be a recommended general strategy. Thiswas first realized by Alisauskas for the usual SU(3) algebra (I 3) (and laterfor the quantum Ui3) algebra(14»).Here the question of the applicabilityof this approach to a derivation of results for a Wigner-Racah algebra forq-deformed objects is considered. The present discussion is restricted to thecalculation of starting Wigner coefficients for the Uq(3) algebra, leaving tothe future a derivation of more general results. As noted above, the corre­sponding Wigner coefficients for the standard U( 3) case can be obtained bygoing to the q ~ 1 limit.

2. WIGNER COEFFICIENTS IN THE PO APPROACH

Let Aik> with i, k= 1,2,3, be a Cartan-Weyl basis for the Ui3) alge­bra(1S)and D[fJ an irrep of this algebra with highest weight [f] = (fI!2f3)'

••_-----------------_!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!=~~~~==~~~~~~=

The Biedenharn-Louck-Hecht Resolution 1037

(Also, DO!') will be used to denote an irrep of SUq(3) where A.= II - IzandJ.1. = Iz- 13') On the space of an irrep D[f], the canonical Gelfand-Zetlinbasis can be realized:

with

e= -(A.+2J.1.)+3(l+k)

A=!(A.+I-k)

MA =/11 - !(f12 +122) =r-A

(2)

(3)

(4)

(5)

(6)

where the integers I, k, r satisfy 0 ~ k ~ A., 0 ~ 1~ J.1., 0 ~ r ~ 2A.A decomposition of the tensor product of irreps into irreducible

components can be written as

D(AI!']) X D(A2!'2) = L v(A!')D(A!')(A!,)

(7)

The outer multiplicity label p = 1, 2,..., vIA!') is used to distinguish multiplyoccurring irreps D(A!'). The Wigner coefficients < (A.IJ.1.I)ai' (A.zJ.1.z)azl

(A.J.1.)a) ~ are by definition the elements of a unitary transformation betweencoupled and uncoupled representations of SUq(3) in the a-scheme,

j(A.IJ.1.d, (A.zJ.1.z):p(A.J.1.)a) q

= L «A.IJ.1.d ai, (A.zJ.1.z)azl(A.J.1.) a)~I(A.IJ.1.1) (1)ql(A.zJ.1.z) aZ)q (8)

Building on the BLH scheme for a resolution of the outer multiplicity,DA developed a practical algorithm for evaluating all SU(3)::) SU(2) xU( 1) Wigner coefficients. The DA prescription includes a straightforwardSchmidt orthogonalization procedure for producing independent orthogonalvectors in the multiplicity subspace. The scheme requires a set of specialstarting (seed) Wigner coefficients with p = Pmax = vIA!,)'

(9)

1038Asherova, Draayer, Kharitonov, and Smirnov

These seeds are associate with a unit tensor operator characterized by amaximal null space and includes the highest weight states IH = (fdZ!3»and /HI) of the irreps (AJ.l) and (A1J.ld, respectively. As indicated above,DA determined a general expression for these special Wigner coefficients byemploying empirical methods. Below an alternative analytic expression isintroduced, one derived using the PO method. (9,15) The latter, therefore,provides an analytically proven result, missing in all previous analyses, forthe seed coefficients upon which the DA prescription rests.

To follow the argument, consider the PO defined by

p~~:~q= L I(AJ.l) a) pp «AJ.l) a'lp

(10)

where a = cAM A' Then the matrix element (ME) of the PO, p~~:!q,has theform

= L «A I J.l I ) alo (A2J.l2) a21 (AJ.l) a) ~p

In particular for the ME of the PO piA.p) q, H.M

«AIJ.lI) HI, (A2J.l2) czAzmzl p<jfli I(AIJ.lI) cIAIAI, (AzJ.lz) M2)

= «AIJ.lI) HI' (AzJ.lz) C2A2m21 (AJ.l) H) ~max

x «AIJ.lI) CIA IA I, (A·ZJ.l2) M21 (AJ.l) M) ~max

(11 )

(12)

where M = (fdd2) is the dominant weight corresponding to the state withk=O, 1=J.l, and r=A+J.l in (4)-(6), namely,

( 13)

where A M and cM are given by

(14)

This configuration satisfies the relations

The Biedenharn--Louck-Hecht Resolution1039

Similarly, Mz is the dominant weight in the irrep (Azpz), i.e.

(16)

The values of quantum numbers A 1 and 81 are chosen as follows:

The last Wigner coefficient on the r.h.s. of (12) corresponds to themaximum possible value of A Me and the maximum change,

(18)

in the angular momentum (AA=~(Az+Pz».

In accordance with the BLH procedure, the only nonvanishing Wignercoefficients with maximum AA in the (Ap) irrep have P =Pmax' In otherwords, they are different from zero only for tensors T(A21'2) with maximalnull space. Therefore, in the particular case (12), the sum over P on ther.h.s. of (11) reduces to the single term. It follows that formula (12) can beused to obtain the starting coefficients (9) (up to a normalization factor)if the ME of the projector on the l.h.s. of this formula can be found.However, it should be noted that for (12) to apply, the condition

(19)

must be satisfied. If (19) does not hold, the symmetry property of Wignercoefficients under (Ap) and (A,PI) interchange must be applied.

As mentioned above, the last coefficient on the r.h.s. of (12) can beconsidered to be the normalizing factor. It can be calculated using the POPC;:>;J which projects out the dominant weight (M = UI!3fz» vectorbel~nging to the D(AI') irrep with P =Pmax:

« (AlP I) 81 A I A" (Azpz) Mzl (Ap) M) ~max)Z

= < (AlP I) 8, A IA I, (Azpz) MzI Pljt>;J IU,pd 81 A IA I, (Azpz) Mz)

(20)

The PO Pljt>;J, in turn, can be obtained III factorized form from theextremal projector( I[)

1040

for the highest weight vector

Asherova, Draayer, Kharitonov, and Smirnov

by the substitution 2~-l>3(13, 14):

p(AI')qM,M

(23)

(24)

( - 1YI + "2 + "J q"i [A + 1]! [J.l + 1]! [A. + Il + 2] != "I .~. "J [n 1]! [nz]! [n;]! [A + 1 + n I]! [Il + 1 + nz]! [A, + Il + 2 + n;] 1

x A"I A"I A"i A"i A"2 A"231 13 ZI IZ Z3 3Z (25)

By using the commutation relations for powers of the generators ofU(3), (15)a more convenient form for calculating ME of the projector onthe r.h.s, of (20) can be found(141:

p(AI') q _ ( -1)"1 +"2 q"J(IIJ + I1[A, + I]! [Il + I]! [A + Il + 2]!M.M- "I'~'''J [n(]! [nz]! [n3]! [A, + 1 +n(]! [Il+ 1 +nZ+n3]!

[A+1l+2+nl +nz+n3]!x .[A+1l+2+nl+n3]! [A+1l+2+nZ+n3]!

(26)

Then by expanding the powers of generators in this expression in accord­ance with coproduct rules,(15)the ME of the PO pfjt,>;J can be deduced:

(( _1)"1 +"2 q"JA2-1121'2[A + I]! [Il + 1]1 )x [A. +Il + 2]1 [A +Il +2 +n] +nz]!

= "~2 ([nl]! [nz]! [A + 1 +nl]! [J.l+ I +nz]! )x[A+1l+2+nl]! [A+J.l+2+nz]1

Let the last ME be denoted G"I"2 and recall that the ME of the U,/3)generators in the canonical Gelfand-Zetlin basis were found in Ref. 16.Using those formulas yields the following result for G" " :1 2

The Biedenharn-Louck-Hecht Resolution

G = [kd! [ilJ +/J -kJ + I]! [ilJ -kJ +nd!

nln2 [A.,-kd! [ilJ +/1-kJ +nJ +n2+ I]! [kJ -nd!

1041

[ill +/1 +n2+ I]! [lJ +n2]! [,uJ -II]! [ilJ +,uJ -kJ +nJ + I]!x---- _[ill +/1 + I]! [ld! [,uJ -II -n2]! [ilJ +,uJ -kJ + I]!

(28)

where

kl = H -2il-,u + 2ill +,ul + 2il2+ ,u2)=,u -,uJ -,u2 + 2n (29)

I, = Hil + 2,u- ilJ +,uJ - il2- 2,u2)=,uJ - n (30)

n = H - il-- 2,u+ ill + 2,u, + il2+ 2,u2) (31)

The integers kl, II> nl> and n2 satisfy the following restrictions: O~kJ ~ill,O~/I ~,ul, O~nl ~kJ, 0~n2~,uJ -I,.

To calculate the starting Wigner coefficient, consider relation (12). Theprojector Pj;':JJ can be constructed as

P(AIJ)q __ 1_AIJ p(AIJ)qH, M - [,u]! 23 M, M

and reduced toe14):

Pj;':JJ=[il+1]! [,u+1]! [il+,u+2]!

(-1 )n, +n2 qu+r(nl-r)

X n" Eu, r [,u + n2 + I]! [il + ,u + 2 + nd!

[il+,u+2+nJ +n2-u]!x---- _([il +,u +2 +n2]! [u]! [n2 - u]! [il + 1+nl - U]!)x [r - u]! [n I - r]! [,u - r + u]!

(32)

By substituting the PO (32) into (12) and expanding powers of the gener­ators in accordance with coproduct rules, (15) the ME on the l.h.s. of (12)can be shown to be given by the following expression:

< (A-Ill I) HI, (Azllz) ezAzmzl P<JtlJ I(AIIlI) elAIAl, (Azllz) Mz>

= [...l+ I]! [Il + 1]! [A + Il + 2]!

r

1042 Asherova, Draayer, Kharitonov, and Smirnov

(-1tl+n2 qu+r(nl-r)

xn n~rs[ll+nz+l]![...l+1l+2+n2]!I' 2· ••

[...l+1l+2+nl+nz-u]!x-----------------([...l +Il +2 +nl]! [u]! [nz -u]! [...1+ 1+nl -U]!)X [r - u]! [n I - r]! [Il - r + u] !

-t

t

Ir:

l.

!,~.

I

I

where

Cs= < (AIIlI) Htl A~3A~~A~~ I(Allld eIAIAI>

-No.li'l) [s]! [Ill +n2-s]!- (kill) [s-n2]! [Ill -s]!

Dr = < (A21l2) e2Azm21 A;I A~\ -r Aitn2-r-s I(A2Ilz) Mz>

=N(A.2i'2) ( [A2-m2]! )1/2 (_I)A2-m2-r <P(k2l2) [2A2]! [A2 +m2]! q

[NI-r]! [1l-N2-r]! [A2+m2+r]!X-----------------[llz-Il+N2+r]! [NI-A2+m2]! [A2-m2-r]!

In (34) and (35)

(A.i')_ ([...1]! [Il]! [...l+ Il + I]! [l + 1+ 1- k]! [k]! [/]!)1/2N (kl) - [...1- k]! [Il -I]! [...l+ Il + 1- k]! [l + 1+ I]!

Recall that this result is valid under condition (19).

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

J

It

lI

t

II

n1 II II ,

I

The Biedenharn-Louck-Hecht Resolution 1043

Note that expressions (27)-(31) suffice to determine the normalizing

1\ Wigner coefficient in (12). Furthermore, the l.h.s. of(12) is known through(33)-(40) and can be rewritten in the final form:

< (AIIlI) HI' (A21l2) G2A2m21(All) H) ~max

x < (AIIl d GIA IA 1> (A21l2) M21 (All) M) ~max

= «AIIlI) HI, (A21l2) G2A2m21PlJi':li I(AIIlI) GIAIA1> (A21l2) M2)

(41 )

(42)

U, r. s

where

A=(_1)NJ-N2+A2-m2 ,,[A+I]! [1l+1]! [A+Il+2]! [2A +1]1/2q [A+Il+2+NI]! 2

x ([AI]! [Ill]! [AI +Ill + 1]! [NI]! [IlI-N2]! [AI-NI +N2+ 1]!)1/2[N2]! PI-Nd! [AI +Ill-NI + 1]! PI +N2+ 1]!

X(([A2+1l2+A2-m2~NI +1]! ))1/2x [A2 +1l2 -Il +A2 -m2 +N2 + 1]!

(_l)s+r qP[A2 +m2 + r]!

Bu.r,s=([A2_m2_r]! [u]! [r-u]! [Il-r+u]! )x[A+NI-u+1]! [1l2-Il+N2+r]!

[A+Il+NI-N2+s-u+2]! [1l-N2+s-r]!x---------------------[1l-N2+s+ 1]! [A+Il-N2+s+2]! [IlI-s]! [s-N2-u]!

(43)

CX=!(NI(A2-AI-lld+N2(1l1 +1l2)-IlIlI)

- (A2 - m2)(A2 - m2 + A2+ 112- Il- NI + N2 + 1)

P =u + r(A2 + 112- Il + III + N2 + 1) - rs + S(Il- 112- N2)

(44)

(45)

(46)

1044 Asherova, Draayer, Kharitonov, and Smirnov

For A + f1 < A2 + f12 it is necessary to use symmetry properties of theWigner coefficients with respect to a permutation of (A)f1)) and (Af1) irrepsor to determine an analytical continuation of the relations found aboveinto the A + f1 < A2 + f12 region (see Ref. 14). From this it follows that theseed Wigner coefficients, (9), required for an application of the DAprescription, are known in analytical form. Thus these results, coupled withthe DA prescription, suffice to determine all Wigner coefficients.

As an example, consider Wigner coefficient for the (A)f1 d =(A2f12) =(22) and (Af1) = (41) case. The coefficients

Xi == < (22) H), (22) c2A2m21 (41) H) ~m",

x < (22) CIA!AI, (22) M21 (41) M)~max

= < (22) HI, (22) C2A2m21 Pji.)>;j 1(22)CIA! A I, (A'2f12) M2) (47)

with C2= 0, m2 = 1, A2 = 1 (i = 1), A2 = 2 (i = 2) are calculated through(41)-( 46):

[2] [2] [2] [5 ]([ 4]! )1/2XI = - [3][4][8][4]! ;

[2][5]([5]!))/2 8 2X2=[3F[4F[6][8]([2][6]+q [4]).

(48)

(49)

The second Wigner coefficient on the l.h.s. of (47) corresponds to thenormalizing coefficient

N== «22) - 3H, (22) 0221(41) - 3H)q

It follows from Eqs. (27)-( 31) that

1

N2 [3][3][4][6][8] ([3][3][ 4][6][8] + [2][2][ 4 ][5][9]

-q2[2][3][ 4 ][5][8] - q-2([2][ 4][ 4][6][8] + [2][2][5][ 10])

+q-4[2][5][6][8]) (49)

These Wigner coefficients are all in agreement with the tables of Ref. 5 forthe classical q = 1 limit.

The Biedenharn-Louck-Hecht Resolution

3. CONCLUSIONS

1045

An analytical formula for (normalized) starting Wigner coefficients,(I), which are required input in the DA prescription for generating Wignercoefficients, and their q-analogs, (9), has been derived. The method hasbeen incorporated into a new DA code. The solution was tested by com­paring results with those from the standard DA code for a large number of(A IIId, (A21l2), and (All) irreps. In all cases the formulas given abovereproduce results from the standard DA code. This exercise confirms thecorrectness of the empirical result expressed by Eq.(20) of Ref. 8 and at thesame time demonstrates the correctness of an alternative starting point thatis based on analytically proven results.

Advantages of the results obtained in this work are: (I) they give nor­malized results for Wigner coefficients, and (2) they are valid for both theusual U(3) and q-deformed U,/3) algebras. The latter means, for example,that it should now be possible to develop a new generation of algebraic

computer codes for calculating U(3) as well as Ui3) Wigner coefficients.This work was inspired by Larry Biedenharn, to whom one of the

authors (J.D.) owes a great deal of gratitude for the questions he raised andthe answers he provided on scientific issues of common interest. ProfessorBiedenharn was particularly helpful during the development of the DAcode, which draws heavily on his work. Two of the authors (R.A. andYu.S.) are grateful for support from the US National Science Foundationfor their visit to Louisiana State University (LSU) and to the scientists andstaff members of the Department of Physics and Astronomy at LSU fortheir kind hospitality. This work was also supported by the Russian Foun­dation for Fundamental Research, Grant NO.96-01-01421.

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2. K. T. Hecht, Nucl. Phys. 62, 1 (1965).

3. L. C. Bicdenharn, A. Giovanni, and J. D. Louck, J. Math. Phys. 8, 691 (1967).

4. L. C. Bicdenharn and J. D. Louck, Commun. Math. Phys. 8, 89 (1968); J. Math. Phys. 13,1985 (1972).

5. J. A. Castilho Alcaras, L. C. Biedcnharn, K. T. Hecht, and G. Neely, Ann. Phys. (Nell'York) 66, 85 (1970).

6. L. C. Bicdenharn, M. A. Lohc, and J. D. Louck, J. Math. Phys. 26, 1458 (1985); 29, 1106(1988).

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Draayer and J. Janecke, cds. (World Scientific, Singapore, 1993).

8. J. P. Draayer and Y. Akiyama, J. Math. Phys. 14, 1904 (1973).

9. R. M. Asherova and Yu. F. Smirnov, Nucl. Phys. B4, 399 (1968); I. Guseva, Yu. F.

Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, preprints L1YAF, Gatchina 1981, 1983.

1046 Asherova, Draayer, Kharitonov, and Smirnov

10. R. M. Asherova, Yu. F. Smirnov, and V. N. Tolstoy, Theor. Math. Phys. 8, 255 (1971);IS, 107 (1973).

II. G. Racah, in Group Theoretical Concepts and Methods in Elementary Particle Physics,F. Gursey, ed. (Gordon & Breach, New York, 1964), p.31; V. Bargmann andM. Moshinsky, Nucl. Phys. 23,177 (1961). M. Moshinsky, J. Math. Phys. 4, 1128 (1963).

12. Z. Pluhat, Yu. F. Smirnov, and V. N. Tolstoy, Development of SU(3) Symmetry Techni-que, preprint, Charles University, Prague, 1982; J. Phys. A: Math. Gen. 19, 21 (1986).

13. S. I. Alisauskas, J. Math. Phys. 33, 1983 (1992).14. S. I. Alisauskas, J. Math. Phys. 37, 5719 (1996).15. Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, Yad. Fiz. 54, 721 (1992).16. A. A. Malashin, Yu. F. Smirnov, and Vi. I. Kharitonov, Yad. Fiz. 58, 651 (1995).

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