The Pattern Calculus for Tensor Operators in Quantum Groups Mark Gould and L

The pattern calculus for tensor operators in quantum groups Mark Gould and L. C. Biedenharn

Citation: Journal of Mathematical Physics 33, 3613 (1992); doi: 10.1063/1.529909 View online: http://dx.doi.org/10.1063/1.529909 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/33/11?ver=pdfcov Published by the AIP Publishing

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Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.42.98 On: Thu, 29 Sep 2016 02:42:08 The pattern calculus for tensor operators in quantum groups Mark Gould Department of Applied Mathematics, University of Queensland,St. Lucia, Queensland,Australia 4072 L. C. Biedenharn Department of Physics, Duke University, Durham, North Carolina 27706 (Received 28 February 1992; accepted for publication 1 June 1992) An explicit algebraic evaluation is given for all q-tensor operators (with unit norm) belonging to the quantum group U&(n)) and having extremal operator shift patterns acting on arbitrary U&u(n)) irreps. These rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns. A more conceptual derivation of these results is discussed using fundamental properties of the q-6j coefficients.

I. INTRODUCTION Schwinger (“boson operator”) map’ to realize explicitly the required general irreps and elementary tensor opera- Quantum groups are of considerable current interest, tors. These results were later verified using the “vector in both mathematics and physics, and have been the focus coherent state” (VCS) method (an algebraic version of of much recent research.le3 Although quantum groups the Borel-Weil construction) by LeBlanc and Hecht6 are not groups, they share many group-like features. For and, using a more conceptual approach, by Gould.’ The generic values of q-the deformation parameter-the rep- extension to B(n) was also given by Gould.8 resentation theory for compact quantum groups is sur- The approach used by Gould’ constructed the re- prisingly close to that of standard compact Lie groups, quired tensor operator matrix elements by abstract pro- and it is of interest to see how far this similarity extends jection operator techniques, and then evaluated these ma- for more complicated structures, such as tensor operator trix elements explicitly, using a projection operator algebras, built on representation theory. In the present realization, originally developed for angular momentum paper we shall demonstrate that a pattern calculus for [SU( 2)] calculations,g and later extended to a general Lie tensor operators in the U&u(n)) quantum group exists group.lot” To understand the concepts underlying this and is very similar in form-but not detail-to the pat- method by an elementary example, consider the total an- tern calculus known for tensor operators on Lie group gular momentum operator J [for the group SU(2)] as representations. being the sum of two independent angular momenta L It appears helpful, before discussing the modifications and u/2, that is, J=u/2+L. Then 2o9L= J2-L2-i is introduced by quantum groups, to review briefly the ideas an invariant operator (that is, [J,crL)] =O), and this op- and methods used in constructing the pattern calculus for erator may be used to construct (a) the (orbital) Casimir ordinary (compact) Lie groups. The pattern calculus is invariant operator: L2 = tr( ( wLj2) and (b) the projection actually three things: (a) an explicit evaluation of all operators: P,:(J+j=lfi):P+= (a-L+Z+ 1)/(21+ l), matrix elements of all elementary tensor operators be- P- = (I- 0.L )/21+ 1. (This latter step uses the eigenval- tween all (unitary) irreps of a given compact Lie group; ues CPL-P Z, - I - 1. ) Matrix elements of P, then yield the (b) a diagrammatic procedure whereby the diagram cor- desired spin-i tensor operator results. relates (or more precisely, implies) the explicit (alge- The required extension to an arbitrary Lie group con- braic) matrix element associated (by the pattern calculus siders the algebra of tensor products, &: rules) to the diagram; and (c) a procedure for the construction of all tensor operators. The elementary tensor a?= U(g) @ U(g), (1.1) operators are given directly by the pattern calculus, and by extending the calculus so that patterns act on patterns, where gcLie (G) and U(g) is the universal enveloping one obtains an algebraic basis for the construction of gen- algebra of Lie (G), and we use the diagonal coproduct eral tensor operators. d(g) = 18 g+g Q 1. (Coproducts are usually denoted by It is the merit of the pattern calculus that it “ex- the symbol A, but for tensor operator theory this symbol plains” and makes accessible structurally the otherwise has a different significance. Hence we use a for the co- difficult-to-comprehend explicit matrix elements (which product to avoid confusion.] The analog of the spin-f re- can be of arbitrarily large complexity and length). alization u/2 is now an irrep /z of G, with the associated The pattern calculus was first constructed [for U(n)] carrier space (module) VA, and Lie algebra generators by Biedenham and Louck4 using the straightforward pro- rA(g) playing the r6le of u/2. Generating elements of the cedure of direct evaluation, exploiting the Jordan- algebra &‘A = End( VA) o U(g) are then given by

J. Math. Phys. 33 (1 l), November 1992 0022-2488/92/l 13613-23$03.00 @ 1992 American Institute of Physics 3613 Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.42.98 On: Thu, 29 Sep 2016 02:42:08 3614 M. Gould and L. C. Biedenharn: Tensor operators in quantum groups

Commutation relations, a,dg) = h 8 mw). (1.2) The Casimir element C of G is C= &x,.x’, where {x,] ~wq = 0, (2.1) is a basis for Lie G and {x3 is the dual basis defined by the Killing form. The analog to PL is then [HisET] = fk,ij,Ei’,

A&~Q(c) 0 I+IQ(I) 63c-a,(c)). (1.3) 1, i=j, (2.2a) where k,, = -l/2, i=j* 1, (2.2b) It is easily seen that An lies in the center of otherwise, (2.2c) End( Vn) o U(g), that is, [a,(g>,,4An]=O. 0, Just as in the defining example, the invariants of the Lie group G, that is, the center of U(g)-denoted Z-is [Ei+,E~] =sO~-‘~~~‘s4[2H,1 [see (2.8)]; generated as an algebra by the (finitely many) elements --4 z,,,= trA( (AJ “). Similarly, projection operators for irre- (2.3) ducible modules of -CBAcan be constructed as polynomials Quadratic q-Serre relations, in Ah. These projection operators can be given explicitly [as in (b) above] when one has determined the eigenvalue [ET,ET] =O, j#ih 1, ltrace operation. As we shall show in detail, this general procedure is sufficient to determine the (ij) = (i,i* l), l

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f3(Hi) =Hj@ 1+ 1 @Hj 9 (2.9) with ((m’) 1X,1 (m)) the matrix elements of the generator X, for the irrep [ml. a(Ei”)=Et+ eqHi/4+q-Hd4@E,F, (2.10) The tensor operator classification is not categoric; this is clear from noting that multiplying a given irreduc- E(l)=& (2.11a) ible tensor operator labeled by [m] by an invariant operator leaves the classification unchanged. To eliminate this E(E,‘)=E(Hi) =O, (2.11b) freedom one can define unit tensor operators whose norm (see below) is unity. For G= U(n) with n)3, distinct unit tensor operators with the same irrep labels y($) = -q”‘Qif, (2.12a) htl * * *rn,J exist, but it has been shown16 that a canonical (natural) labeling splitting this multiplicity exists for y(Hi)=-Hi. (2.12b) n=3, and is conjectured” to exist for all n. The quantum group V&(n)) is defined by the gen- Applications of symmetry structures in quantum erators above plus the additional generator H,,, which physics are heavily dependent on the fundamental theo- commutes with all other generators. The Hopf algebra rem for tensor operators (the generalized Wigner-Eckart operations for H,, are the same as those for the other HP theorem). Such a theorem need not exist for a given symmetry structure, but depends rather upon the specific way in which the symmetry is realized.5 The two required III. RESUME OF TENSOR OPERATOR CONCEPTS properties of the realization are equivariance and adjoint action (or, more generally, the derivative property). A given symmetry group has the implication for It is a consequence of these considerations that the physics, not only of distinguishing possible physical states generalized Wigner-Clebsch-Gordan (WCG) coeffi- by symmetry based quantum numbers, but also of classi- cients for a given symmetry group G occur in two logi- fying (and partially determining) physical transitions be- cally distinct ways: (a) as coupling coeficients for the tween these states. This latter structure is abstracted as Kronecker product of irreps carried by kinematically in- the aIgebra of tensor operators of the symmetry group. dependent constituent systems (the Clebsch-Gordan Consider a compact Lie group G to be a physical problem), and (b) as matrix elements (up to an invariant symmetry. This symmetry allows one to determine a scale factor) of physical transition operators (the Hilbert space basis for physical states using vectors car- Wigner-Eckart problem ) . Conversely, if equivariance and rying the irreps of G and, in general, additional parame- derivation are not valid for a given realization, then this ters from configuration space. To discuss the problem of latter result (b) fails. It is not obvious, but nonetheless interactions, which induce transitions within and between true, that one can indeed extend both (a) and (b) to irreps, it is useful to abstract from the physical Hilbert quantum groups.‘8.‘9 Let us summarize how the tensor space a model space,‘4T’5 M, defined to be the direct sum operator structure is extended to quantum groups (q- of vectors carrying unitary irreps of the group G, each tensor operators). equivalence class of irreps occurring once and only once. Definition I: Let T denote the vector space of opera- The operators on M are defined to belong to the linear tors mapping the model space M of the compact quantum space T with the action T: M -+ M. Using equivariance, the group V,(g) into itself, T:M + M. An irreducible q-tensor symmetry G can be exploited to give a partial classifica- operator is a set of operators, {+}eT that carries a tion of the operators belonging to T. Let gEG and finite-dimensional irrep E, with vectors 5, of the quantum M -t U(g) M. The equivariance condition is group V,(g). That is,

U(g)TJJ-‘(g) = C Tpji(g), (3.1) i E&$ = c (W I Ea I ~&k,5’ , where D(g) is a representation. It follows that a tensor 5’ operator {Ti}-a set of operators--can be classified by an irrep label of G, with individual operators in the tensor where E, is a generator of U,(g), E,(t,,6) denotes an operator set being labeled as vectors carrying the irrep. action of E, on T, and ( * * * ) denotes the matrices of the At the Lie algebra level the action of the generator X,Eg generators for the irrep E. A q-tensor operator is accord- on the irreducible tensor operator set T(,), with (m) a ingly a linear combination of irreducible q-tensor opera- Gel’fand-Weyl pattern, [cf. Eq. (3.5)], is the adjoint ac- tors, with coefficients invariant under the q-group action. tion Theorem’* 2: If {tz,{} is a q-tensor operator of the compact quantum group U,(g) such that the coproduct [X,,T(,,] = C (Cm’) I&l (m))T(mt)9 (3.2) of U,(g) is compatible with the action E,( tz,5), then the Cm’) matrix elements of {ta,6} in M are proportional to the

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q-WCG coefficients of V,(g) with the constant of pro- For U( 3) and for Ue(u( 3)) the operator pattern (r) portionality an invariant. has been proved to be canonical, that is, unique to within It is clear from the above definition that tensor oper- equivalence and free of any arbitrary choices. [It is con- ators form an algebra with invariant operators as scalars. jectured” that this is true for all U(n) and U&(n)).] This algebra is generated algebraically by the unit tensor The individual elements of an irreducible unit tensor operators, that is, tensor operators that are normed to operator set are labeled by the symbol unity on all irreps not in the characteristic null space of the operator. This algebra of tensor operators contains as a subalgebra the universal enveloping algebra. Expressed differently, the algebra of tensor operators is a generali- zation of the concept of a universal enveloping algebrasI Unit q-tensor operators are characterized by two structures: The angular brackets denote a unique unit tensor opera- (a) The quantum group action, by which the unit tor, labeled by the U&(n)) irrep labels [Ml-common to tensor operators may be divided into irreducible sets car- both the operatorpattern (I?) and the Gellfand- Weylpat- rying the labels of a quantum group irrep; this is the label tern (M) . For brevity, the operator pattern r is inverted W,?&f2” * . *M,,,J for the q group U,$u( n)). The operators (this leaves the betweenness relations invariant). The sig- belonging to a finite-dimensional irrep may be further nificance of these labels can be seen from the following split (using the induced q-group action) by labels char- properties. acterizing the various components of the operators, which (a) The weights ( W) = ( WI+*= W,) associated by the carry representation labels in the canonical group- equivariant quantum group action on subgroup chain, U,(u(n))~U4(u(n-l))~...>U~ x (u ( 1 )), written in the form of a Gellfand- Weyl pattern: I tr) \ Ml, MI,, ... Mm Ml,,- 1 ... Mn-I,,-I (M) = . . . are given by M** *** M22

Ml1 1’ Wi= 2 (Mj,i-Mj,i-l), Mu=Oo, i> j. (3.5) (3.3) i Note that these weights are independent of (I’). which is a triangular pattern of integers obeying the be- (b) The shifts associated to the operator tweenness constraints

M+Mij- 1ZMi+ IJ * (3.4)

The Gel’fand-Weyl pattern is a “geometric” realization, via constraints, of the Weyl branching theorem for U(n), which carries over to the quantum group UJ u( n ) ) when q is generic (not a root of unity), in consequence of the that is, the action by the operator on the irrep labeled by Lusztig-Rosso theorem.20V21 [m,,,m2,,,...,m,,,,] yields the irrep [m,,+Al,m2,, (b) The Gel’fand-Weyl patterns in (a) do not fully +A 2,...,mnn+An] (or the null irrep), where the shifts characterize a unit tensor operator, and a further splitting CAi) = ( Al,A2,...,A,) are given by is necessary. A suitable labeling (unique to within f 1 phase conventions) is afforded by a second pattern, called Ai= 1 (I’j,i-I’j,i-i), ru=O, i>j. (3.6) an operator pattern ( I’), which is structurally similar to a i Gel’fand-Weyl pattern in being integer valued and obeying the betweenness constraints Note that the shifts (A) are independent of (M). The subset of unit tensor operators belonging to the Ml, M,, ,-. Mtl, universal enveloping algebra is the subset characterized r1,fF-l *** **’ rn-l,n-1 by (A) = (k), that is, constant shift k for all labels (M), (l-)r . . . which [for Ue(su(n)) model space] is equivalent to zero shift (A) = (0). (Strictly speaking, this requires normal- h izing by the appropriate invariant norm.)

It is a consequence of the definition of a unit tensor operator that the matrix elements of the operator,

(i) +c’$y,(ij) ,f$j considered as an explicitly determined algebraic object, is clearly enormously complicated and can hardly be understood from looking at detailed algebraic matrix elements. What is needed is an understanding of the structure im- are exactly the same as the matrix elements of the matrix, plicit in such an object. For the quantum group U&u(n)), which brings the tensor product of the irreps this desired structure is implied by the subgroup chain [M] 8 [minitial] ~ [m6nd ] to fully reduced (block-diagonal) u&(n))> u&b--l))I ***U,(l). For this chain we form, ordered by the standard ordering on irrep labels have the result [M]. In other words, the construction of explicit unit tensor operators is equivalent to the complete resolution of the Clebsch-Gordan (tensor) product, including the (q( g)l;$;) resolution of all multiplicities. To simplify the notation we will refer to the tensor operator set =(F) ([$;:,I [;;I I,$::,)

x (y&-l1 (($y)J I’$)- 1). (3.7)

This result, stated in words, asserts that a matrix as ([Ml) or simply as (/2) with il denoting an irrep, if the element for a unit tensor operator belonging to U&u(n)) meaning is otherwise clear. can be expressed as a sum of products of two factors: (a) It is a matter of convenience as to whether or not one a reduced operator coeflcient: chooses to calculate matrix elements algebraically as tensor operators acting on model space or group theoreti- cally as the reduction of the tensor product of irreps. For (3.8) calculations of reduced tensor operators it proves concep- tually very convenient to work with tensor product spaces formalizing the concept of U,(u( n)):U,(u( n - 1)) which, by construction, is an invariant under the U&u (n spaces characterized by extended weight vectors (see Sec. - 1)) subgroup; and (b) an irreducible unit tensor oper- VI. ator matrix element, A particularly simple class of unit tensor operators is the class of such operators for which the shift labels (A) (Q)-11((g:)_*) 1‘ $)-I), (3.9) are a permutation of the operator irrep labels [Ml. For this class, there is no multiplicity and a direct determina- which belongs to a unit tensor operator in the sub-q group tion of the associated operator elements via the reduction U,(u(n - l)), where the irrep labels are contained in the of the tensor product is feasible. It is this class of opera- patterns (I’) and (M),-,. tors that the pattern calculus handles most expeditiously, Thus, one has a recursive approach to the explicit and this is the class fully determined in this paper. We determination of tensor operators in which one need only remark that this class includes all elementary unit tensor obtain generic matrix elements for the reduced operators, operators, where an elementary unit tensor operator (dis- denoted by tinct from the unit operator) is defined to be an irreducible unit tensor operator with irrep labels of the form - u-1 - [Ml , [M] = [ l~~*lO--~O] (k= 1,2,..., n-l). u-7 k n-k abstracting the operator from the matrix elements above. The operator (This abstraction is unique since the labels of the reduced

operator determine completely the final state, given the WI @ mu) initial state irrep data.) Note that the reduced operators act in a U,(u(n)):U&u(n- 1)) subspace, labeled by a =@ 63 Woe~ol~o+A(~‘)). truncated (two-rowed) Gellfand- Weyl pattern. (We char- hcA ~+A(~‘)~J@F,J acterize this truncated space as a space of extended PO=lr weight vectors in Sec. V.) (3.14) Matrix elements for these reduced operators can be We now demonstrate that the overlap coefficients be- most easily obtained using projection operator techniques tween the two decompositions (3.11) and (3.14) give the in tensor product modules, which we now develop. desired matrix elements of the reduced tensor operators. Consider the reduction of the tensor product module Denote the orthogonal projection onto the submodule Y= v(n) o V(p), where v(n) and V(p) denote tinite- V(,U+ A(y) 1vo) of Eq. (3.13) by P,. Similarly, we denote dimensional irreducible U&u(n)) modules with highest the orthogonal projection onto the submodule weights il and CL,respectively. There are two natural de- V&,epoIpc,+A(y’)) of Eq. (3.14) by P,s. compositions of V into irreducible U&u (n - 1)) submod- Now apply Schur’s lemma to this situation. We note ules. We may Grst decompose V into irreducible U&u(n)) that both of the projection operators P, and Ps intertwine modules to give the Clebsch-Gordan reduction with the action of U&( n - 1)). It is thus clear that we may write V(A) 8 V(p) = e v(cl+A(r>), (3.10) pfA(yW@~ Paq&=APa, (3.15) where il ep denotes the set of distinct highest weights occurring in the decomposition of v(n) @ V(,u), p qA&3=~p~ > (3.16) + A( y) denotes the irreps occurring in this decomposition, and y is the operator pattern (multiplicity) label used for some scalar A. It is easily seen that the constant ,l is to distinguish the equivalent U&u (n )) modules occurring the (square of) the overlap of the two projections, and in the decomposition [an explicit labeling pattern for hence vanishes unless the two U&u (n - 1)) modules have U&(n)) was given above]. Next we may decompose the same irrep labels. Moreover, the overlap is precisely each space V@+A(y)) into irreducible U,(u(n- 1)) the (square of) the reduced tensor operator matrix ele- submodules [labeled by v. with the subscript 0 denoting ment, that is, U&u( n - 1)) to give the following decomposition into irreducible U&A (n - 1)) submodules] /+q+Wr”. V(A) 0 V(p) = a3 es f’(p++(y) Jvo). /Lt~~~~ [(g]l I~3 /). p+A(yW@p ~0 (3.17) (3.11) Remark The determination of the square of the de- (This latter decomposition is multiplicity-free from the sired reduced matrix elements rather than the matrix el- Weyl branching theorem. ) ements themselves, involves no loss of information for q On the other hand, we may decompose each space groups, since the ( f ) phase of the square root is unde- v(n) and V(p) into irreducible U,(u(n- 1)) submod- formed for generic (continuous) values of the deforma- ules to give tion, and hence this ( f ) phase can be taken from earlier work.4*6*7 V(A) 0 V(p)= @ V(AlAo) Q Wlpo). (3.12) WA POCP IV. INVARIANT OPERATORS The construction of the projection operators-to Now we may write the Clebsch-Gordan decomposition which our problem has been reduced-involves as a prior of the tensor product module ?‘(A,) o V(po> according step the construction of the ring of invariant operators, to ZxZ,, where Z is the centralizer of U&(n)) and Z. is the centralizer of the sub-q group, U&u (n - 1)). This is V(&) 0 V(pd = @ Wo~~oIluo+A(~‘)), ~o+A(r’)~ek~ the subject of the present section. (3.13) Consider the q group U&( n )). The explicit construction of the centralizer Z can be carried out by means where y’ is again an operator pattern labeling possible of the R matrix, a fundamental object in the theory of multiplicity. Thus we obtain the following reduction of quantum groups. The universal R matrix has been de- the space I’(n) o V(,u) into irreducible U,(u(n- 1)) sub- fined by Drinfeld.13 Consider the Hopf subalgebras modules: UJb,), U&b-)C U,(g) generated by {hi,ei\i=1,2,...,r}

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and {h;fi] i= 1,2,...,r}, respectively. Define a basis {e,]s [rQ =o, vud,(g). (4.9) =1,2,...} for U&b-); then there exists a basis {eS]s = I,2 ,... } dual to {e,]s= I,2 ,... } for the Hopf subalgebra Let p denote half the sum of the positive roots of g UJb+ ). The operator R is then defined by and let TV be the operation of taking the diagonal sum of any operator belonging to End( VA). Denoting by h,EU,(g) the Cartan element satisfying a( hp) = (a,~) R= 2 e,@gEU,(g) 8 U,(g). (4.1) s Vad%, we have the following. TheoremW 4: Given a r satisfying (4.8), the operator It has been shown by Drinfel’d’3 that R satisfies the equa- l?‘kUq(g), defined by tion rA=7,{(rA(q-% Q i)r), Rd( u) =g( u)R, ‘&U,(g) (4.2) satisfies Eq. (4.9). (as well as the Maguire-Yang-Baxter equation, which, Proofi Given in Appendix A. however, will not be needed in the following). In Eq. Remark This result shows that the q analog of the (4.2), 2 is defined as follows: Introduce the twisting op- trace operation for an operator @-an operation required eration T, which changes the order of the elements in the to produce an invariant-is the diagonal sum of qehpU. tensor product U,(g) 8 U,(g), that is, for any element We call this operation the q-trace and denote it by ZiUiB U,EU9(g) 8 U,(g), we have T(XiuiB Vi) =ZP~B uk q-trA ( B ) = TA( qmhpB ), with rn the diagonal sum. with T’=id. Using the operator T we may then define a Let us consider now the invariant operator I”, where by r in Theorem 4 is obtained from the centralizer of the algebra a,( U4( u (n)) by using ( rA Q l)%‘. We denote this a= T(d). (4.3) operator as Ce’. Lemma 5: The eigenvalues of ‘%” acting on a finite- Let dimensional irreducible U,(g) module Y(p) are given by

RT=TT(R)= ~@ee,. (4.4) 5 It follows directly from Eq. (4.2) that RT satisfies the equation Prooj Given in Appendix B. Remark It is interesting to consider the limiting eigenvalues of %” when q -+ 1. Defining E by q = e’, we have R%(u) =d(u)RT, Vud,(g), (4.5)

and hence the operator %‘df,&g) Q U,(g), defined by w? lq+*= dim[A] +E~% (n,P+p) +2 ,C, (/J+pJi) t Ce=RTR, (4.6) X(ni&+p)+@($)* (4.10) has the property that Since Z++(&p) = 0, for any simple Lie algebra the %il(u)=R%(u)R=d(u)%, VgdJg) o U,(g). second term vanishes. For the third term one has (4.7) (p,v)(A++p,~Zdim[Al Accordingly, we have the following result: 2A,~(/-J~Ai)(ni9Y) = Theorem22 3: %’ E RTR is in the centralizer of the 2dimg ’ algebra ?l(U,(g)). The central element V may be used to construct in- Thus we find the following. variant operators of U&(n)) in this way: Consider a Lemma 6: In the limit q-+ 1, one finds finite-dimensional irreducible representation of U,(g) %‘-dim[;l] furnished by the irreducible module Y(il), with ,%being 12(A)dim[A1 ‘(I2@L) + @@I), the highest weight. Given an operator (q1’2-q-1’2)2 d 2 dim g REnd V(A) 8 U,(g), satisfying where I2 is the Casimir invariant of the simple Lie algebra Ir,(Qe 1mu>1 =Q ~=Uq(g), (4.8) 8. Since we want to work with U&u(n)), which is not a we want to construct from I’ an operator &U,(g), such simple Lie group, the condition that X+(&p) = 0 no that longer holds, and we find

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%‘--dim(A) 2. (4.15b) d/2-d’2 q=l (4.11) and d is the number of distinct weights of A. The results developed above assume a much more Pmoj Given in Appendix C. perspicuous form if we specialize the irrep ;1 to be the Remark: This is the quantum analog” of the charac- fundamental irrep of U&u (n )), that is, we choose 2 to be teristic identities23P24for Lie algebras, which, in turn, is a [ 1 b]. For this case, we find that the eigenvalues of %” are reformulation and extension of the resultsiO’” for U(n). Since our objective is to obtain invariant elements in the ring ZxZ,, we now turn to constructing elements in (4.12) Z,, having already obtained in B an element of Z (and hence of Zo). [We use systematically from here on, the notation that the subscript 0 denotes an element in U&u (n The weights of ,u+p appearing in Eq. (4.12) are the -1))C U&u(n)) correIuted to a similar element in partial hooks, Pi, of U&u(n)), which play a basic role in U&(n)).1 the representation theory. The partial hooks are defined Consider then the operator B,, belonging to by End V(k) Q U&(n)), defined by

pin (p+P)i=pi+?Z-1, i=1,2 ,..., n. (4.13) Bo= (q,cnczo l)cXC+). (4.16)

Thus By construction, B, is in the ring a,(ZxZ,) and defines U&u (n - 1)) invariants a( &) which satisfy polynomial identities on U&u (n - 1)) irreps V(po). . $$[I cl+ i q-Pi, (4.14) Lemma 8: Acting on a finite-dimensional irreducible i=l U,(u(n- 1))) module &do) C V(p), the matrix B. satisfies the following identity: which defines the characteristic roots. The characteristic

roots for U&u (n - 1)) are defined similarly. In this form d the invariance-of %” to transformations of the translated 11 PO-L$(po)h,,(l) o l)l=O. (4.17) Weyl group W are obvious, since these transformations t=1 simply permute the partial hooks in the sum. Remark: A basis for the centralizer Z of U&(n)) Prooj? The same (Appendix C) as for the lemma can be constructed using: (4.15). %~=q-tr~+ cl((Cn;ze l)%‘)k), k= 1,2,..., n. Their eigenvalues determine W invariant polynomial functions V. EXTENDED WEIGHTS, LEXICALLY INTERTWINED WEIGHTS of the U&u(n)) roots q-‘i. Cf. explicit results in Ref. 22. Let us now return to the task of constructing a basis In order to discuss eigenvalues of centralizer ele- for the ring of invariant operators ZXZ,, where Z is the ments, which is basic to our approach, it is useful to centralizer of U&u(n)) and Z. is the centralizer of the introduce the concept of extended weights. Although the U,(u(n-l))CU&(n)). This ring is to operate on the concept applies more generally, we will limit our discus- tensor product of V(1) Q V(p), which is to model the sion to the q group U&u(n)) and the sub-q group U&u (n action of the tensor operator (A) acting on the irrep p. - 1)). An extended weight is then a pair of weight vec- Accordingly, we consider End V(h) o U,(g) and de- tors, denoted (A In,), with 1 a U&u(n)) weight and ilo a fine the invariant operator, U,(u(n - 1)) weight. An inner product between extended weights can be defined as B+rAel)d(%“>. (4.15) .(~l~o)‘(fJlPo) =(k) + (no&oh (5.1) Lemmaa 7: Acting on a finite-dimensional irreduc- with (,) the inner product on the weight vectors of ible U,(g) module V(p), the matrix operator B satisfies U&(n)) [respectively, U&24( n - l))]. the following polynomial identity: The group-subgroup relation between the two weight vectors &lo in an extended weight @,A,) implies that there are restrictions on extended weights. For both U(n) (4.15a) i, [B--Pt(p)h(l) o (1))l =O, and U&u(n)) these restrictions are consequences of the Weyl branching rules, which, as we have seen earlier, are with encoded by lexicality conditions. In fact, it is easily seen

that an extended weight for U,#(n>):U&( n - 1)) is Lemma 9: Every element of ZXZ, determines a p- necessarily of the form of a truncated Gel’fand-Weyl pat- invariant function on extended weights. tern containing only the two top rows. Remark: This generalizes Chandra’s result on central Iexically intertwined extended weight elements of simple Lie algebras. (~,~~~~U~u(n)):(i,(u(n- 1)) has the form We utilize this result below to simplify the determination of reduced tensor operator matrix elements.

)~i+l,,>;li+l,n-1>...>~2,, 9 (5.2) VI. CONSTRUCTION OF THE PROJECTION where a= hd2n*~ *&J and OPERATORS =(jZl,n-1,~.2,n-l...aZn--,n--l). Equivalently, the lexica$ intertwined extended weight (&lo) fits into the two- Using the elemknts B and B. of the centralizer ZxZ, we can now construct the projection operators discussed rowed pattern: in Sec. III above, necessary for determining the reduced tensor operator matrix elements. (We relabel more sug- [h” a 2n -** ;In--ln 4tn\ gestively the operators P, and Pa of Sec. III.) \ &t-l Azn- 1 an-In-l I ’ The projection operator, denoted P,-, is to project the (5.3) irreducible module V@+ A( r))-where A is a weight of with the lexicality constraints of a Gel’fand-Weyl pat- il and r is an operator pattern d-from the tensor prod- tern. uct module F’(n) Q Y(p). (We assume “general posi- We denote the set of all lexically intertwined weights tion,” that is, cLi)ili SO as to avoid null spaces.) by 3’. These weights play a role analogous to those of The operator P,- is difficult to construct for the gen- dominant integral weights familiar in the theory of simple eral case because of the multiplicity problem. If, however, Lie algebras, as can be seen from the easily established the shift A( I’) corresponds to a permutation of the high- fact that (A I~o)~3’~;1,~o are dominant integral est weight, then the associated I’ pattern is unique weights of U&u(n)), U&( n - 1 )), respectively. [A(l?)jr] and extremal [A(I’)EPerm(&)]. Since the The extended translated Weyl group .YF= WX K. pattern calculus-at least initially-is limited to such acts on an extended weight by cases we assume that l? is extremal and label the projection operator as PA. This operator can be easily con- o(alan,)=(a(a> I&), is@, (5.4) structed from B by using

~obc~l~o>=(~l~obc~o>), g)E w, . (5.5) W-PA) PA= n (6.1) Given any function f(n I no), we then define A+ (flA-&,) * #A av 1A,) =fG- ’ (A 1 In,), 0-s @, (5.6) Now let ilo be a dominant U,(u(n - 1)) weight so that (A I ilo)4? is lexically intertwined. We can then con- ~~c~I~o)=f(;lI~~‘(~,,), =j30. (5.7) struct the projection operator PA0 using the invariant element B,,. (We are assuming on&e again that the weight The previous results, Lemmata (7) and (8), may now be A0 is a permutation of the highest weight ilo, so as to reformulated as follows. avoid multiplicity problems.) We find Given (A l;lO)&‘, we denote by V(,l I ;1,) the irreducible module of U,(u(n - 1)) with highest weight A0 (Bo-&I,,),) contained in the irreducible module V(A) of U&u (n )) ph= n (6.2) with highest weight A. Then, given ceZxZo, it must take (&)~1, (~A,,-&J,) * a constant value on V(n I no), denoted c(n 1A,). Our previous results state that this eigenvalue is invariant under The operator PA0 projects the U&u (n - 1)) tensor prod- %“, that is, uct module V(n,) Q V(po) C V(n) @ V(p) onto the module V(,uo+Ao)C V(p+A). ~c(d~~o)=c(~-l(;l~~o)=c(il~ilo)), Vi%?, (5.8) The projection operators PA and PQ which have been constructed in Eqs. (6.1) and (6.2), suffice to de- aoc(~ Ino) =4/z 1G; ‘(n,>=c(np,)), vz($wo. termine the desired reduced tensor operator matrix ele- (5.9) ments. To demonstrate this let us form the product op- Thus we have shown (cf. remarks preceding Lemma 6) erator PAP% and then take the (partial) q trace over the that: module V(n,) C V(n):

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The projection operator PA0 projects onto the ~2"qq-trY(~)CY(L)(PAPAo). (6.3) V(po+Ao) module in the tensor product module The operator Z-which acts on the U?(u(n)) module V(n, o po), so that the q trace is simply dim,ko+ ho]. V(,u)-is, by this construction, an invariant operator in This establishes the stated result [Eq. (6.4)] for the the centralizer ZX Z. of U&U (n - 1)) in U&(n)). Thus eigenvalues of 9. n 9 assumes a constant value on the module V(,uc) C V(,u) and hence has an eigenvalue on the ex- VII. DENOMINATOR POLYNOMIALS tended weight (p I&E-Y’. Theorem 10: The eigenvalues of 9’ defined in Eq. It is convenient now to split the terms in the operator (6.3) are given by Z/-see Eq. (6.6)-into a numerator and a denominator part. Explicitly we define for the denominators of PA and Pb the operators dA and dh: ~&o+Aol . dim&d &= rl: (fiA--/%,)P (7.1) -w #A where the shifts A,A, are determined by the extremal operator patterns (I’) and (r,) with A G A( I), ho= A( I?,) and dim,bo] is the U,(U( n - 1)) q dimension db= n U&,,-P(n,,,). (7.2) given by wr #ho [ (Po+Pota)1 As we will prove in Sec. IX there are common (ei- dim&d =trv,,tbj C@“Y = a?0 [(POP)1 ’ genvalue) factors between the product operator dAd& (6.5) and the numerator polynomial arising from the product operator PAPS These common factors can be described with the product over the positive roots of U(U( n - 1)). in a Lie theoretic way in terms of sets of weight vectors in Prooj We first construct S’, the q trace of 9’ over A and 1,. Let us define the following sets of weights: the module V(,U~) C V(p): ~A={Az,~I&-A=~~, O#ka and 9'~q-try(~oe~o)cY(~eCc)(PAPao). (6.6) a a positive root of Lie G) (7.3a) Since 9? is constant on Y(po), this yields for the eigenvalues of 9 the eigenvalues of 9’ multiplied by ~~,,“{(&)r&l (&)i-A=kao, oZk% dim&a]. Next we observe that Pb in 9’ can be replaced by a0 a positive root of Lie G). (7.3b) (Pz,), since Ph, is a projection operator, and then Letting w(A) denote the set of weights of ,J.we define Pb(pZ,) can be replaced by P~PAPQ since cyclic per- the complementary sets by mutation is permitted under the q trace. From Eq. (3.21) we have established that iiA=ti(A) -tiA (7.3c)

- u-1 - 2 and b+Al [Al [PI P&PAP& = bo+Aol tnol 1~01 pAo’ oAo=C-d(A) -a&. (7.3d) .(ro). ) (6.7) In addition, we denote the cardinality of @A by nA and Of WA, by nAo. It iS not difficult to show that nA Thus =2(&p) and nho = 2 ( &J~), where p is half the sum of

2 the positive roots of U&(n)), and similarly for pe. - (r) - In Sec. IX we deduce divisibility of the numerator by g’= b+Al [Al [PI the factors bo+AoI [A01 bol go). F= n (PA+$) n (/&,,-&I& (7.4) +&(A) c+o4cn, (6.8) #A +a,

which implies that we have the relation [cf. Eq. (6.6)] Den

(7.5) = n &--&i) n (~A,-~(~,,,), +EoA(A) a&yp) where Num,( * * *) is a polynomial in the roots of degree (7.7b) nA + nb = 2(&p) + 2(~0,p0) = 2@1~0)*(plp0) andK and Cn,no is a numerical constant in C[q1’2,q-“2] inde- is a numerical coefficient [independent of representation labels (,u 1~~) on which the invariant acts]. We call pendent of the representation labels (p]po) of the ex- Num,( - * - ) the numerator polynomial, which will be de- tended weight vector on which Numq( ** *) and termined explicitly in Sec. IX. [The subscript q denotes Den,( * * * ) act in Eq. (7.7a). that NumJ * * +) has an overall q-phase factor.] We call By construction, we note that

the denominator polynomial. [The subscript q denotes that Den,( * * * ) has an overall q-phase factor.] At this point we can deduce the explicit form of the (7.6) denominator polynomial for the quantum group Uq(#(n)). For the set @A(n) we have the weights: Substituting Eq. (7.6) into the result given by Theorem 10, we find li=A+ko, a=Ei-ej (i

= _ [k]q-(‘/2)(~+P+A,~i+Ei) =CA,A,. dif~$ “pko](Denq i t))-( ’ x [(p+p+A,a) +kl, (7.9) where [* . *] denotes a q integer. There is a similar result (7.7a) for flAo(pO) - &o(,uo). Substituting into the definition of the denominator polynomial Den,( * +. ), Eq. (7.7b), we where obtain

I dim,bol AcAj Den q (,u[,u~)= “fiA’ [Ai-Aj]!(-q-“/‘)‘~+p+A,Ei+Ej)(A.Ei-Ej)). kgl [(P++P,Ei-ej)+k-I] dimqh+Aol i

X {similar terms for U,(u( n - 1 ))factors}. (7.10)

This result can be simplified to give an explicit manic polynomial form for the denominator function Den,( ** * ) acting on the lexically intertwined extended weight (p 11~~):

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[(p+p+A,a)-111 ‘A’fi

(Ao,qh>O [(po+po+Ao,adl! (bff’o [(pO+pO,aO)--ll! x rI (7.11) ao>o [ (p0+p0,a0) I! no,o [(rUO+pO+AO,aO)--II!’ I

In this result the q phase is given by Den a a0 = a&A + @sio,Ao9 (7.12) ’ ( A A0 )

with . n [(Po+Po+Ao,~o)I! @a,A=-i AiiT (~+p+AtEi+Ej)(A,Ei-ej) cro,o [(pO+pOtaO>l! ’ ‘

1 (&o)i< CL\o)j %,,,A0 = - -2 C (~o+~o+Ao,Ei+~j)(AotEi-~j) a a0 i (4)O,j -5 C (Po+~o+&sEi+~j) (&,Ei-ej)* i

(7.14) (8.2) Remark: In the formulas developed above for the denominator function, the components coming from the irreps ;1 and lo [in U&(n)) and U&( n - 1 )), respec- More explicitly, we have, for the semimaximal case, tively] appear to play an equivalent role (for example, in dA and d&) . In the final result for Den ( +* * ) the role of ilo is distinguished. The reason for this is the ratio of q di- mensions that enters in Theorem 10. This fact is of inter- (II 1~01 est because it underlies the curious (but essential!) “tail rule,” which is a key component of the pattern calculus rules (Rule 4, Sec. XI). [pi+ hi-pj-Aj+j-i- I]! i

a a0 after we have found all numerator factors by using the 4, a /z. (PIPO) extended Weyl group symmetry, that the numbers of ze- ( ) ros implied by the betweenness constraints is exactly nA + nh, [defined after Eq. (7.3d)l. We must now prove that each of these factors in Eqs. = -k i (/-ff+Ai-/Lj-Aj+n+ 1 -i-j) (,%i-Aj) icj (9.2) and (9.3) implies a q-number factor in the evaluation of the Num& 9.e) function on the extended weight 0-JIPO). -k “il ((PO)i+ (AO)i-(/JO)j-(AO)j+n-i-n To find a linear factor in the Uq(u(n- 1)) and i

IX. NUMERATOR POLYNOMIAL

The numerator polynomial has been defined implic- a( q~2~~:,~) +P( $(:)d_l:l) +q~,2~q-m itly, Eq. (7.7a), on the assumption-to be proven in this section-that the common factors F, defined in Eq. (7.4), cancel. [The original numerator polynomial, containing the factors F, is defined by the projection operators in eqs. =0, a,j3,y&[q1’2,q-1’2], (9.4) (6.1) and (6.2) .] We will first prove the existence of the common factors F and then evaluate explicitly the nu- whenever case (i) occurs, that is, whenever (po) r=pr- k. merator function acting on an arbitrary extended weight. Substituting (9.2) into Eq. (9.4) implies The structure of the numerator function is determined by the betweenness relations for lexically intertwined extended weights. It is known, from the explicit results for U(n), that the betweenness constraints are, in (y-a-p) +q-‘m’G?+aq-k) =O. (9.5) fact, dejinitiue4’5 for the numerator function, and hence (from the Lusztig-Rosso theorem) for the q group This is to hold identically in q for all possible extended u,(O)). weights (p 1po), from which we deduce that From the betweenness constraints we see that a reduced tensor operator matrix element must vanish whenever (p I po) is lexically intertwined but (CL+ A I po+ A,) P=-aqek and a+/?-Y=o. (9.6) is not lexically intertwined. This situation can occur in two possible cases. 0) (po+Ao),> @+A), for some r, l

(pO+Ao),-(p+Ah),-k=O, l, refer to the initial We thus deduce divisibility of the numerator polyno- extended weights. mial by factors (ii) @+A),+,> (po+AO)r for some r, l A,. k=l pr+,--(po)r+k---l=Q la%A,+,--((ho),. (9.3) (9.8) Each of the factors in Eqs. (9.2) and (9.3) correlates one-to-one with a factor in the denominator function [as Applying exactly the same argument to the Weyl conju- is known to be true for the U(n) case415].It will follow, gate polynomial

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In case (ii) we similarly get further factors corre- a 20 Num sponding to a(A) ao(Ao) we deduce divisibility of the latter by factors

o&),-a(A), rI q-(1/2)bo),+P>[ (Po),-p,+k], k=l *oW,+,--ho,, (9.13) when ae(Ac),>o(A), . (9.9) or equivalently, Since, from Weyl group symmetries,

-(‘~2)((po),+p,+,+(k-1))(4(1/2)do,,l-(po~,+k-l) a. 4 =?‘Z;’ Num, a(A) _q-(‘/2)(p,+I-(Po),+k-‘). (9.14) (9.10) Since q -(1’2)(k-1)~C[q1’2,q-1’2] is just a (representa- we then deduce divisibility of tion independent) scalar we thus deduce divisibility by factors A’+j@lb’q- by the further set of factors: (1’2)((pO)~+~‘+1)[pr+l-(po)r+m-~]. q(4$,--dA), (9.15) ~-lG1 n 4-( ‘/2)((~)‘+P,)~PO,-P,+klApplying all possible permutations of Uq(u(n)) roots k=l pr+ 1 and Uq(u( n - 1)) roots (po), as before, we deduce ~o@,,),-~(A), divisibility of the numerator polynomial by the further set = n 4-‘moL7,-~c ’,+P~z)) of factors: k=l

Ant>& (9.11) (9.16) T@ing into account arbitrary permutations of k and Wo, we thus deduce divisibility of the numerator by the If we count up the number of factors obtained above factors we find precisely the number nA + nQ which, as discussed earlier, is exactly the number of factors in the denominator polynomial. Accordingly, since all remain- (1’2)(Wr+Pk) [ (PO),-pk+m]. ing factors [from F, Eq. (7.4)] must cancel, we can now give an explicit form for the general numerator function (A,$,> Ak (9.12) acting on the extended weight (p 1~~):

(1/2)(h)r+Pk)[Pk- (po),+m- 11 Ak> (ho),

n-1 (&j),-Ak x fi n fl q-(“2)((PO)‘+Pk’[(po)r-pk+m] k=l r=l m=l (b)r>‘h

n n-1 Ak-(&)r n-l &)r-Ak (9.17) k=l r=l ,rl, bk-(PO)r+m-l] kfi, ,;[, ,‘1, [(&‘O)~-~k+m]~ A/‘> (ho), (‘h)r>Ak where the numerator q phase is defined to be

n-1 n-1 ,r; t?‘k+ (Po)r)@k-- (Ao)k; kz, c (P/c+ (Po),)((Ao),--‘kc). k=l r=l hk> (J&J), Ak> (ho), Ak< (ho), Ak< (&I), (9.18)

X. SUMMARY OF EXPLICIT RESULTS

A. The semimaximal case We are now in a position to determine the semimaximal case-begun in Sec. VIII for the denominator function- completely. For the numerator function, using Sec. IX, we have the result n &,),-(L)k =pm(’ 2) ryk mt, [(pO)r-pk+m] b.: n (‘)k-(‘b)r [pk-(po)r+m-l], m=l

where

-’ i dDk+ (pO)r)(@O),-~k~-; 2; (Pk+ @O),)(Ak- (A,>,). (10.1)

Combining this result with the results, Eqs. (8.3) and (8.4), we find for the reduced (squared) tensor operator matrix element:

2 [P+nl Y [PI =C~,,;q’$ 2)-N um(: ::) (Den(: :I))-‘. (10.2) b+~ol LAOI [PO1 ( max : ) [Note that the q factors are now explicit-hence the subscript q on Num( * * * ) and Den ( +* * ) have been dropped.] The explicit results are ,I n-1 A,-(&I, tt- 1 (&O)k-Ak (plpo)= n n mt, [Pk-@‘O)r+m-l] ,I,n r=,n $, [(Po)r-Pk+m]~ (10.3) k=l r=l kk> (‘b)r (&),>Ak

n [~~+~i-rui-Aj+j-i- l]! ‘5’ [(/1o)i-(~lo)j+(ao)i-(Ao)j+i-i]! (10.4) [pi-/Jj+j-iI! i

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=i E, (Pi+4-Pj-;li+n+ l-i-j)(Ai-Ajzi) -i “i’ (&)i+ (&Ii- (/&)j- (&)i+n-i-j) ‘

x((aO)i-(aO)j)-~ i (Pk+ (PO),)((aO)~)((aO),-ak)-~ Lz: (Pk+ (PO)r)(~k-(~O)r). (10.5) r

The constant CjLo is determined by the requirement that

-max.

(:;;I :;A lr:;,=l~ (10.6) ,max

which yields the result

[ar-ak+k-r- i]! ‘fil [(aO>r-(&)k+k-+ =!? (10.7) r

where

(010). (10.8)

SimpliJcation of q phases,

-k 12 @k+ (PO)r-r)(~k-(&)r)-

At this point, it is convenient to work with the overall phase

( lO.lOa)

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which simplifies to

&+Pk) &-ak>-bk+ (~O)r)(bwr-&))

( 10.10)

Redefine the constant

(10.11)

For the semi-maximal case we have, finally,

max

(,:‘,~~d,~‘~~,ll’l&ll) ‘=q ‘(i%,t,~Num(: t:)(Den(: :I))-‘, , max

with @(*a-), CA,& defined in Eqs. (lO.lOb) and (lO.ll>, respectively, and Num( a-*) and Den(***) given in Eqs. (10.3) and ( 10.4)) respectively.

B. The general result

The general reduced tensor operator matrix elements can now be obtained from the results above, Eq. ( 10.12), by using extended Weyl group symmetries. We find

(10.13)

where

a(; ;)(,I,)=*(; ;I)(a- ‘(p) I~&d), A=a(A), Ao=oo(a), ( 10.14)

and

‘ff [Pk-(Po)r+Ak-(Ao)r-ll! fi “ii’ [ t&r-Pk+ (A,),--A,]!

[Pk-(PO)r-ll! k=l t-1 [ (PO)r-Pkl! (Ao),> Ak (ho)+ Ak (10.15)

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II-1 [Pr-Pkl! [(PO),-(PO)k+(AO)r-(hO)kI! II [Pr-P/c+Ar-&I! lack [(PO)r-(PO)kl! A,> Ak Ar (hg)k

n-1 [(pO>r-(PO)k-ll! x II (10.16) lack [(Po)r-(~O)k+(A~)r-(A~)k-ll!‘ &)r< (ho)k

Xl. THE PATTERN CALCULUS FOR THE QUANTUM 0 GROUP U&I(~)) 1 0 We have obtained in Sets. IX and X the matrix ele- 1 0 0 ttshift pattern= ments for all tensor operators in U&u(n)) having ex- 1 0 tremal shifts and acting on arbitrary irreps. In the present section we will demonstrate that these matrix elements 0 are characterized by a very simple structure, which can 0 1 0 be directly related to the shifts themselves. These rela- = tions constitute the pattern calculus rules. ( 0 1 ) * The pattern calculus rules determine the explicit reduced matrix element in the form We can now state the pattern calculus rules for elementary tensor operators.

A. The pattern calculus rules for elementary tensor operators in U@(n)) Rule 1: Write out two rows of dots with n dots in row n and n - 1 dots in row n- 1, in the matter displayed below:

directly from the initial state data-b] and hoI-using 0 0 0 0 a l row n the operator data [a], [ao], (I), and (I,). It is important 0 a 0”’ 0 0 row n-l’ to note that the final state data are uniquely determined from this information. Note that the operator patterns are Rule 2: In row n, assign to the ith dot the U&u(n)) (I’)=(r.$, ia=n-1 and (r,)=(r,)? i(j=n-2, partial hook: Pi,nEmi,,+n-i; do the same in row n - 1 [with (r,),,-, = (a,)J, as defined in Sec. III; the shifts using the U&U (n - 1)) partial hooks: Pi,n_ i = mi+- 1+ n A(T), A(I’,) are defined by Eq. (3.7). -1-i. The pattern calculus rules apply to extremal operator Rule 3: Draw arrows between the dots as follows: patterns, that is, operator patterns, such that the shift A Select a dot i in row n and a dot j in row n - 1. If A,, implies the pattern I’. For such patterns, the shifts A are > Aj,“- 1, draw ZUIarrow from dot i to dot j; if Ai,+ < Aj,,, permutations of the maximal weight of the irrep charac- draw the arrow from j to dot i. Carry out this procedure terizing the operator. for all dots in rows n and n - 1. (If Ain = Aj,“- 1 go to Now arrange the two shifts A and A0 in the form of another pair.) This yields a numerator arrow pattern with a two-rowed pattern, similar to a (truncated) Gel’fand- arrows going between rows. Weyl pattern, but without the betweenness constraints. We Carry out this same procedure for dots within row n call this the shift pattern. and dots within row n - 1. This yields a denominator ar- An example will make these steps clear. Take, for row pattern with arrows going within rows. example, the tensor operator ([l 0 01) and consider the Rule 4: Assign to each arrow the q-integer factor reduced matrix element associated to the r pattern (, i) and the I” pattern (loo). That is, MW -p(head) +e(til) I,

wherep( tail/head) is the partial hook associated with the is then given by dot (tail/head), and e( tail) 1[m~~~~~~!~~)o] j[ ‘ i$jikl]j[;:I:]) 1, if the tail of the arrow is on row II - 1, F I0, if the tail of the arrow is on row n. Rule 5: Write out the products: where each of the terms have been given by earlier rules. iV2=product of all q-integer factors These pattern calculus rules determine explicitly all for numerator arrow pattern, reduced matrix elements of all elementary tensor opera- @=product of all q-integer factors tors acting on arbitrary extended weight vectors in the for denominator arrow pattern. quantum group structure U&(n)): U&u (n - 1)). In the limit q-+ 1, they become exactly the pattern calculus Rule 6: sign convention. The ( * ) sign for the matrix rules4 for the family of unitary groups U( n ) . element is obtained in this way. Take the shift A-which consists of k ones and (n -k) zeros in some order-and assign the k integers il B. Extension of the pattern calculus rules to q- Aj,,-,(ro), draw A,(r) -Ai,-, arrows from dot which we assign (cf. rule 6) the sequences ( il,i2* * *i,J and i to dotj; if A,(r)

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Rule 9: normalization. For nonelementary extremal the matrix elements for U@(n)) and the linear factors operators, one uses the normalization CA,b given by Eq. for U(n) become q-integer factors for U&u(n)). ( 10.11) to multiply the term found for the product of the This result leaves only the q-phase factor to be deter- elementary operators. mined by an a priori argument. To develop this argument, let us consider the 6j coefficients of the quantum group U&(n)). (These coefficients are also known as Racah coefficients or “tetrahe- XII. AN ALTERNATIVE DERIVATION dral invariant operators;” see below. ) The explicit results for the pattern calculus for The 6j coefficients are most directly understood from U&(n))-as given in Sec. XI and proved in Sets. VII- the tensor operator viewpoint. The q-tensor operator al- X-are structurally similar to the pattern calculus results gebra has as an algebraic basis the unit tensor operators, for the Lie group U(n)-the most significant difference classified by Gel’fand-Weyl and operator patterns, with being the presence of a “q-factor” (factors of the form invariant operators as scalars. The q- 3j coefficients de- qa). This close structural similarity strongly suggests that fine a Kronecker multiplication in this algebra, such that there should be a more direct route to these results for the the q-3j product of unit tensor operators yields an irre- q-group pattern calculus. In this section we develop ducible tensor operator: ( W’I) X WI)) briefly such an alternative, more conceptual, approach. = (invariant) * ([M”]). The product operator is not a unit For generic values of q, the Lusztig-Rosso theo- tensor operator but an invariant multiple of such an op- rem2o721imposes strong constraints on the representation erator. This invariant multiplien is none other than the theory of compact q groups, and, as we have often used q-6j coefficient, which accordingly is defined as above, the labeling of individual vectors in a U,@(n)) irrep is forced, by this theorem, to be identical in form to a Gel’fand-Weyl pattern of integers. (This same con- straint requires the operator patterns to be integral and = (q- 6j invariant operator). (12.1) also structurally identical to Gel’fand-Weyl patterns.) In view of these strong constraints, and the require- The action of this q - 6j invariant operator is on the it-reps ment of continuity in q, we see that the argument pre- [m] comprising model space M. Thus we see that sented in Sec. IX, determining (from betweenness constraints and generalized Weyl symmetry) the numerator polynomial, is valid a priori. The same argument, moreover, implies that these zeros appear as q-integer factors = numerical q - 6j coefficient. (12.2) vanishing for nonlexically intertwined final states. From continuity arguments, and the invariance of dimension to This abbreviated notation suppresses intermediate deformation, we can conclude that these nonlexicality ze- coupling labels (operator patterns I’) and the structure of ros and their Weyl transforms are, in fact, the only zeros the q-6j coefficient is not apparent. A more accessible of the reduced operators. form is obtained if we make explicit all the various cou- It is a known result that the zeros of the elementary plings involved. Reading from right to left (as is custom- operators of the Lie group U(n) are definitive in determining the pattern calculus [to within ( * ) signs, which ary in quantum mechanics for the action of operators on state vectors) these couplings are of the form: operator are conventional]. (M) acts on the irrep [m] to yield the new it-rep [m’], etc. Knowledge of these zeros thus determines the linear factors of the numerator in the U(n) case, and the q- That is integer factors in the q-group case. The denominator factors (in both cases) can be uniquely determined from this ;L: :[ml+ [m’l, (12.3a) information by choosing special cases for which the re- 1 1 duced matrix element is known to be unity. By using permutational symmetry of the partial hooks (generalized Weyl symmetry) the general case may be obtained :L,; :[m’] -+ [m”], (12.3b) ( 1 from the special cases. These arguments suffice to prove the following. Lemma II: The explicit monomial reduced matrix :Li)] :[m”] + [ml, (12.3~) elements for the q-group U&(n)) are determined by ab- ( 1 stract structural arguments (symmetry and continuity) to within a multiplicative q-phase factor (qa). The pattern calculus rules for U(n) determine the ( f ) sign of (12.3d)

[Here the [ml’s belong to model space M. Since the over- mined in Sec. X and given by rule 8 of the pattern cal- all action is that of an invariant operator, we must return culus rules. to the initial irrep [m] as shown in ( 12.3~). The last This determination by abstract arguments of the q- coupling (12.3d) is effected by a q-3j coefficient, as in phase factor completes our aim of determining the pat- (12.1).] tern calculus for monomial matrix elements by concep- Each of these couplings requires an operator pattern tual arguments. We conclude that the pattern calculus for (l?) to be uniquely defined. Thus we see that the struc- U&(n)) can be determined completely, using abstract ture of the q-6j coefficient is determined by four triples structural arguments, from the pattern calculus for U(n). of irreps (each triple uniquely defined by an associated We have sketched this argument very briefly, but the operator pattern r) and by six irreps. We may accord- details can be filled in as required. The point of our dis- ingly associate this structure with a tetrahedron, by asso- cussion is not to obtain these coefficients explicitly (this ciating each triple and operator pattern to a face of the has already been carried out completely in the sections tetrahedron, and the six irreps to the lines. (The symme- above), but rather to demonstrate that an alternative, try of the tetrahedron suggests corresponding symmetries noncalculational, conceptual approach does exist yielding of the q-6j invariant, which, however, have not as yet precisely the same results. been fully proved for U&(n)), n>3. Hence one should regard the tetrahedron as only a convenient mnemonic ACKNOWLEDGMENT for the actual coupling relationships.) It is important for the q-group properties of this tet- This work was supported in part by the National rahedral coefficient to recall that each of the four cou- Science Foundation, Grant No. PHY-9008007. plings deBnes an ordering, and moreover that the opposite ordering is the replacement: q+q-I. If we now try to impose a consistent ordering on the four faces of the tet- APPENDIX A rahedron, we see that no consistent ordering is possible, Theoremss 4: Given a r satisfying (4.9)) the operator and, in fact, the only satisfactory ordering of the four I”kU,(g), defined by faces requires that each edge be ordered oppositely for each of the two faces to which the edge belongs. Thus rLTA{(77A(q-hf) 8 i)r), each irrep is associated to two couplings in which the orderings are opposite. This requirement is therefore satisfies Eq. (4.9). [Here T~( * * * ) denotes the diagonal equivalent to the following result. sum.] Lemma 22: The q-6j coefficient is invariant to the Proo$ For convenience we repeat Eq. (4.8) here: substitution: q-q-‘. This is a remarkable result-known25 to be true for [mAmx4i=o, vuqw. (4.8’) Ue(su( 3))--and from it we can deduce that the following. Now consider Eq. (4.8’) with g=h,. Multiplying it by Lemma 13: A monomial q-6j coefficient has no q- rA(qmhp) @ 1, then taking the sum rn, we obtain phase factor. Monomial 6j coefficients are known26 from U(n) to O=r,t{[r,(n;le l)a(hi)] (r,t(qehp) o 1) be uniquely characterized (to within *signs) by their zeros and symmetries, which, in turn, leads to a pattern (Al) calculus for these coefficients. By an application of the = [~~{b-~Cqehp)ED1N?,hl, same argument as in Sec. IX, and as used above, we can since [h,hd]=O and the matrices of End V(A) cyclically conclude that the monomial q-6j coefficients are deter- commute under the sum 7;1. mined by this same pattern calculus, but with linear fac- Next consider Eq. (4.8’) with g = e&-(1’4)hi Mul- tors replaced by q integers. There is no undetermined q- tiplying by (rA(q-(hp-(“2)hi) @ 1) and taking the diagonal phase factor, and the ( f ) signs are undeformed. sum, we find How does this information help in our problem of determining the q-phase factors for reduced matrix ele- o=Tn{[r,(TAB l)d(efl-(“4)(hi))] ments? The answer lies in another fundamental structural relation for the tensor operator algebra: x(7r~(q-(%-(1’2)hi) 8 1)). (A21 Lemmazs.2z28 14: The q - 3j coefficients are limits of the q-6j coefficients in the limit that mnn-+ - CO. Using the result that: In particular, monomial reduced matrix elements for elementary tensor operators are the limits of monomial (IT-~~ l)d(egmh14) =(r~(eg-h~4) 8 1)(rA(qMhf2) q- 6j coefficients. This limit operation defines the q-phase factor, which is found to be precisely the q phase deter- 8 (eiq -h/4)), (A3)

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and introducing the first term of this result, Eq. (A3), nF==,(h’)“i(f In q)“Q, m,Ez+, with the hi defined by Xrj=, into Eq. (A2), we obtain p(h’)v(hi)=(p,v), v~,e- Evaluating the rhs of Eq. (BZ), we obtain

(A4) - T&A ( efl -hi4) ~0l)r(~~(q-(~~lp-~i~))601)). $$A#= As q--(4@-P)#= AZ q--(P+PJi), 033)

from which one finds that the eigenvalue of 55’ on V(p) Using the fact that [ei,q-‘hp-h”2’] = 0, one finds that is as stated. n these two terms cancel. Introducing the second term in Eq. (A3) into Eq. APPENDIX c: (A2), we obtain Lemma** 7: Acting on a finite-dimensional irreduc- [7-A{(77L(q-hf>0 l)r),eg-“14]. (A5) ible U,(g) module V(p), the matrix operator B satisfies the following polynomial identity: This establishes the desired result:

(A61 ii [B-Pt(p)(q(l) @(1))l =Q [nCh(q’-h~‘P)8 W3,eil =O, t=1 since the factor q -h#4 has already been shown to com- with mute and can be removed. The result for fi is established similarly. n

APPENDIX 6 and d is the number of distinct weights of il. Pro& The tensor product V(A) Q V(p) will decom- Lemma 5: The eigenvalues of 55” acting on a finite- pose into dimensional irreducible U,(g) module V(p) are given by V(A) 8 V(p) = 8 V&+A(y)), (Cl)

VA+ & q-(P+PJi)e p+A(yWelr where V(p + A( y)) is an irreducible U,(g) module. Since B lies in the centralizer of (r~ Q 1 )a( U,(g)), when acting Prooj We let $5” act on the lowest weight vector 9 of on V(p+A(y))C V@.) 8 V(p), it takes the eigenvalue V(p):

which, using Lemma 5, can be expressed as Since {e,(s=l,2,...) is the basis for U;, efl vanishes identically, except for those e&U; made up entirely of (C3) Cartan elements of U,(g). Thus

%%+ These are the desired eigenvalues, and noting that p + A ( y) has only d distinct values we have

={& 2) y...L!pg Xp+a(y,(B) =&&), t= W,...,d, (C4) and hence, B satisfies the indicated polynomial identity.W

’ C. N. Yang and M. L. Ge, in Braid Group, Knot Theory and Statistical Mechanics, Advanced Seria in Mathematical Physics (World Scien- tific, Singapore, 1989), Vol. 9. . ..(h.)“r(h’)‘1...(h’)‘,])vS, U32) *T. Curtright, D. Fairlie, and C. Zachos, in Quantum Groups, Proceed- ings of the Argonne Workshop, April-May 1990 (World Scientific, where we have used the fact that a basis for the Cartan Singapore, 1991) . ‘H.-D. Doebner and J.-D. Hennig, in Quantum Groups, Lecture Notes elements of U; is given by n~x,[~:,)m./,l!], m&+, in Physics, Proceedingsof 8th International Workshop on Mathemat- while a basis for those in is given by ical Physics, July 1989 (Springer-Verlag, Berlin, 1990), Vol. 370.

‘L. C. Biedenham and J. D. Louck, Commun. Math. Phys. 8, 89 16G. E. Baird and L. C. Biedenharn, J. Math. Phys. 5, 1730 (1964). (1968). “L C. Biedenham, A. Giovannini, and J. D. Louck, J. Math. Phys. 8, sL. C. Biedenham and J. D. Louck, Angular Momentum in Quantum 691 (1967). Physics, Theory and Application, Encyclopedia of Mathematics and its ‘*L. C. Biedenham and M. Tarlini, Lett. Math. Phys. 20, 271 (1990). Applications (Addison-Wesley, Reading, MA, 1981) (reprinted by 19V. Rittenberg and M. Scheunert, J. Math. Phys. 33,436 (1992). Cambridge University, Cambridge, 1989). *‘G. Lusztig, Adv. Math. 70, 237 (1988). 6R. LeBlanc and K. T. Hecht, J. Phys. A 20, 4613 (1987). *‘M. Rosso, Commun. Math. Phys. 117, 581 (1988). ‘M. D. Gould, J. Math. Phys. 27, 1944 (1986). ‘*M. D. Gould, R. B. Zhang, and A. J. Bracken, J. Math. Phys. 32, ‘M. D. Gould, J. Math. Phys. 27, 1964 (1986). 2298 (1991); M. D. Gould, J. Links, and A. J. Bracken, J. Math. 9B. L. Van der Waerden, Die Gruppentheoretische Methode in der Phys. 33, 1008 (1992). Quantenmechanik, Berlin 1932, Die Grundlchren d. Math. Wiss. 23R B. Zhang, M. D. Gould, and A. J. Bracken, Commun. Math. Phys. Band XXXVI. li7, 13 (1991). “G. E. Baird and L. C. Biedenham, J. Math. Phys. 5, 1723 (1964); cf. 24A. J. Bracken and H. S. Green, J. Math. Phys. 12,2099 ( 1971); H. S. the Appendix. Green, ibid. 12, 2106 (1971). I’M. D. Gould, Ann. Inst. H. PoincarC A 32,203 (1980); J. Phys. A 17, 2sM. Nomura, J. Phys. Sot. Jpn. 57, 3653 (1988); J. Math. Phys. 30, 1 (1984). 2397 (1989). “M. Jimbo, Lett. Math. Phys. 11, 247 ( 1986). 26L. C . Biedenham and J. D. Louck, The Racah-Wigner Algebra in “V. G. Drinfeld, Quantum Groups, Proceedings of the International Quantum Theory, Encyclopedia of Mathematics and Its Applications Congress of Mathematics (MSRI Berkeley, CA, 1986), pp. 798-820; (Addison-Wesley, Reading, MA, 1981), Vol. 9. Sov. Math. Dokl. 36, 212 (1988). *‘L. C. Biedenham, J. Math. Phys. XxX1, 287 (1953). “I. M. Gel’fand and V. Zelevinsky, Societe Math. de France, Asttrique, **L. C. Biedenham, in Quantum Groups, Proceedings of the 8th Inter- Hors Series 117, 1985. national Workshop on Mathematical Physics, Lecture Notes in Phys- ‘sD. E. Flath and L. C. Biedenham, Can. J. Math. 37, 710 (1985). ics Clausthal, Federal Republic of Germany (1989), p. 67ff.