Clebsh-Gordan coefficients for the algebra \mathfrak{gl}_3 and hypergeometric functions
aa r X i v : . [ m a t h . R T ] J a n Clebsh-Gordan coefficients for the algebra gl and hypergeometric functions ∗D. V. ArtamonovJanuary 5, 2021 Abstract
The Clebsh-Gordan coefficients for the Lie algebra gl in the Gelfand-Tsetlin base are calculated. In contrast to previous papers the result isgiven as an explicit formula. To obtain the result a realization of a repre-sentation in the space of functions on the group GL is used. The keystonefact that allows to carry the calculation of Clebsh-Gordan coefficients isthe theorem that says that functions corresponding to Gelfand-Tsetlinbase vectors can be expressed through generalized hypergeometric func-tions. Let U , V — be finite dimensional irreducible representation of the Lie alge-bra gl N . Let us take their tensor product and split it into a sum of irreduciblerepresentations U ⊗ V = X α W α . (1)Let { u i } , { v j } be bases in U , V , and let { w αk } be a base in W α . One has anrelation w αk = X i,j C i,jk ( α ) u i ⊗ v j , C i,jk ( α ) ∈ C . (2)The coefficietns C i,jk ( α ) in this relation are called the Clebsh-Gordan coeffi-cients . ∗ Это препринт Произведения, принятого для публикации в журнале «Алгебра иАнализ», 2021 год. Владелец прав на распространение – ПОМИ РАН. This is a preprintof a paper submitted to "St. Petersburg Mathematical Journal". All rights belong to POMIRAS N = 2 , . In the representations we take aGelfand-Tsetlin base, since this type of base appeares naturally in applicationsthat are discussed below.The Clebsh-Gordan coefficients for gl play an important role in the quantummechanics in the theory of spin. These coefficients were calculated explicitly byvan der Waerden and Racah (see [1, 2]).The Clebsh-Gordan coefficients for the algebra gl play an important rolein the theory of quarks (see [3]). But the problem of calculation of the Clebsh-Gordan coefficients in the case of the algebra gl is a much more difficult problemthan in the case gl . Nevertheless in some sence the formulas were obtained.Firstly it was done by Biedenharn, Baird, Louck and others in the series ofpapers [4, 5, 6, 7, 8] . But the obtained results are bulky. Also the papers donot contain a direct formula of type C i,jk ( α ) = . . . . It is only clear that such aformula can in principle be obtained from the results of these papers.At the same time there were appearing numerous papers where these coef-ficients were calculated in some particular cases (see [9, 10, 11, 12, 13]). Thereare attempts to make the formulas easier (in some particular cases) by usingspecial functions (see [14, 15]).Since an explicit result was not obtained investigations were continuing. Oneshould mention the paper [16], where the authors announced the discovery ofan explicit formula for the Clebsh-Gordan coefficients for the algebra я gl , butin this paper an answer is bulky and actually it is not explicit. The paperdoes not contain a formulas of type C i,jk ( α ) = . . . . So even later there wereappearing papers (see [17]) devoted to the search of an explicit formula for theClebsh-Gordan coefficients.Since a simple or at least an explicit formula was not obtained there appearedpapers devoted to algorithmic caculation of Clebsh-Gordan coefficients for gl ,see [18, 19]. Let us especially note the paper [20], where a code of a program thatcalculates an arbitrary Clebsh-Gordan coefficient for gl N is given. Moreover thisalgorithm is realized as an online calculator [21].A starting point for the present paper is the following. In [4] a very inter-esting formula was derived. If one realizes an irreducible representation of gl in the space of functions on GL , then a function corresponding to a Gelfand-Tsetlin base vector can be expressed using a Gauss’ hypegeometric function F , .Also in [4] a derivation of formulas for the action of generators of the algebra issketched, which is bases on using of contiguity relations for the function F , .The aim of the present paper is to obtain explicit for the Clebsh-Gordancoefficients for gl using the formulas expressing a function corresponding to a Mostly in these papers the case gl N is considered, but only in the case gl the obtainedresults allow to obtain in principle a formula for an arbitrary Clebsh-Gordan coefficient. F , but the hypergeometric Γ -series.It’s construction has an important advantage over the function F , : such aseries remembers all it’s parameters. This means the following. The consid-ered Γ -series satisfies a system of PDE called the Gelfand-Kapranov-Zelevinskysystem (GKZ shortly). The parameters of a Γ -series give asymptotic of it’s be-haviour near a components of a singular locus of the GKZ system . This fact isused in the present paper to obtain relation for the Γ -series that allow to obtainexplicit formulas for the Clebsh-Gordan coefficients.The structure of the paper is the following. In §2 the basic notions areintroduced . In 2.3 we presnet an important result: an explicit constructionof highest vectors of representations W α from (1). Before a close problem ofmultiplicities was discussed (см. [22, 23]), i.e. a problem of construction of anindex α in (1). However explicit construction of highest vectores was not known.These vectors are of two types.The case when the representation W α has the highest vector of the firsttype is considered in §3. The result of the calculation of the Clebsh-Gordancoefficients in presented in Theorems 3 and 4.The case when the representation W α has the highest vector of the first typeis considered in 4.2. Note that this case is reduced to the previous one. Theresult of the calculation in presented in Theorems 5 and 6.The resulting formulas are not simple but they are really explicit. Γ -series Information about a Γ -series can be found in [24].Let B ⊂ Z N be a lattice, let γ ∈ Z N be a fixed vector. Define a hypergeo-metric Γ -series in variables z , . . . , z N as follows F γ,B ( z ) = X b ∈ B z b + γ Γ( b + γ + 1) , (3)where z = ( z , . . . , z N ) . We use a multi-index notation: z b + γ := N Y i =1 z b i + γ i i , Γ( b + γ + 1) := N Y i =1 Γ( b i + γ i + 1) . We need the following properties of a Γ -series.3) A vector γ can be changed to γ + b , b ∈ B , under this transformation theseries does not change.2) ∂∂z i F γ,B ( z ) = F γ − e i ,B ( z ) , where e i = (0 , . . . , at the place i , . . . , .Below we put N = 4 , γ = ( γ , γ , γ , , and B = Z h (1 , − , − , i .3) Let F , ( a , a , b ; z ) = P n ∈ Z ≥ ( a ) n ( a ) n ( b ) n z n , where ( a ) n = Γ( a + n )Γ( a ) , is aGauss’ hypergeometric series. Then F γ,B ( z , z , z , z ) = cz γ z γ z γ F , (cid:16) − γ , − γ , γ + 1; z z z z (cid:17) ,c = 1Γ( γ + 1)Γ( γ + 1)Γ( γ + 1) .
4) One has F , ( a, b, c ; 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) . Put = (1 , , , , then for γ = 0 F γ,B ( ) = 1Γ( γ + 1)Γ( γ + 1)Γ( γ + 1) · Γ( γ + 1)Γ( γ + γ + γ + 1)Γ( γ + γ + 1)Γ( γ + γ + 1) . (4)Also note that F γ,B ( z , z , z , z ) | z z = z z = z γ z γ z γ F γ,B ( ) . (5)5) A Γ -series satisfies the Gelfand-Kapranov-Zelevinsky system of PDE (cid:16) ∂ ∂z ∂z − ∂ ∂z ∂z (cid:17) F γ,B = 0 ,z ∂∂z F γ,B + z ∂∂z F γ,B = ( γ + γ ) F γ,B ,z ∂∂z F γ,B + z ∂∂z F γ,B = ( γ + γ ) F γ,B ,z ∂∂z F γ,B − z ∂∂z F γ,B = ( γ − γ ) F γ,B . (6)Note that a singular locus of this system is defined by the equations z z z z ( z z − z z ) = 0 . (7)A sum of a Γ -series is called a A -hypergoemetric function. Note that the Γ -series considered in the paper are actually finite sums. In the paper a realization of a representation of the Lie algebra gl in the spaceof function on the Lie group GL . On a function f ( g ) , where g ∈ GL , anelement X ∈ GL acts by left shifts ( Xf )( g ) = f ( gX ) . (8)Passing to an infinitesimal action we obtain an action of gl on the space offunctions. 4et a ji be a function of a matrix element , occurring in the row j and thecolumn i . Introduce determinants a i ,...,i k := det( a ji ) j =1 ,...,ki = i ,...,i k . (9)In other words this is a determinant of a submatrix in the matrix ( a ji ) , formedby rows , . . . , k and columns i , . . . , i k . As a ji , this a function on GL .An operator E i,j acts onto a determinant by transforming the column indices E i,j a i ,...,i k = a { i ,...,i k }| j i , (10)where . | j i a substitution of an index j by i , if the index j does not occur in { i , . . . , i k } , then we put . | j i equal to zero. One has. Proposition 1.
The function a m − m ( m − m )! a m , m ! (11) is a highest vector with the weight [ m , m , . We divide in (11) by ( m − m )! m ! to obtain simpler formulas below.Also we use a realization in the space of function on a subgroup Z of matricesof type x y z , (12)i.e. in the space of functions of type f ( x, y, z ) ; see [25]. A tensor product of representations can be realized in the space of functions ona product GL × GL . Let a ji be a matrix element of the first factor GL , andlet b ji be a matrix element of the second factor GL .In the previous Section we introduced determinants a i ,...,i k , analogously onecan define determinants b i ,...,i k . Let us also introduce the following expressions ( ab ) i ,i = det a i b i ! i = i ,i , ( aabb ) i ,i ,i ,i = a i ,i b i ,i − a i ,i b i ,i , ( aab ) = det a i a i b i i =1 , , , ( abb ) = det a i b i b i i =1 , , . The notation a ji for the matrix elements is chosen by analogy with the papers [4, 5, 6, 7, 8];in these papers a bosonic realization is used, but this realization is equivalent to a realizationin the space of functions on GL . E i,j act onto these determinants by changing the column indices bythe formulas analogous to (10).Take a tensor product of representations of gl with the highest weights [ m , m , and [ ¯ m , ¯ m , with the highest vectors of type (11). Split thistensor product into a sum of gl -irreducibles. Theorem 1.
In the space of gl -highest vectors there exists a base consisting ofvectors a α b β a γ , b δ , ( ab ) ω , ( abb ) ϕ ( aabb ) θ , , , ; (13) where α + ω + ϕ = m − m , γ + θ = m ,β + ω = ¯ m − ¯ m , δ + ϕ + θ = ¯ m . (14) Proof. The vectors (13) belong to the tensor product of representations withthe highest vectors a m − m a m , and b ¯ m − ¯ m b ¯ m , . Indeed a space of a rep-resentation with the highest vector a m − m a m , is a space of polynomials in a , a , a , a , , a , , a , , such that for each monomial one has deg a +deg a +deg a = m − m , deg a , +deg a , +deg a , = m (see [25]). Analogously one can describea representation with the highest vector b ¯ m − ¯ m b ¯ m , . Due to relations (14), thefunction (13) is a linear combination of products of a polynomial in a and apolynomial in b , with satisfy the written conditions. From a formula of type (10) it immediately follows that (13) is a highestvector for gl . The vectors (13) are linearly independent. Indeed, let us pass to a realizationon the group Z × Z . Introduce the coordinate x , y , z и x , y , z on the factorsby analogy with (12), one has: a , b , a , , b , , ( abb ) ( x − x ) z + x − y , ( ab ) , ( x − x ) , ( aabb ) , , , ( z − z ) . (15)Hence functions ( ab ) , ( abb ) , ( aabb ) , , , are algebraically independent, that iswhy the functions (13) are linear independent. Let us prove that linear combinations of (13) give all gl -highest vectors. Todo it let us prove the equality of dimensions of the space of gl -highest vectorsand of the span vectors (13). Let us give an explicit description of the space of gl -highest vectors usinga technique from [25]. Elements of the tensor product we realize in the space offunctions f on Z × Z . This function belongs to a representation if and only if6 satisfies the indicator sysytems in variables x , y , z and x , y , z : L m − m +11 f = 0 ,L m +12 f = 0 , where L = ∂∂x + z ∂∂y , L = ∂∂z , L ¯ m − ¯ m +11 f = 0 , L ¯ m +12 f = 0 , where L = ∂∂x + z ∂∂y , L = ∂∂z . (16)The function f is a gl -highest vector if and only if it is invariant underthe right action of the group diag ( Z × Z ) . Such a function can be written as f ( ζ, z ) = g ( ζh − ) , z × ζ ∈ Z × Z . Note that h = zζ − = x − x ( y − y ) − z ( x − x )0 1 y − y . (17)The conditions (16) for the function f ( ζ, z ) are equivalent to the followingconditions for the function g ( h ) of a matrix h = x y z : L m − m +11 g = 0 ,L m +12 g = 0 , where L = ∂∂x + z ∂∂y , L = ∂∂z ; R ¯ m +11 g = 0 ,R ¯ m − ¯ m +12 g = 0 , where R = ∂∂z + x ∂∂y , R = ∂∂x . (18)To find a base in the space of solution of (18) introduce variables u = y − x z , v = y + x z , w = z , w = x . (19)Note that ( u + v ) = 2 y , ( v − u ) = 2 x z . Instead of variables x , y , z onecan use variables u, v, w or u, v, w . These two collections are related by theequality w = v − u w .Introduce function w A ( u + v ) B w C = w A − C ( u + v ) B ( v − u ) C = w C − A ( u + v ) B ( v − u ) A . (20)The space of polynomial solution of the system L m +12 g = L m − m +11 g = 0 has a base consisting of functions (20), such that A, B, C ≥ , A ≤ m , B + C ≤ m − m . (21) In [25] the derivation of this result contains a mistake. R ¯ m − ¯ m +12 g = R ¯ m +11 g = 0 has a base consisting of functions (20), such that A, B, C ≥ , C ≤ ¯ m − ¯ m , B + A ≤ ¯ m . (22) Note that the vectors (13) are defined by nonnegative integers ω, ϕ, θ , suchthat ω + ϕ ≤ m − m , ω ≤ ¯ m − ¯ m ,θ ≤ m , ϕ + θ ≤ ¯ m . (23) To the inequalities (23) their correspond inequalities (21), (22). The cor-respondence A ↔ θ, B ↔ ω, C ↔ ϕ is a bijection between the solution spaces of (23) and (21), (22). Hence thedimension of the span of vectors (13) equals to the dimension of the space of all gl -highest vectors.The formula (13) is non-symmetric: it involves ( abb ) , but it does not in-volve ( aab ) . To obtain a symmetric formula one can operate as follows. Using arelations a a , − a a , + a a , = 0 , b b , − b b , + b b , = 0 , one can obtain a relation ( ab ) , ( aabb ) , , , = ( aab ) a b , + ( abb ) a , b . (24)From these relations one can make the following conclusion. If one introducesa notation f ( ω, ϕ, ψ, θ ) = a α b β a γ , b δ , ( ab ) ω , ( abb ) ϕ ( aab ) ψ ( aabb ) θ , , , ,α + ω + ϕ = m − m , γ + θ + ψ = m ,β + ω + ψ = ¯ m − ¯ m , δ + ϕ + θ = ¯ m , then one has a relation f ( ω, ϕ, ψ, θ ) = f ( ω − , ϕ + 1 , ψ, θ −
1) + f ( ω − , ϕ, ψ + 1 , θ − . Applying this transformation one concludes that every highest vector can beexpressed through the vectors of type f (0 , ϕ, ψ, θ ) and f ( ω, ϕ, ψ, . The factthat these vector are linear independent can be obtained by restriction of thecorresponding functions onto the subgroup of upper-triangular matrices.8 roposition 2. |in the space of gl -highest vectors one has a base , consistingof vectors of type : f (0 , ϕ, ψ, θ ) , ϕ, ψ, θ ≥ ,α + ϕ = m − m γ + θ + ψ = m ,β + ψ = ¯ m − ¯ m , δ + ϕ + θ = ¯ m , (25) f ( ω, ϕ, ψ, , ω, ϕ, ψ ≥ ,α + ω + ϕ = m − m , γ + θ + ψ = m ,β + ω + ψ = ¯ m − ¯ m , δ + ϕ + θ = ¯ m . (26) Let us return to representations of gl realized in the space of function on GL with the highest vector (11). Let us give a formula for the functionscorresponding to the Gelfand-Tsetlin base vectors. To fix a normalization wetake in the space of gl -vectors the following base a m − k ( m − k )! a k − m ( k − m )! a m − k , ( m − k )! a k , k ! . (27)Note that this function can be rewritten as follows [25]: ( E , ) m − k ( m − k )! ∇ m − k , ( m − k )! f, ∇ , = E , + ( E , − E , + 1) − E , E , , where the highest vector f is given by the formula (11). Now let us find a vectorcorresponding to an arbitrary Gelfand-Tsetlin diagram: m m k k s . (28) Theorem 2.
Put B = Z h (1 , − , − , i , γ = ( s − m , k − s , m − k , , then tothe diagram (28) there corresponds a m − k ( m − k )! a k , k ! F γ,B ( a , a , a , , a , ) . (29)In [4] a close formula is given but instead of a Γ -series the function F , isused. Using formula relating F , and F γ , one immediately obtains the Theorem9 The case of the vector f ( ω, ϕ, ψ, In the space of gl -highest vectors we consider the base vectors of type f = a α b β a γ , b δ , ( ab ) ω , ( abb ) ϕ ( aab ) ψ α ! β ! γ ! δ ! ω ! , α + ω + ψ = m − m , γ + ϕ = m ,β + ω + ϕ = ¯ m − ¯ m , δ + ψ = ¯ m . (30) The weight of this vector equals to [ M , M , M ] = [ α + β + γ + δ + ω + ϕ + ψ, ϕ + ψ + ω, ϕ + ψ ] . To write a formula for a Clebsh-Gordan coefficient we need a formula foran arbitrary Gelfand-Tsetlin base vector in a representation defined by (30). Avector corresponding to a diagram M M M M − T M − T M − T − S , (31)can be written as follows: E S , S ! E T , T ! ∇ T , T ! f . Let us find a function on GL × GL corresponding to this diagram. Note thatall generators gl , that are not Cartan element, act onto ( aab ) and ( abb ) as zero.The main difficulty is to write a formula for the action of ∇ T , onto f .Firstly in Section 3.1 we write a formula for the action of ∇ , . Then to obtaina formula for the action of ∇ T , , we derive some new relations for Γ -series. Usingthem in Section 3.3 we write a formula for function corresponding to (31).Below we consider functions of type F γ,B ( a , a , a , , a , ) for different γ .That is why we use a shorter notation F γ ≡ F γ,B ( a , a , a , , a , ) . (32) ∇ , Let us give a formula for the action of operators e ∇ , = (( E , − E , + 1) E , + E , E , ) onto a function g = a α b β a γ , b δ , ( ab ) ω , . Lemma 1. e ∇ , g = ( α + β + γ + δ + ω + 1) (cid:16) a ∂∂a + b ∂∂b (cid:17) g − ( aab ) ∂ ∂a , ∂b g − ( abb ) ∂ ∂a ∂b , g. roof. The operator E , can be written as follows: a ∂∂a + b ∂∂b + a , ∂∂a , + b , ∂∂b , + ( ab ) , ∂∂ ( ab ) , . Onto E , g the operator ( E , − E , + 1) acts as a multiplication onto ( α + β ) .The operators E , E , are written as follows: a ∂∂a + b ∂∂b + a a , ∂ ∂a ∂a , + a b , ∂ ∂a ∂b , + b a , ∂ ∂b ∂a , + b b , ∂ ∂b ∂b , + a ( ab ) , ∂ ∂a ∂ ( ab ) , + b ( ab ) , ∂ ∂b ∂ ( ab ) , . Summing these two operatora one obtains after a simplification: ∇ , g = ( α + β + γ + δ + 1) × (cid:16) αa a α − b β a γ , b δ , ( ab ) ω , + βa a α b β − a γ , b δ , ( ab ) ω , (cid:17) − βγ ( aab ) a α b β − a γ − , b δ , ( ab ) ω , − αδ ( abb ) a α − b β a γ , b δ − , ( ab ) ω , . Put O = a ∂∂a + b ∂∂b , O = ( aab ) ∂ ∂a , ∂b + ( abb ) ∂ ∂a ∂b , . Note that ( E , − O = O E , , ( E , − O = O E , , O O = O O . (33)Thus one has, e ∇ n , f = X k c α + β + γ + δ + ωk O k O n − k f,c hk = ( − n − k X ≤ i < ···
One has a relation : a u F γ = X τ Y τ ( a a , − a a , ) p τ F γ τ , (34) ( abb ) λ λ ! ( aab ) µ µ ! ( ab ) ω , ω ! = X τ X τ a u τ a v τ , ( a a , − a a , ) p τ × b g τ b h τ , ( b b , − b b , ) q τ F θ τ ( a ) G ϑ τ ( b ) , (35) E n , n ! a m − k ( m − k )! a k , k ! F γ = X τ Z τ a i τ a j τ , ( a a , − a a , ) r τ F ε τ . (36) The index τ runs through some set. We need explicit formulas for the coefficients X , Y , Z . Introduce notations.Let γ = ( γ , γ , γ , , γ , ) , put Π p,γ := Y t = p (cid:0) t ( t + 1) + t ( γ + γ + γ , + γ , ) (cid:1) . (37) Proposition 3. In (34) for p τ = p ∈ Z ≥ one has a unique summand. For it γ τ = γ + u, γ τ = γ − p,γ τ , = γ , − p, γ τ , = 0 ,Y τ = Y uγ,p = Γ( u )Γ( u − k ) · F γ ( ) F γ α ( ) · p,γ τ . Proposition 4. In (35) one has p τ = q τ = q ∈ Z ≥ . To obtain , a formula for τ , fix a partition q =: q + q + q , ϕ + ϕ + ϕ := λ − q − q , ψ + ψ + ψ := µ − q − q, and put u = ϕ + q , v = ψ + q ,g = ψ + q , h = ϕ + q ,θ = ϕ + ω − ψ , θ = ϕ + ω + ψ ,θ , = ψ + ψ , θ , = 0 ,ϑ = ψ + ω − ϕ , ϑ = ψ + ω + ϕ ,ϑ , = ϕ + ϕ , ϑ , = 0 ,X τ = X θ,ϑ,q i ψ i ,ϕ i ,ω i = ( − ϕ ψ ω q q Π q,θ Π q,ϑ · h λ,µ,ωq ,q ,q F θ ( ) F ϑ ( ) × q ! ϕ ! ϕ ! ϕ ! ψ ! ψ ! ψ ! ω ! ω ! , where h λ,µ,ωq ,q ,q = P { ,...,q } = I ⊔ I ⊔ I , | I j | = q j × Q j / ∈ I ⊔ I (( q − j )+( λ − { the number of i s ∈ I , such that i s < j } )+( µ − { the number of i s ∈ I , such that i s < j } ) + ω ) . (38) Proposition 5.
In the formula (36) one has r ∈ Z ≥ and i τ = m − k + n , j α = k − n ,ε τ = γ , ε τ = γ − n − r,ε τ , = γ , + n − r, ε , = 0 ,Z τ = Z γ,m − k n ,n ,r = 1Π r,ε τ · F γ − n e ( ) F γ τ ( ) · m − k )! n !( k − n )! n ! . Th proofs of the results of this Section in given in Section 5.
Apply an operator E S , S ! E T , T ! ∇ T , T ! (39)to the vector (30). Note that α + β + γ + δ + ω = M − M . The result of application of (39) to (30) can be calculated by several steps.13 ) After application of ∇ T , T ! to (39) one gets X k + k + ϕ ′ + ψ ′ = T d M − M k + k ,ϕ ′ + ψ ′ λ ! ϕ ! ϕ ′ ! µ ! ψ ! ψ ′ ! × a α − k − ϕ ′ ( α − k − ϕ ′ )! b β − k − ψ ′ ( β − k − ψ ′ )! a γ − ψ ′ , ( γ − ψ ′ )! b δ − ϕ ′ , ( δ − ϕ ′ )! × ( ab ) ω , ω ! a k k ! b k k ! ( aab ) λ λ ! ( abb ) µ µ ! ,λ = ϕ + ϕ ′ , µ = ψ + ψ ′ . (40)The summation is taken over all decompositions of T , written below P .In the sums written below the summation also is taken over all partitionsand some integer parameters. When we move from a sum to the next one thenumber of partitions increases and it not possible to write them all under thesign P . We describe them in the test following the sum. Apply to the product ( ab ) ω , ω ! ( aab ) ( ψ + ψ ′ ) ( ψ + ψ ′ )! ( abb ) ( ϕ + ϕ ′ ) ( ϕ + ϕ ′ )! , containing in (40), the sec-ond main equality, using the Plucker relation a a , − a a , = − a a , , b b , − b b , = − b b , , one obtains X d M − M k + k ,ϕ ′ + ψ ′ λ ! ϕ ! ϕ ′ ! µ ! ψ ! ψ ′ ! × X θ,ϑ,q i ψ i ,ϕ i ,ω i ( α − k − ϕ ′ )!( β − k − ψ ′ )!( γ − ψ ′ )!( δ − ϕ ′ )! k ! k ! × a u + q + k a v + q + γ − ψ ′ , b g + q + k b h + q + δ − ϕ ′ , × a α − k − ϕ ′ F θ ( a ) · b β − k − ψ ′ G ϑ ( b ) , (41)The summation in (41) is taken over all partitions of T , that appeared inthe first step and over parameters q , ψ i , ϕ i , ω i defined in (38) According to(38), the parameters of the Γ -series are the following: u = ϕ + q , v = ψ + q , g = ψ + q , h = ϕ + q ,θ = ϕ + ω − ψ , θ = ϕ + ω + ψ , θ , = ψ + ψ , θ , = 0 ,ϑ = ψ + ω − ϕ , ϑ = ψ + ω + ϕ , ϑ , = ϕ + ϕ , ϑ , = 0 . Apply the first main equality to the following products from (41): a α − k − ϕ ′ F θ ( a ) = X t Y θ,α − k − ϕ ′ t ( a a , − a a , ) t F γ t ( a ) ,b β − k − ψ ′ G ϑ ( b ) = X s Y ϑ,β − k − ψ ′ s ( b b , − b b , ) s G δ s ( b ) . X d M − M k + k ,ϕ ′ + ψ ′ λ ! ϕ ! ϕ ′ ! µ ! ψ ! ψ ′ ! × ( − t + s X θ,ϑ,q i ψ i ,ϕ i ,ω i · Y θ,α − k − ϕ ′ t · Y ϑ,β − k − ψ ′ s ( α − k − ϕ ′ )!( β − k − ψ ′ )!( γ − ψ ′ )!( δ − ϕ ′ )! k ! k ! × ( u + q + k + t )! × ( v + q + γ − ψ ′ + t )!( g + q + k + s )!( h + q + δ − ψ ′ + s )! × a u + q + k + t ( u + q + k + t )! a v + q + γ − ψ ′ + t , ( v + q + γ − ψ ′ + t )! b g + q + k + s ( g + q + k + s )! × b h + q + δ − ϕ ′ + s , ( h + q + δ − ϕ ′ + s )! · F γ t ( a ) · G δ s ( b ) . (42)The summation is taken over partitions that appeared in the previous stepsand also over integer non-negative parameters t , s . The parameters of the Γ -series involved in this formulas are the following: γ t = ϕ + ω − ψ + α − k − ϕ ′ , γ t = ϕ + ω + ψ − t,γ t , = ψ + ψ − t, γ t , = 0 ,δ s = ψ + ω − ϕ + β − k − ψ ′ , δ s = ψ + ω + ϕ − s,δ s , = ϕ + ϕ − s, δ s , = 0 . Apply to (42) the operator E T , T ! , which action onto the tensor product canbe written as follows: E T , T ! = X N + M = T E N , N ! ⊗ E M , M ! . One obtains the following expression: X const · i ! j ! o ! e ! · a i i ! a j , j ! b o o ! b e , e ! · Z γ t ,u + q + k + tN ,N ,P Z δ s ,g τ + q + k + sM ,M ,P · F ǫ ( a ) · G ε ( b ) . (43)The summation is taken over the partitions and integer parameters form theprevious steps and also over new non-negative integer parameters r , l . Here15 onst is a constant from (42) and i = u + q + k + t + N + r, j = v + q + γ − ψ ′ + t − N + r,o = g + q + k + s + M + l, e = h + q + δ − ϕ ′ + s − M + l,ǫ = ϕ + ω − ψ + α − k − ϕ ′ , ǫ = ϕ + ω + ψ − t − N − r,ǫ , = ψ + ψ − t + N − r, ǫ , = 0 ,ϑ = ψ + ω − ϕ + β − k − ψ ′ , ϑ = ψ + ω + ϕ − s − M − l,ϑ , = ϕ + ϕ − s + M − l, ϑ , = 0 . Apply to (43) the operator E S , S ! , which acts onto a tensor product as follows: E S , S ! = X H + J = S E H , H ! ⊗ E J , J ! . As a result one obtains a linear combination of products a u a v , u ! v ! F ρ ( a ) · b g b h , g ! h ! G ̺ ( b ) , (44)где u = ϕ + q + q + k + t + N + r,v = ψ + q + q + γ − ψ ′ + t + N + r,N = N + N ,g = ψ + q + q + k + s + M + l,h = ϕ + q + q + δ − ϕ ′ + s + M + l,M = M + M ,ρ = ϕ + ω − ψ + α − k + ϕ ′ − H,ρ = ϕ + ω + ψ − t − N − r + H,ρ , = ψ + ψ − t + N − r ρ , = 0 ,̺ = ψ + ω − ϕ + β − k + ψ ′ − J,̺ = ψ + ω + ϕ − s − M − l + J,̺ , = ϕ + ϕ − s + M − l, ̺ , = 0 . (45)16he factor a u a v , u ! v ! F ρ ( a ) corresponds to the diagram m m m − t m − t m − t − s , t = ( ϕ + q + q + k ) + ( t + N + r ) t = ( ψ + ψ ) − ( t − N + r ) s = ( ϕ + ω + ψ ) − ( t + N + r ) + H. (46)The factor b g b h , g ! h ! G ̺ ( b ) corresponds to the diagram ¯ m ¯ m m − ¯ t ¯ m − ¯ t ¯ m − ¯ t − ¯ s , ¯ t = ( ψ + q + q + k ) + ( s + M + l )¯ t = ( ϕ + ϕ ) − ( s − M + l )¯ s = ( ψ + ω + ϕ ) − ( s + M + l ) + J. (47)The coefficient at the product (44) equals to d M − M k + k ,ϕ ′ + ψ ′ λ ! ϕ ! ϕ ′ ! µ ! ψ ! ψ ′ ! ( − t + s + r + l i ! j ! o ! e ! × X θ,ϑ,q i ψ i ,ϕ i ,ω i Y θ,α − k − ϕ ′ t Y ϑ,β − k − ψ ′ s Z γ t ,u + q + k + tN ,N ,P Z δ s ,g + q + k + sM ,M ,P ( α − k − ϕ )!( β − k − ψ )!( γ − ψ )!( δ − ϕ )! k ! k ! × ( u + q + k + t )!( v + q + γ − ψ ′ + t )! × ( g + q + k + s )!( h + q + δ − ϕ ′ + s )! . (48)The calculation of this expression should be done as follows. Initially we aregiven α, β, γ, δ, ω, ϕ, ψ and (31). From (31) we know M − K , M − K , K − S .1. Fix a partition k + k + ϕ ′ + ψ ′ = M − K and then calculate d M − M k + k ,ϕ ′ + ψ ′ .2. Using ω , λ = ϕ + ϕ ′ , µ = ψ + ψ ′ , calculate θ , ϑ and X θ,ϑ,q i ψ i ,ϕ i ,ω i according tothe Proposition 4.3. Calculate γ t , δ t , Y θ,α − k − ϕ ′ t , Y ϑ,β − k − ψ ′ s according to the Proposition 3.4. Fix a partition M − K = N + N + M + M .Using the Proposition 5,find Z γ t ,u + q + k + tN ,N ,P и Z δ s ,g + q + k + sM ,M ,P , i , j , o , e . Let us give the final answer in a simpler form. Take in (1) a representation withthe highest vector (30). Take in this representation a vector corresponding to17he diagram M M M M − T M − T M − T − S . (49)In the previous Section we fixed partitions T = k + k + ϕ ′ + ψ ′ , T = N + N + M + M , S = J + H , integers q, r, s, r, l ∈ Z ≥ , and the partitions q =: q + q + q and ϕ ′ + ϕ − q − q =: ϕ + ϕ + ϕ , ψ ′ + ψ − q − q =: ψ + ψ + ψ .Let us give another description of parameters involved in formulas (46), (47),(48). For the given diagram (49) we fix partitions T =: T ′ + T ′′ + T ′′′ + T ′′′′ , T =: T ′ + T ′′ ,T ′′ + T ′′ + ω =: L + L , S =: S ′ + S ′′ , fix arbitrary A, B ∈ Z ≥ . (50) Theorem 3.
Take a gl -highest vector in U ⊗ V of type f ( ω, ϕ, ψ, . Take avector (49) in the corresponding representation W α . Then the vector (49) is alinear combination of tensor products of vectors m m m − T ′ − A m − T ′′′ − T ′ + Am − T ′ − S ′ − L и ¯ m ¯ m m − T ′′ − B ¯ m − T ′′′′ − T ′′ + B ¯ m − T ′′ − S ′′ − L . To write a coefficient we fix partitions T ′ =: k + ϕ + q + q , T ′′ =: ϕ + ϕ ,T ′′ =: k + ψ + q + q , T ′′′′ =: ψ + ψ ,ω := L − ϕ − ψ , ω := L − ψ − ϕ A =: t + r + U, B =: s + l + V. (51)Introduce notations q := q − q − q , H := S ′ , J = S ′′ ,N = U, N = T ′ − U, M = V, M = T ′′ − V. (52)The calculations , made in the previous Section , give the following theorem. Theorem 4.
The coefficient at the tensor product form the Theorem equals tothe sum of expressions (48) over all partitions (51), (52), where symbols X, Y, Z о are defined in Propositions
4, 3, 5, and parameters in the arguments
X, Y, Z are defined by partitions (50), (51), (52) . The case of the vector f (0 , ϕ, ψ, θ ) Take a representation V with the highest vector a m − m a m , . (53)The exists an invariant scalar product on the space V ( · , · ) , such that the Gelfand-Tselin base is orthogonal with respect to this base.Using this scalar product one can identify V and V ∗ : v ↔ ( v, · ) . The actionof the algebra ontoа V ∗ is the following: g : ( v, · ) ( − g t v, · ) . (54)The obtained representation is called contragradient. Proposition 6.
The base in V ∗ , dual to the Gelfand-Tsetlin base in V, is abase proportional to the Gelfand-Tsetlin base in V ∗ .Proof. Let z + ∈ gl be an element of the algebra corresponding to a positiveroot. If v ∈ V ∗ is the highest vector then − z t + v , v ) = ( v , z + v ) , i.e. v is orthogonal to all vectors of type z + v . But span of the vectors oftype z + v is a span of all non-lowest Gelfand-Tsetlin vectors. An orthogonalcomplement to this span is generated by the lowest vector. Hence v underthe identification of V ∗ and V is mapped to the vector proportional to thelowest vector. Considering elements z + ∈ gl ⊂ gl and z + ∈ gl ⊂ gl ⊂ gl ,one obtains that a base dual to the Gelfand-Tselin base is proportional to theGelfand-Tselin base in V ∗ .Thus V ∗ is a representation with a highest vector proportional to a ( m − m )3 a m , . (55)If one uses an ordinary realization of the contragradient representation thenit is realized as a representation with the highest vector proportional to a − m , a − m , , . Multiply this representation onto a representation with the highest vector a m , , .Then one obtains a representation with the highest weight [ m , m − m , andthe highest vector a m a m − m , . (56)19onsider a mapping a ↔ a , , a ↔ a , , a ↔ − a , , (57)which a bijection between the space of the contragradient representation (i.e. arepresentation with the highest vector (55)) into the space of a representationwith highest vector (56). This mapping conjugates the actions v
7→ − E ti,j v и w ( m δ i,j + E i,j ) w. Proposition 7.
Under the mapping (57) the Gelfand-Tselin vector is mappedinto a Gelfand-Tselin vector up to a sign m m m − t m − t m − t − s ( − s + t m m − m m − m + t t t + s . (58) Proof.
To the diagram on the left side of (58), there corresponds a function a t t ! a t , t ! F ( m − m − t − s,s,t , ( a , a , a , , a , ) . Under the mapping (57) this function is transforms to a t , t ! a t t ! · F ( m − m − t − s,s,t , ( a , , − a , , − a , a ) . Using a relation between a Γ -series and a Gauss’ hypergeometric functions F , ,one obtains const · a t , t ! a t t ! a m − m − t − s ( − a ) s ( − a , ) t F , (cid:18) . . . ; ( − a )( − a , ) a a , (cid:19) = ( − s + t a t t ! a t , t ! F ( m − m − t − s,s,t , ( a , , a , , a , a )= ( − s + t a t t ! a t , t ! F ( m − m + t + s,m − m − t − s + t ,m − m − t , ( a , a , a , , a , ) . This function corresponds to a diagram on the right side in (58).The lowering operators ∇ , , ∇ , , ∇ , , acting onto a diagram on the leftare conjugated to the operators ∇ , = − E , + − E , − E , +1 E , E , , ∇ , = E , , ∇ , = E , acting on a diagram on the right. We call them the “raisingoperators”. Applying these operstors to the lowest vector one can obtain a vectorcorresponding to an arbitrary diagram.Note that under the mapping (57) (and analogous mapping for b ) one has ( aabb ) θ ( abb ) φ ( aab ) ψ ( − θ ( ab ) θ ( aab ) φ ( abb ) ψ . (59)20ccording to previous considerations we come to the following conclusion.Take a tensor product of representations with the highest weights [ m , m − m , and [ ¯ m , ¯ m − ¯ m , . Split it into a sum of irreducibles. Take a summandwith the lowest vector of type f ( θ, ϕ, ψ, . Apply the mapping (57) to all thisconstruction. We obtain a tensor product of representations with the highestweights [ m , m , and [ ¯ m , ¯ m , . This representation is splitted into a sumof irreducibles. A chosen lowest vector of type f ( θ, ϕ, ψ, is mapped to thehighest veftor of type f (0 , ψ, ϕ, θ ) , multiplied by ( − θ . Since the actions ofthe lowing and of the raising operators are conjugated we conclude that theGelfand-Tsetlin base vector in the chosen irreducible summand is mapped to aGelfand-Tselin base vector in the corresponding irreducible summand up to asign ( − θ + S + T . Consider a tensor product of representations with the highest weights [ m , m , and [ ¯ m , ¯ m , , split it into a sum of irreducibles and take a summand with thelowest vector of type . . . ( aabb ) θ , , , ( aab ) ϕ ( abb ) ψ . Take a vector in this representation corresponding to a diagram M M M + T T rrrrrrrrrr T T ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) T + S S ; ; ①①①①①①①①① (60)Fis a decomposition (50). One has a Theorem. Theorem 5.
Fix a gl -highest vector in U ⊗ V of type f (0 , ϕ, ψ, θ ) . Take avector (60) in the corresponding representation W α . It is written as a linearcombination of tensor product of diagrams m m m + T ′′′ + T ′ − A T ′ + AT ′ + S ′ + L and ¯ m ¯ m m + T ′′′′ + T ′′ + B T ′′ + BT ′′ + S ′′ + L . Theorem 6.
A copefficient at the product form the Theorem equals to thesum of expressions (48) over partitions (51) , (52), multiplied by the sign : ( − θ +( T + S )+( T ′′′ + T ′ + S ′ + L )+( T ′′′′ + T ′′ + S ′′ + L ) = ( − T ′′′ + T ′′′′ + T ′′ + T ′′ + ω + θ . Note that in the case θ = ω = 0 the highest vector is both of the first andof the second type. But theorems 3, 4 and 5, 6 give different answers sincedifferent approaches are used. Γ -series In the GKZ system (6) the second, the third, the forth equations describe thehomogeneity property of the function F γ . Using them let us prove the followingequation. Proposition 8. a , F γ = X F γ + ( a a , − a a , ) X F γ , where X , X — are some constants , γ , γ — are new parameters of a Γ -series.Proof. Using the homogeneity equation from the GKZ system one can obtain: a F γ = − a , F γ + e − e , + const · F γ + e ,a , F γ = − a , F γ + e , − e , + const · F γ + e , ⇒ a a , F γ = a , F γ + e , − e , + e − e , + const · a , F γ + e + e , − e , + const · F γ + e + e , . Also a F γ = − a F γ + e − e + const · F γ + e ⇒ a F γ = a , F γ + e − e + e − e , + const · F γ + e − e + e + const · F γ + e = a , F γ + e − e , + const · F γ + e ⇒ a a , F γ = a , F γ + e − e , + const · a , F γ + e . Let us use F γ + e , − e , + e − e , = F γ + e − e , , F γ + e + e , − e , = F γ + e . Hence, ( a a , − a a , ) F γ = const · a , F γ + e + const · F γ + e + e , . (61)22nalogously the following propositions can be proved. Proposition 9. a F γ = X F γ + ( a a , − a a , ) X F γ , where X , X — are some constants ( other then in p revious Proposition ) . Proposition 10. a F γ,B = X F γ + ( a a , − a a , ) X F γ . Proposition 11. a F γ = + X F γ + X ( a a , − a a , ) F γ . From these Propositions one gets a Lemma.
Lemma 3.
For arbitrary u, v, w, t one has a u a v a w , a t , F γ = X α X α ( a a , − a a , ) p α F γ α . (62)Since for γ = (0 , , , one has F γ ≡ , then the following statement takesplace. Lemma 4.
For arbitrary u, v, w, t one has a u a v a w , a t , = X α X α ( a a , − a a , ) p α F γ α . (63) Let us introduce a notation for the first operator from (6): O := ∂ ∂a ∂a , − ∂ ∂a ∂a , . (64) O (cid:0) ( a a , − a a , ) k F γ (cid:1) = ( k ( k +1)+ k ( γ + γ + γ , ))( a a , − a a , ) k − · F γ . (65) Proof.
One has O (cid:0) ( a a , − a a , ) k (cid:1) = k ( k + 1)( a a , − a a , ) k − . (66)Now let us find (cid:16) ∂∂a ( a a , − a a , ) k (cid:17) ∂∂a , F γ + (cid:16) ∂∂a , ( a a , − a a , ) k (cid:17) ∂∂a F γ (cid:16) ∂∂a ( a a , − a a , ) k (cid:17) ∂∂a , F γ − (cid:16) ∂∂a , ( a a , − a a , ) k (cid:17) ∂∂a F γ = k ( a a , − a a , ) k − (cid:16) a ∂∂a + a ∂∂a + a , ∂∂a , + a , ∂∂a , (cid:17) F γ = k ( a a , − a a , ) k − ( γ + γ + γ , ) F γ . Using OF γ = 0 , one gets O (cid:0) ( a a , − a a , ) k F γ (cid:1) = ( k ( k + 1) + k ( γ + γ + γ , ))( a a , − a a , ) k − F γ . We are going to use the following idea. If one knows that a function f ( z ) can bepresented as a power series then the coefficients of this series can be calculatedby the following ruler f ( z ) = c + c z + c z + c z . . . ⇒ c = f (0) , c = ddz f ( z ) | z =0 , . . . . (67)Let us use the ruler (67), where analogs of ddz and | z =0 are the following ddz O = ∂ ∂a ∂a , − ∂ ∂a ∂a , , . | z =0 . | a a , = a a , . The fact that the relation (34) takes place follows from (62). Let us prove thestatement 3, which gives a formula for the coefficients of this relation.Let us find terms with p τ = p . For this purpose let us apply to both partsof (34) the operator O p , and then let us make a substitution . | a a , = a a , . On the left one has O p ( a u F γ ) = ua u − F γ − e , = Γ( u )Γ( u − p ) a u − p F γ + pe − pe − pe , . (68)After the substitution . | a a , − a a , one obtaines Γ( u )Γ( u − p ) a u + γ a γ − p a γ , − p , F γ + pe − pe − pe , ( ) . (69) On the right after application of O and the substitution . | a a , = a a , thereremains only the summands (34) for which p τ = p . According to the Proposition12 at every summand there appeares a coefficient Π p,γ τ = p Y t =1 (cid:0) t + t ( γ τ + γ τ + γ τ , + γ τ , ) (cid:1) . p τ = p one has γ τ = γ + u, γ τ = γ − p,γ τ , = γ , − p, γ τ , = 0 ,Y γ τ = Γ( u )Γ( u − k ) · F γ ( ) F γ τ ( ) · p,γ τ . The fact that the relation (35) takes place follows from (63). Let us prove theProposition 4.Let us use the principle (67). We need to apply the oparators O pa O qb , where O a = ∂ ∂a ∂a , − ∂ ∂a ∂a , , O b = ∂ ∂b ∂b , − ∂ ∂b ∂b , . To find summands with p τ = p, q τ = q , let us apply O pa O qb and make substi-tutions . | a a , = a a , , b b , = b b , on the left and on the right.On the right in (35) we obtain summands with p τ = p, q τ = q , transformedto Π p,θ τ · Π p,ϑ τ · X τ a u a v , b g b h , · a θ a θ a θ , , F θ ( ) · b ϑ b ϑ b ϑ , , G ϑ ( ) . (70)Consider the left side. Since O pa and O qb commute we can apply them in anarbitrary order. Introduce a notation f ( λ, µ, ω ) := ( abb ) λ λ ! ( aab ) µ µ ! ( ab ) ω , ω ! . (71)This expression equals to ( a b , − a b , + a b , ) λ λ ! ( b a , − b a , + b a , ) µ µ ! ( a b − a b ) ω ω != X a ϕ + ω a ϕ + ω a ϕ · a ψ , a ψ , a ψ , · b ψ + ω b ψ + ω b ψ · b ϕ , b ϕ , b ϕ , × ( − ϕ + ψ + ω ϕ ! ϕ ! ϕ ! ψ ! ψ ! ψ ! ω ! ω ! . (72)The summation is taken over all indices ϕ i , ψ i , ω i , such that ϕ + ϕ + ϕ = λ,ψ + ψ + ψ = µ,ω + ω = ω. Using (72), we obtain that application of O b to (71) gives a a , f ( λ − , µ − , ω ) + a ( − a ) f ( λ − , µ, ω − ( − a )( − a , ) f ( λ − , µ − , ω ) − ( − a ) a f ( λ − , µ, ω − a a , − a a , ) f ( λ − , µ − , ω ) . Thus an application of O qb to (71) gives ( a a , − a a , ) q f ( λ − q, µ − q, ω ) . (73)Now apply O a to this expression. Using (66), one obtains that O a (( a a , − a a , ) q f ( λ, µ, ω )) equals q ( a a , − a a , ) q − f ( λ, µ, ω )+ ( b b , − b b , )( a a , − a a , ) q f ( λ − , µ − , ω )+ q ( a a , − a a , ) q − (cid:16) a , b f ( λ, µ − , ω )+ a b , f ( λ − , µ, ω ) + a b f ( λ, µ, ω − − ( − a , )( − b ) f ( λ, µ − , ω ) − ( − a )( − b , ) f ( λ − , µ, ω ) − ( − a )( − b ) f ( λ, µ, ω − (cid:17) = q ( a a , − a a , ) q − f ( λ, µ, ω )+ ( b b , − b b , )( a a , − a a , ) q f ( λ − , µ − , ω )+ q ( a a , − a a , ) q − (cid:16) (( aab ) − a , b ) f ( λ, µ − , ω )+ (( abb ) − ( a b , )) f ( λ − , µ, ω ) − ( ab ) f ( λ, µ, ω − (cid:17) = q ( a a , − a a , ) q − (cid:16) ( q + λ + µ + ω ) f ( λ, µ, ω ) − a , b f ( λ, µ − , ω ) − b , a f ( λ − , µ, ω ) (cid:17) + ( b b , − b b , )( a a , − a a , ) q f ( λ − , µ − , ω ) . (74)Hence in the case p < q after the substitution . | a a , = a a , , b b , = b b , oneobtains . Since O pa and O qb commute, then in the case p > q one also obtains .Now put p = q . Then instead of λ and µ take λ − q and µ − q as in (73).According to (74) one obtaines q ! X q + q + q = q ( − q + q f ( λ − q − q , µ − q − q , ω )( a b , ) q ( b a , ) q h λ,µ,ωq ,q ,q + . . . , (75)26here h λ,µ,ωq ,q ,q = X { ,...,q } = I ⊔ I ⊔ I , | I j | = q j Y j / ∈ I ⊔ I (cid:16) ( q − j )+( λ −{ the number of i s ∈ I , such that i s < j } )+( µ −{ the number of i s ∈ I , such that i s < j } )+ ω (cid:17) . (76)Here . . . in (75) is a sum of terms that vanish after the substitution . | a a , = a a , , b b , = b b , .After this substitution one gets X a ϕ + ω − ψ a ϕ + ω + ψ a ϕ + q · a ψ + q , a ψ + ψ , × b ψ + ω − ϕ b ψ + ω + ϕ b ψ + q · b ϕ + q , b ϕ + ϕ , × q ! · ( − q + q · h ϕ,ψ,ωq ,q ,q · ( − ϕ + ψ + ω ϕ ! ϕ ! ϕ ! ψ ! ψ ! ψ ! ω ! ω ! . (77)Compare (70) and (75), (77). We conclude that in (35) one has p τ = q τ = q , andfor these summands u = ϕ + q , v = ψ + q , g = ψ + q , h = ϕ + q ,θ = ϕ + ω − ψ , θ = ϕ + ω + ψ , θ , = ψ + ψ , θ , = 0 ,ϑ = ψ + ω − ϕ , ϑ = ψ + ω + ϕ , ϑ , = ϕ + ϕ , ϑ , = 0 ,X τ = ( − ϕ ψ ω q q Π q,θτ Π q,ϑτ · q ! ϕ ! ϕ ! ϕ ! ψ ! ψ ! ψ ! ω ! ω ! h Φ , Ψ ,ωq ,q ,q F θ ( ) F ϑ ( ) , where h λ,µ,ωq ,q ,q was defined in (76).The summation is taken over all indices such that q = q + q + q ,ϕ + ϕ + φ = λ − q − q,ψ + ψ + ψ = µ − q − q,ω + ω = ω. .2.5 The third main statement We need a formula for the action of E n , n ! = P n + n = n a n a n , ∂ n ∂ n a ∂ n ∂ n a , ontoa function a m − k ( m − k )! a k , k ! F γ , associated with a Gelfand-Tselin diagram. E n , n ! a m − k ( m − k )! a k , k ! F γ = X n = n + n a m − k + n ( m − k )! n ! a k − n , ( k − n )! a n , n ! F γ − n e . (78)According to (62), one has X n = n + n a m − k + n ( m − k )! n ! a k − n , ( k − n )! a n , n ! F γ − n e = X τ Z τ a i τ a j τ , ( a a , − a a , ) r τ F ε τ . (79)Thus the relation (36) takes place. Let us prove the Proposition 5, that givesformulas for coefficients in (36).Let us use the ruler (67). Apply the operator O r to the left side of (79).Using OF γ = 0 , one gets a m − k + n ( m − k )! n ! a k − n , ( k − n )! a n − r , ( n − r )! F γ − ( n + r ) e . Apply O r to the right side of (79), one gets X r τ = r Π r.ε τ · Z τ · a i τ a j τ , F ε τ + . . . , where . . . iis a sum of terms that vanish after the substitution . | a a , = a a , .For summands with r τ = r one has i τ = m − k + n , j τ = k − n ,ε τ = γ , ε τ = γ − n − r,ε τ , = γ , + n − r, ε τ , = 0 ,Z τ = 1Π r,ǫ τ F γ − n e ( ) F ε α ( ) · m − k )! n !( k − n )! n ! . References [1] van der Waerden B. L.,
Die gruppentheoretische Methode in der Quanten-mechanik , Julius Springer, Berlin, 1932.[2] Racah G.,
Theory of complex spectra . II, Phys. Rev. (1942), 438–462.[3] Greiner W., Muller B., Quantum mechanics. Symmetries , 2nd ed., SpringerVerlag, Berlin, 1994. 284] Biedenharn L. C., Baid G. E.,
On the representations of semisimple LieGroups . II, J. Math. Phys. (1963), 1449–1466.[5] Louck J. D., Biedenharn L. C., A pattern calculus for tensor operators inthe unitary groups , Comm. Math. Phys. (1968), 89–131.[6] Louck J. D., Biedenharn L. C., Canonical adjoit tensor operators in U ( n ) ,J. Math. Phys. (1970), 2368–2411.[7] Chacon E., Ciftan M., Biedenharn L. C., On the evaluation of thmultiplicity-free Wigner coefficients of U ( n ) , J. Math. Phys. (1972),577–589.[8] Louck J. D., Biedenharn L. C., On the structure of the canonical tensoroperators in the unitary groups . II.
An extension of the pattern calculusrules and the canonical splitting in U (3) , J. Math. Phys. (1972), 1957–1984.[9] Moshinsky M., Wigner coefficents for the SU (3) group and some applica-tions , Rev. Modern Phys. (1962), 813–828.[10] Chew C. K., von Baeyer H. C., Explicit computation of the SU (3) Clebsch–Gordan coefficients , Nuovo Cimento A (1968), 53.[11] Hecht K. T., Suzuki Y., Some special SU (3) ⊃ R (3) Wigner coefficientsand their application , J. Math. Phys. (1983), no. 4, 785–792.[12] Klink W. H., SU (3) Clebsch–Gordan coefficients with definite permutationsymmetry , Ann. Phys. (1992), no.1, 54–73.[13] Biedenharn L. C., Baid G. E.,
On the representations of the semisimpleLie groups. V . Some explicit Wigner operators for SU (3) , J. Math. Phys. Explicit canonical tensor operators and orthonormal couplingcoefficients of SU (3) , J. Math. Phys. (1992), no. 6, 1983–2004.[15] Louck J. D., Biedenharn L. C., Special functions associated with SU (3) Wigner–Clebsch–Gordan coefficients
A calculus for SU (3) leading to analgebraic formula for the Clebsch–Gordan coefficients , J. Math. Phys. (1996), no. 12, 6530–6569. 2917] Grigorescu M. SU (3) Clebsch–Gordan coefficients , arXiv preprintmath-ph/0007033[18] Williams H. T., Wynne C. J.,
A new algorithm for computation of SU (3) Clebsch–Gordan coefficients , Comput. Phys. (1994), 355–359.[19] Rowe D. J., Repka J., An algebraic algorithm for calculating Clebsch–Gordan coefficients ; application to SU (2) and SU (3) , J. Math. Phys. (1997), no. 8, 4363–4388.[20] Alex A., Kalus M., Huckleberry A., Delft J., A numerical algorithm forthe explicit calculation of SU ( n ) and SL ( N, C ) Clebsh–Gordan coefficients ,arxiv-math-ph-1009.04437v1.[21] htttp://homepage.physik.uni-muenchen.de/ von-delft/Papers/ClebshGordan.[22] Baird G. E., Biedenharn L. C.,