Central Values for Clebsch-Gordan coefficients
CCentral Values for Clebsch-Gordan coefficients
Robert W. Donley, Jr.
Abstract
We develop further properties of the matrices M ( m , n , k ) defined by theauthor and W. G. Kim in a previous work. In particular, we continue an alternativeapproach to the theory of Clebsch-Gordan coefficients in terms of combinatorics andconvex geometry. New features include a censorship rule for zeros, a sequence of 36-pointed stars of zeros, and another proof of Dixon’s Identity. As a major application,we reinterpret the work of Raynal et al. on vanishing Clebsch-Gordan coefficientsas a “middle-out" approach to computing M ( m , n , k ) . In the representation theory of SU ( ) , the Clebsch-Gordan decomposition for tensorproducts of irreducible representations yields a uniform pattern for highest weights,generally, as an arithmetic progression of integers with difference two, symmetricabout zero. Curiously, at the vector level, if one tensors two vectors of weight zero,a similar arithmetic progression of weights occurs, but now with difference four. Inthe theory of spherical varieties, extensions of this problem consider minimal gaplengths in the weight monoid associated to spherical vector products. An elementarycalculation for the case of SU ( ) initiates the present work, given in Proposition 15,and we review the open problem of vanishing beyond the weight zero case.Continuing the work began in [3], this approach to Clebsch-Gordan coefficientssubstitutes the use of hypergeometric series [12] and the like with elementary com-binatorial methods (generating functions, recurrences, finite symmetry groups, Pas-cal’s triangle). Of course, hypergeometric series are fundamental to the theory; a longrange goal would be to return this alternative approach, once sufficiently developed,to the hypergeometric context with general parameters. At the practical level, certaincomputer simulations in nuclear physics and chemistry may require excessively large Robert W. Donley, Jr.,Queensborough Community College, Bayside, New York, e-mail:
[email protected] a r X i v : . [ m a t h . R T ] N ov Robert W. Donley, Jr. numbers of Clebsch-Gordan coefficients, possibly with large parameters. Methodsfor an integral theory have immediate scientific applications.As with [3], this work contains many direct elementary proofs of known, butperhaps lesser known at large, results and gives them a new spatial context. Onecentral item of concern, the domain space, is a set of integer points inside of afive-dimensional cone, equipped with an order 72 automorphism group, which inturn consists of the familiar symmetries for the determinant.Key observations in this work depend on the fixed points of the symmetry groupin the cone, a distinguished subgroup of dihedral type D , polygonal subsets ofthe domain invariant under the subgroup action, and the simultaneous use of re-currences with the subgroup’s center. One byproduct of the theory is yet anotherproof of Dixon’s Identity and some variants, and, with this identity, our computa-tional viewpoint shifts from the specific “outside-in" algorithm of [3] to a universal“middle-out" approach, adapted from the work in [9].Section 2 recalls the algorithm of [3] for M ( m , n , k ) , and section 3 develops basicproperties of M ( m , n , k ) . Sections 4 and 5 review the theory of the so-called “trivial"zeros, and sections 6–9 reconsider the results of [9] as computations near the centerof M ( m , n , k ) . As a function in five variables, c m , n , k ( i , j ) is defined on the integer points of the cone0 ≤ i ≤ m , ≤ j ≤ n , ≤ k ≤ min ( m , n ) , ≤ i + j − k ≤ m + n − k . It is convention to extend this domain by zero; here it is enough to do so at least forthe matrices M ( m , n , k ) defined below and for clarity we often omit the corner zeros.In general, the various Clebsch-Gordan coefficients C m , n , k ( i , j ) and c m , n , k ( i , j ) differby a nonzero factor, and our approach to vanishing of Clebsch-Gordan coefficientsis through vanishing of the sum c m , n , k ( i , j ) . From [3], all c m , n , k ( i , j ) in (3) below may be computed algorithmically as amatrix M ( m , n , k ) . With m , n , k fixed, Proposition 1 and Theorem 1 below produce M ( m , n , k ) , with columns corresponding to coordinate vectors for weight vectorsin the subrepresentation V ( m + n − k ) of V ( m ) ⊗ V ( n ) . Consideration of Pascal’sidentity gives an explicit formula for c m , n , k ( i , j ) in Proposition 2.For non-negative integers a , b , c , multinomial coefficients are defined by (cid:18) a + ba (cid:19) = ( a + b ) ! a ! b ! and (cid:18) a + b + ca , b , c (cid:19) = ( a + b + c ) ! a ! b ! c ! . (1)First the highest weight vectors for V ( m + n − k ) are defined by Proposition 1 (Leftmost column for M ( m , n , k ) ) With ≤ i ≤ k , define the ( i + , ) -th entry of M ( m , n , k ) by entral Values for Clebsch-Gordan coefficients 3 c m , n , k ( i , k − i ) = (− ) i (cid:18) m − ik − i (cid:19) (cid:18) n − k + ii (cid:19) . (2)Repeated application f to these highest weight vectors produces coordinates forgeneral weight vectors in the corresponding V ( m + n − k ) , recorded as Proposition 2 (Entry at coordinate ( i + , i + j − k + ) in M ( m , n , k ) ) c m , n , k ( i , j ) = k (cid:213) l = (− ) l (cid:18) i + j − ki − l (cid:19) (cid:18) m − lk − l (cid:19) (cid:18) n − k + ll (cid:19) . (3)We also note the following alternative expression from [3]: c m , n , k ( i , j ) = k (cid:213) l = (− ) l (cid:18) i + j − ki − l (cid:19) (cid:18) m − ik − l (cid:19) (cid:18) n − jl (cid:19) . (4)With this expression, results in later sections may be interpreted by way of Dixon’sIdentity and its extensions for binomial sums. Theorem 1 (Definition of Matrix M ( m , n , k ) ) To calculate the coordinate vectormatrix M ( m , n , k ) :1. initialize a matrix with m + rows and m + n − k + columns,2. set up coordinates for the highest weight vector in the leftmost column usingProposition 1, and extend the top row value,3. apply Pascal’s recurrence rightwards in an uppercase L pattern, extending byzero where necessary,4. for the zero entries in lower-left corner, corresponding entries in the upper-rightcorner are set to zero, and5. the ( i + , i + j − k + ) -th entry is c m , n , k ( i , j ) . In this work, we often remove corner zeros for visual clarity. Many examples of M ( m , n , k ) will be given throughout this work. A useful parametrization for the domain of c m , n , k ( i , j ) is given by the set of Reggesymbols (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − k m − k ki j m + n − i − j − km − i n − j i + j − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5)Note that each row or column sums to J = m + n − k . That is, the domain space isin one-one correspondence with all 3-by-3 matrices with nonnegative integer entrieshaving this magic square property. Here we follow [12], while [9] switches rows 2and 3.
Robert W. Donley, Jr.
In turn, one may define the Regge group of symmetries; each of the 72 deter-minant symmetries, generated by row exchange, column exchange, and transpose,correspond to a transformation of c m , n , k ( i , j ) . Of particular interest here is the dihedral subgroup D of twelve elements gen-erated by column switches and the interchange of rows 2 and 3. Some elements,including a generating set, are given as follows: Proposition 3 (R23 Symmetry - Weyl Group)
With the asterisks defined by (1), (cid:18) m + n − ki , j , ∗ (cid:19) c m , n , k ( m − i , n − j ) = (− ) k (cid:18) m + n − km − i , n − j , ∗ (cid:19) c m , n , k ( i , j ) . (6) Proposition 4 (C12 Symmetry)
When defined, c n , m , k ( j , i ) = (− ) k c m , n , k ( i , j ) . (7) Proposition 5 (C13 Symmetry)
With i (cid:48) = i + j − k and m (cid:48) = m + n − k , c m (cid:48) , n , n − k ( m (cid:48) − i (cid:48) , j ) = (− ) n − j c m , n , k ( i , j ) . (8) Proposition 6 (C123 Symmetry)
With i (cid:48) = i + j − k and m (cid:48) = m + n − k , c m (cid:48) , m , m − k ( m (cid:48) − i (cid:48) , i ) = (− ) m − k + i c m , n , k ( i , j ) . (9)Since the Weyl group symmetry preserves m , n , and k , it transforms M ( m , n , k ) toitself. In fact, the net effect of this symmetry is to rotate M ( m , n , k ) by 180 degrees,change signs according to the parity of k , and rescale values by a positive scalar,rational in the five parameters. In particular, the zero locus of c m , n , k ( i , j ) in M ( m , n , k ) is preserved under this symmetry.In the complement of the corner triangles of zeros, the upper, left-most, andlower-left edges in M ( m , n , k ) have non-vanishing entries by construction. By theWeyl group symmetry, the remaining outer edges also have this property. Thus theseedges trace out a polygon, possibly degenerating to a segment, with no zeros on itsouter edges. Definition 1
We refer to the complement of the corner triangles of zeros in M ( m , n , k ) as the polygon of M ( m , n , k ) . A zero in the interior of the polygon of M ( m , n , k ) iscalled a proper zero . Other terminology for zeros will be noted below.Opposing edges of this polygon have the same length. The horizontal edges havelength n − k +
1, the slanted edges have length m − k +
1, and the vertical edges havelength k +
1. When the polygon is a hexagon, one notes that the absolute values ofthe entries along the top, lower-left, and right-sided edges are constant and equal to,respectively, (cid:18) mk (cid:19) , (cid:18) nk (cid:19) , and (cid:18) m + n − km − k (cid:19) . (10)These binomial coefficients are composed of parts m − k , n − k , and k . Thesevalues apply to the degenerate cases (parallelogram or line segment) accordingly. entral Values for Clebsch-Gordan coefficients 5
The remaining edges link these values through products of binomial coefficients;the indices in the starting coefficient decrease by 1 as the indices in the terminalcoefficient increase likewise. See Proposition 1 for the leftmost vertical edge.We note the decomposition D (cid:27) S × C , (11)where S represents the permutation subgroup generated by column switches and theWeyl group symmetry generates the two-element group C . While column switchesdo not preserve M ( m , n , k ) in general, the polygon of M ( m , n , k ) maps to the polygonof values in another M ( m , n , k ) , differing only by sign changes. Thus there is a well-defined correspondence between the proper zeros of M ( m , n , k ) and M ( m , n , k ) under a column switch.Of the column switches listed, the C12 symmetry changes sign according to k andinverts proper values in each column, and the C13 symmetry changes sign accordingto n − j and reflects values across the northeasterly diagonal. The column switchC123 rotates values by 120 degrees counter-clockwise and changes sign accordingto m − k + i . For example, the polygons of M ( , , ) , M ( , , ) , M ( , , ) below permuteamong themselves under the D symmetries: − − − −
36 6 3 , − − − − − − , − − − −
33 3 . Next, we note two additional recurrences from [3]; these impose further restric-tions on proper zeros within M ( m , n , k ) . Proposition 7 (Pascal’s Recurrence)
When all terms are defined, c m , n , k ( i , j ) = c m , n , k ( i , j − ) + c m , n , k ( i − , j ) . Proposition 8 (Reverse Recurrence)
When all terms are defined, a c m , n , k ( i , j ) = a c m , n , k ( i + , j ) + a c m , n , k ( i , j + ) where1. a = ( i + j − k + )( m + n − i − j − k ) ,2. a = ( i + )( m − i ) , and3. a = ( j + )( n − j ) . The reverse recurrence is Pascal’s recurrence after application of the Weyl groupsymmetry. It follows a rotated capital-L pattern, only now weighted by positiveintegers when 0 < i < m and 0 < j < n .These recurrences immediately yield Robert W. Donley, Jr.
Proposition 9 (Censorship Rule)
Suppose c m , n , k ( i , j ) = properly in M ( m , n , k ) .Then adjacent zeros in M ( m , n , k ) may occur only at the upper-right or lower-leftentries. That is, in the following submatrix of M ( m , n , k ) , the bullets must be nonzero: • • ∗• •∗ • • . Proof
First note that, in either recurrence relation, if any two terms in the relationare zero, then so is the third. Now suppose a pair of proper zeros are horizontallyadjacent. Then, alternating the two relations, one begins a sequence ∗ ∗ ∗∗ ∗ ∗∗ → ∗ ∗ ∗∗ ∗∗ → ∗ ∗ ∗ ∗∗ → ∗ ∗ ∗∗ → · · · . Eventually this serpentine must place a zero at a nonzero entry on the top rowof M ( m , n , k ) , a contradiction. The case of two vertically adjacent zeros followssimilarly. Finally, for the diagonal case (cid:20) ∗∗ (cid:21) , either recurrence rule begins the serpentine. (cid:3) As an adjunct to censorship, certain pairs of zeros severely restrict nearby values.
Proposition 10
With X nonzero, the following allowable pairs of zeros fix adjacentvalues: X X − X − X , X − X − X − X , X X − X X . In particular, each triangle implies one of the following three equalities: ( i + )( m − i ) = ( i (cid:48) + )( m (cid:48) − i (cid:48) ) = ( j + )( n − j ) , where i (cid:48) = i + j − k , m (cid:48) = m + n − k , and c m , n , k ( i , j ) corresponds to the middleentry of the leftmost column.Proof The restriction on values follows immediately from Pascal’s recurrence. Theparameter conditions follow from the second recurrence. (cid:3)
Finally, it will be convenient to note four degenerate cases of M ( m , n , k ) ; the firstthree cases give all conditions for when the polygon is not a hexagon.1. When k = n > , the parallelogram of nonzero entries of M ( m , n , ) consistsof entries in Pascal’s triangle, with columns corresponding to segments of thetriangle’s rows. When n = k = , M ( m , , ) is an identity matrix of size m + , entral Values for Clebsch-Gordan coefficients 7
2. When k = m with n ≥ m , M ( m , n , m ) contains no zeros; ignoring signs, theupper-right corner corresponds to the peak of Pascal’s triangle, with diagonalscorresponding to segments of the triangle’s rows,3. When m = n = k , M ( m , m , m ) degenerates to a vertical segment of length m + m ≥ n = k =
1, a central vertical triplet of zeros occurs, butonly the central zero is proper.In fact, with n ≥ m , the C23 symmetry carries M ( m , n − m , ) to M ( m , n , m ) Cases 1–4 are represented below by M ( , , ) , M ( , , ) , M ( , , ) , and M ( , , ) , respectively: , − − −
16 3 1 , − , − − − − − . Consideration of dihedral reflections leads one to a large family of proper zerosthrough Propositions 4, 5, and the C23 symmetry. In particular, these zeros occurwhen a column switch in the Regge symbol fixes an entry of M ( m , n , k ) and changesparity. Zeros of this type always occur as diagonal subsets in M ( m , n , k ) .To see this, first observe that when n = k , M ( m , k , k ) is a square matrix of size m +
1, and the C13 symmetry preserves the northeasterly diagonal. In this case, thesubgroup of the Regge group generated by C13 and R23, of type C × C , preservesboth the polygon and the diagonal. Specifically, the top row of the general Reggesymbol for these parameters (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k m − k ki j m − i − j − km − i k − j i + j − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (12)is unchanged by the R23 and C13 symmetries.With m (cid:44) k , four edges of the polygon now have length k + , upon which thesubgroup acts transitively. When m = k , the polygon is a regular hexagon preservedby the dihedral subgroup D of Section 3. Proposition 11
Suppose k > and m + k is odd. Then c m , k , k ( i , m + k − i ) = . (13) Proof
Note that coordinates in M ( m , n , k ) are indexed by x = i + j − k + y = i + , and the indices for the diagonal in question are solutions to Robert W. Donley, Jr. ( i + j − k + ) + ( i + ) = m + j = m + k − i . (14)Substituting n = k into Proposition 5, one obtains c m , k , k ( m + k − i − j , j ) = (− ) j c m , k , k ( i , j ) . (15)For entries on the diagonal, this equation reduces to c m , k , k ( i , j ) = (− ) m + k c m , k , k ( i , j ) , (16)and c m , k , k ( i , j ) vanishes under the given parity condition. (cid:3) For example, we have M ( , , ) and M ( , , ) , respectively: − − − − − −
20 0 8 8 4 − − − − ,
10 10 10 − − − − − − − −
106 12 10 . Next we give a formula for the entries below the diagonal as single term expres-sions. There are k proper zeros on the diagonal, and we denote the position of sucha zero as measured from lower-left to upper-right. Proposition 12
The value of the entry directly below the t -th proper zero on thediagonal is given as a single term expression by (− ) k + t + (cid:18) kk (cid:19) (cid:18) k − t − (cid:19) ( k − t + ) ! ( k − ) ! ( m + k + ) ! ( m − k − + t ) ! ( m − k − ) ! ( m + k + − t ) ! . (17) Proof
The coordinates for the s -th proper zero on the diagonal are ( i + , i + j − k + ) = (cid:18) m + k − s + , m − k + s + (cid:19) (18)with i = m + k − s + , j = s − . (19)The entry directly below the first zero has value c m , k , k (cid:18) m + k + , (cid:19) = (− ) k (cid:18) kk (cid:19) . The second recurrence allows us to compute values below the diagonal by a sequenceof factors: at the s -th proper zero, (cid:20) x y y (cid:21) −→ x = − a a y = − ( m + k − s + )( m − k + s − ) s ( k − s + ) y . entral Values for Clebsch-Gordan coefficients 9 Thus the value of the entry below the t -th zero on the diagonal is (− ) k + t − (cid:18) kk (cid:19) t − (cid:214) s = ( m + k − s + )( m − k + s − ) s ( k − s + ) . (20)Since t − (cid:214) s = ( N − s ) = t − ( N − ) ! ( N − t ) ! , t − (cid:214) s = ( N + s ) = t − ( N + t − ) ! N ! , (21)and t − (cid:214) s = ( N − s + ) = ( N − ) ! ( N − t ) !2 t − ( N − t + ) ! ( N − ) ! , (22)the result follows by substitution into (20). (cid:3) In particular, when m even, k odd and t = k + , the entry below the central zero is (− ) k + ( k + ) m ( m + ) (cid:32) m + k + k + , k + , m − k − (cid:33) . (23)When m is odd, k even and t = k + , there is a central square (cid:20) X X (cid:21) (24)with X = (− ) k k + m + (cid:32) m + k + k , k + , m − k − (cid:33) . (25)Two other types of zero diagonals may be obtained by applying column switchesto (13). Applying the C12 symmetry yields Proposition 13
Suppose k > and n + k is odd. Then c k , n , k ( n + k − j , j ) = . (26)This diagonal of zeros connects the midpoints of the horizontal edges of the polygon.By the censorship rule, no zeros on this diagonal are adjacent, and the diagonal isfixed by the C23 symmetry. Applying the C13 symmetry to the parameters ofProposition 13 gives Proposition 14
Suppose k > is odd. Then c m , m , k ( i , i ) = . (27) This diagonal of zeros connects the midpoints of the vertical edges of the polygon.Again, no zeros on this diagonal are adjacent, and this diagonal is fixed by the C12symmetry.From above, M ( , , ) transforms into M ( , , ) and M ( , , ) , respectively:
20 20 −
20 0 2012 − − − − − − − − , − − − − − − − − − − . Next suppose m = n = k with k odd; see Figure 1 below. The polygon in M ( k , k , k ) is now a regular hexagon with sides of length k + , and entries onthree sides have constant absolute value (cid:18) kk (cid:19) . The full D subgroup preserves M ( k , k , k ) , as the general entry has Regge symbol (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k k ki j k − i − j k − i k − j i + j − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (28)In particular, when i = j = k , the central entry is a fixed point under the full Reggegroup. Since k is odd, the polygon now possesses three diagonals of zeros (six-pointed star) with several interesting consequences; first, when k ≥ X X − X X − X − X X X − X − X − X X X − X − X − X X X . By (23), the nonzero entry X is equal to c k , k , k ( k + , k − ) = (− ) k + k + k (cid:32) k + k + , k + , k + (cid:33) . Furthermore, the large triangle is an aggregation of the three smaller triangular zeropair patterns in Proposition 10.Next, as each non-central zero on a diagonal is fixed by an order two subgroupin the Regge group, each orbit under the full Regge group contains 36 zeros. Sinceeach Regge symmetry induces a linear change in indices, we have entral Values for Clebsch-Gordan coefficients 11
Theorem 2
For each odd k > , c k , k , k ( k , k ) is a fixed point of the full Reggegroup and is at the center of both M ( k , k , k ) and a 36-pointed star of zeros in thefive-dimensional Clebsch-Gordan domain space.
252 252 252 252 252 252 -756 -504 -252 0 252 504 756 1176 420 -84 -336 -336 -84 420 1176 -1176 0 420 336 0 -336 -420 0 1176 756 -420 -420 0 336 336 0 -420 -420 756 -252 504 84 -336 -336 0 336 336 -84 -504 252 -252 252 336 0 -336 -336 0 336 252 -252 -252 0 336 336 0 -336 -336 0 252 -252 -252 84 420 420 84 -252 -252 -252 -504 -420 0 420 504 252 -252 -756 -1176 -1176 -756 -252
Fig. 1
M(10, 10, 5) with m = n = k and k = Following [9], we now focus on behavior of c m , n , k ( i , j ) near the center of M ( m , n , k ) . The shape of the center is determined by parities of m , n , and k ; these are determinedby the top line of the Regge symbol, and, through the D symmetries, we can narrowour results to four cases: Table 1
Central area of M ( m , n , k ) based on parity m − k n − k k CenterEven even even single entry ( (cid:44) = Suppose both m and n are even. Recall that M ( m , n , k ) is a matrix of size m + m + n − k + . Thus there is a central entry, c m , n , k ( m , n ) , which we refer to asthe central value of M ( m , n , k ) , and, with its adjacent entries, we obtain a squaresubmatrix of size 3, which we refer to as the central square of M ( m , n , k ) . As seenin the fourth degenerate case, a central square for k > n = k = , and m ≥ Proposition 15
Suppose m and n are even. The central value c m , n , k ( m , n ) = if andonly if k is odd.Proof If k is odd then the Weyl group symmetry implies c m , n , k ( m , n ) = − c m , n , k ( m , n ) , (29)and the central value vanishes.In the other direction, suppose the central value vanishes. Consider the centralsquare − Y ∗ ∗ Y ∗∗ X X with X and Y nonzero. From the lower left hook and the second recurrence rule with i = m and j = n −
1, we have a Y = a X , and positivity of a i implies that X and Y have the same parity. Since − Y and X haveopposite parity, the Weyl group symmetry switches signs and k is odd. (cid:3) Remark 1
This condition is equivalent to the Regge symbol having matching bottomrows with J = m + n − k odd. This result also corresponds to the linearization formulafor products of Legendre polynomials, as seen, for instance, in Corollary 6.8.3 in[1], and it may be shown directly using the Weyl group and the Casimir operator. Inthe physics literature, zeros of this type, or their translates under the Regge group,are referred to as “trivial" zeros; all diagonal zeros are of this type. Trivial zerosalso correspond to indices outside the polygons and those omitted under raising orlowering operators.In turn, the proposition may be interpreted as a “gap 4" result when tensoringvectors of weight zero, in contrast with the usual Clebsch-Gordan decomposition,which corresponds to “gap 2." Since a contribution only occurs for k even, f m / φ m ⊗ f n / φ n = min ( m , n )/ (cid:213) k (cid:48) = C m , n , k (cid:48) ( m / , n / ) f m / + n / − k (cid:48) φ m , n , k (cid:48) . (30)Note that the vectors in the sum have nonzero coefficients and correspond to irre-ducible constituents V ( m + n − k (cid:48) ) for 0 ≤ k (cid:48) ≤ min ( m , n ) . A first step to computing near the center of M ( m , n , k ) requires knowing eitherthe central value or a near central value. To compute these values inductively, onefirst notes some values of c m , n , k ( i , j ) for small k and a four-term recurrence relation.One obtains the following directly from either summation formula: c m , n , ( i , j ) = (cid:18) i + ji (cid:19) , c m , n , ( i , j ) = (cid:18) i + ji (cid:19) m j − nii + j , (31) entral Values for Clebsch-Gordan coefficients 13 c m , n , ( i , j ) = (cid:18) i + ji (cid:19) jm ( j − )( m − ) − i j ( m − )( n − ) + in ( i − )( n − )( i + j )( i + j − ) (32)Zeros corresponding to k = , Lemma 1
When all terms are defined, c m + , n + , k + ( i + , j + ) + c m , n , k ( i , j ) = c m , n + , k + ( i , j + ) + c m + , n , k + ( i + , j ) . Proof
Using formulas (7.2) through (7.5) in [3], we have c m + , n + , k + ( i + , j + ) = c m + , n + , k + ( i , j + ) + c m + , n + , k + ( i + , j ) = c m , n + , k + ( i , j + ) + c m , n + , k + ( i , j ) + c m + , n , k + ( i + , j ) − c m + , n , k + ( i , j ) = c m , n + , k + ( i , j + ) + c m + , n , k + ( i + , j ) − c m , n , k ( i , j ) . See [10] for a normalized version of the lemma, along with normalized versionsof [3], (7.2), (7.3). As a special case of the lemma, we obtain another proof of Dixon’sIdentity, first proven in [5]. Many alternative proofs exist; see, for instance, [6], [4],and [13].
Theorem 3 (Dixon’s Identity)
When m , n , and k are even, the central value c m , n , k ( m , n ) = k (cid:213) l = (− ) l (cid:18) m + n − k m − l (cid:19) (cid:18) m k − l (cid:19) (cid:18) n l (cid:19) = (− ) k (cid:18) m + n − k m − k , n − k , k (cid:19) . Proof
We prove the theorem by induction on N = m + n + k . For the base case, if k =
0, the result follows by (31). Now suppose the theorem holds for N ≤ m + n + k + c m + , n + , k + (cid:18) m + , n + (cid:19) = c m , n + , k + (cid:18) m , n + (cid:19) + c m + , n , k + (cid:18) m + , n (cid:19) − c m , n , k (cid:18) m , n (cid:19) = (− ) k + (cid:18) (cid:32) m + n − k m − k − , n − k , k + (cid:33) + (cid:32) m + n − k m − k , n − k − , k + (cid:33) + (cid:32) m + n − k m − k , n − k , k (cid:33) (cid:19) = (− ) k + m + n − k + k + (cid:32) m + n − k m − k , n − k , k (cid:33) = (− ) k + (cid:32) m + n − k + m − k , n − k , k + (cid:33) . Thus the induction step holds and the theorem is proved. (cid:3)
With similar proofs, the remaining three cases follow:
Proposition 16
With m and n even and k odd, the value below the central zero isgiven by c m , n , k ( m + , n − ) = (− ) k + ( k + )( m − k + ) m ( m + ) (cid:18) m + n − k + m − k + , n − k − , k + (cid:19) . Proposition 17
With m odd and n and k even, the lower-right central value is givenby c m , n , k ( m + , n ) = (− ) k m − k + m + (cid:32) m + n − k + m − k + , n − k , k (cid:33) . Proposition 18
With m and k odd and n even, the lower-right central value is givenby c m , n , k ( m + , n ) = (− ) k + k + m + (cid:32) m + n − k m − k , n − k − , k + (cid:33) . m , n , and k even We now turn our attention to the first of four cases of “non-trivial" zeros. Non-trivialzeros are also called “polynomial" or “structural" in the literature.For fixed m , n , and k , the use of exponents i and j allow for indexing of M ( m , n , k ) more or less according to usual matrix notation. When m and n are both even, thepolygon rotates or reflects about the central value under the D symmetries; in theother two cases, the central area may change shape. Positioning the central value asthe origin, we develop two infinite lattices of rational functions of m , n , and k , onefor each case in the table when m and n are even. entral Values for Clebsch-Gordan coefficients 15 To compute a given M ( m , n , k ) , one extracts the appropriate polygon from thelattice, evaluates the corresponding rational function at m , n , and k , and rescalesby the central or near-central value from Section 5. An algorithm to construct thislattice follows:1. obtain a recursive formula to compute down two columns from the center,2. use Pascal’s recurrence to compute down-and-to-the-right from the center, and3. apply the D symmetries to extend to the entire lattice.First we consider the case with k even. The central value X is given by Theorem3. To begin, we have Proposition 19
Suppose m , n , and k are even, and M ( m , n , k ) has central square Z X ∗ ∗
Y X X B X ∗ A X B X for nonzero X . Then B = λ m (cid:48) − λ m + λ n λ n , A = λ m (cid:48) − λ m − λ n λ m , B = λ m (cid:48) + λ m − λ n λ m , where m (cid:48) = m + n − k and λ s = s ( s + ) . Proof
Since k is even, X is nonzero. The following four equations follow fromPropositions 7 and 8 and the Weyl group symmetry (6):1. Z + Y = ,
2. 1 + A = B , λ m (cid:48) Y = λ m A + λ n , and4. λ m (cid:48) Z = λ m B . These equations reduce to a linear system in A and B with the above solutions.The reverse recurrence yields the formula for B . (cid:3) Consider the following fourth-quadrant submatrix with X in the central position: XA X B XA X B X C XA X B X C X D X . (33)Alternating between the two main recurrences as in Proposition 19 immediatelyyields Theorem 4
Let X be the central value determined by Theorem 3, and define λ s = s ( s + ) . The first two columns of matrix (33) are computed recursively by A = λ m (cid:48) − λ m − λ n λ m , B = λ m (cid:48) + λ m − λ n λ m , (34) A s + = ( λ m (cid:48) − λ m + λ s ) A s + ( λ s − − λ n ) B s λ m − λ s , (35) B s + = λ m (cid:48) A s + ( λ s − − λ n ) B s λ m − λ s . (36)In the triangle from the first column to the diagonal, unreduced denominators areequal along rows and increase by a factor of λ m − λ s as we pass from the s -th to the ( s + ) -st row. That is, the denominator for index s + d s + = s (cid:214) l = ( λ m − λ l ) = s + ( m + s + ) ! ( m − s − ) ! . (37)This implies immediately Corollary 1
For s ≥ , let N ( A s ) and N ( B s ) be the numerators in the unreducedexpressions of A s and B s , respectively. Then N ( A s ) and N ( B s ) are computed recur-sively by N ( A ) = λ m (cid:48) − λ m − λ n , N ( B ) = λ m (cid:48) + λ m − λ n , (38) (cid:20) N ( A s + ) N ( B s + ) (cid:21) = (cid:20) λ m (cid:48) − λ m + λ s λ s − − λ n λ m (cid:48) λ s − − λ n (cid:21) (cid:20) N ( A s ) N ( B s ) (cid:21) . (39)To proceed towards the diagonal, for instance, we have for s ≥ , C s + = B s + B s + , N ( C s + ) = ( λ m − λ s ) N ( B s ) + N ( B s + ) . (40)The denominator is non-vanishing for s < m , and vanishing in a coordinaterelative to the the central value reduces to solving the corresponding Diophantineequation, say N ( A s ) = , in m , n , and k . In [9], three families of zeros corresponding to orders 1, 2, and 3, with 6, 12, and17 subfamilies, respectively, are classified. Here order is a measure of “distance"using 3-term hypergeometric contiguity relations. It may be computed from theRegge symbol directly (section 4 of [9]). Order 1 subfamilies are indexed I through V I , and, in particular, these zeros admit a full parameterization, as do subfamilies 2.7and 2.8. Each subfamily of order 2 zeros contains infinitely many zeros. Cardinalityin order 3 is an open question, with infinitely many zeros known in types 3.1 and3.2. We further note that the conjecture by Brudno in footnote 7 of [7] is case I of[9].For m , n , k even, these subfamilies correspond to positions around the centralvalue as follows: . . . . . . . . . . . I I . . . . I • I . . . . I I . . . . . . . . . . . entral Values for Clebsch-Gordan coefficients 17 with Diophantine equations in the subcentral triangle given by• I : N ( A ) = λ m (cid:48) − λ m − λ n = , • 2.1: N ( A ) = ( λ m (cid:48) − λ m + )( λ m (cid:48) − λ m − λ n ) − λ n ( λ m (cid:48) + λ m − λ n ) = , • 2.2: N ( B ) = λ m (cid:48) ( λ m (cid:48) − λ m − λ n ) − λ n ( λ m (cid:48) + λ m − λ n ) = , • 3.1: N ( A ) = ( λ m (cid:48) − λ m + ) N ( A ) − ( λ n − ) N ( B ) = , • 3.2: N ( B ) = λ m (cid:48) N ( A ) − ( λ n − ) N ( B ) = . Under the D symmetries, the numerators change by permuting m (cid:48) , m , and n andrescaling as in Propositions 3–6.Now suppose m = n = k with k even. The central value reduces to the originalDixon Identity Corollary 2 (Dixon [2])
When m = n = k and k even, the central value c k , k , k ( k , k ) = k (cid:213) l = (− ) l (cid:18) kl (cid:19) = (− ) k (cid:32) k k , k , k (cid:33) . (41)Although we no longer have the diagonals of zeros from the odd k case in section4, there is a central equilateral triangle, with sides of length 4, given by − XX / − X / X / X X / − X − X / X / X . We note two conjectures, which hold experimentally for k < k is odd, zeros only occur on one of the three diagonals in the polygon of M ( k , k , k ) , and2. when k is even, no zeros occur in the polygon of M ( k , k , k ) unless k = , inwhich case there are six doublets forming a hexagon (Figure 2). m and n even, k odd Assume m and n even, and k odd. This section proceeds in a manner similar, butsomewhat simpler than the previous section.Consider the following fourth-quadrant submatrix, with central value 0 and nearcentral value X given by Proposition 16: X XA X B X C XA X B X C X D XA X B X C X D X E X . (42) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Fig. 2
M(16, 16, 8): m = n = k , k even with proper zerosentral Values for Clebsch-Gordan coefficients 19 Alternating between the two main recurrences as in Proposition 19 immediatelyyields
Theorem 5
Let X be the sub-central value determined by Proposition 16, and define λ s = s ( s + ) . With s ≥ , the first two columns of (42) are computed recursively by A = , B = , (43) A s + = ( λ m (cid:48) − λ m + λ s ) A s − ( λ n − λ s − ) B s λ m − λ s , (44) B s + = λ m (cid:48) A s − ( λ n − λ s − ) B s λ m − λ s . (45)In the triangle from the first column to the diagonal, unreduced denominators areequal along rows and increase by a factor of λ m − λ s as we pass from the s -th to the ( s + ) -st row. That is, with s ≥ , the denominator for index s + d s + = s (cid:214) l = ( λ m − λ l ) = s + λ m ( m + s + ) ! ( m − s − ) ! . (46)This implies immediately Corollary 3
For s ≥ , let N ( A s ) and N ( B s ) be the numerators in the unreduced ex-pressions of A s and B s , respectively. Then N ( A s ) and N ( B s ) are computed recursivelyby N ( A ) = , N ( B ) = , (47) (cid:20) N ( A s + ) N ( B s + ) (cid:21) = (cid:20) λ m (cid:48) − λ m + λ s λ s − − λ n λ m (cid:48) λ s − − λ n (cid:21) (cid:20) N ( A s ) N ( B s ) (cid:21) . (48)To proceed towards the diagonal, for instance, we have for s ≥ , C s + = B s + B s + , N ( C s + ) = ( λ m − λ s ) N ( B s ) + N ( B s + ) . (49)As before, when m is large enough, the denominator is non-vanishing, and van-ishing in a coordinate relative to the the central value reduces to solving the corre-sponding Diophantine equation, say N ( A s ) = , (50)in m , n , and k . In the classification of [9], diagonal zeros from Section 4 closest to thecentral zero are denoted by R . For m , n even and k odd, these subfamilies correspondto positions around the central value as follows: .
15 3 .
16 3 .
17 3 .
16 3 . .
16 2 .
11 2 .
12 2 .
12 2 .
11 3 . .
17 2 . I I R I I .
12 3 . .
16 2 . R • • R .
12 3 . .
15 2 . I I • • I I .
11 3 . .
16 2 . R • • R .
12 3 . .
17 2 . I I R I I .
12 3 . .
16 2 .
11 2 .
12 2 .
12 2 .
11 3 . .
15 3 .
16 3 .
17 3 .
16 3 . . Diophantine equations in the subcentral triangle are given by•
I I : N ( A ) = λ m (cid:48) − λ m − λ n + = , • R : N ( B ) = λ m (cid:48) − λ n = ( m − k )( m + n − k + ) = , • 2.11: N ( A ) = ( λ m (cid:48) − λ m + ) N ( A ) − ( λ n − ) N ( B ) = , • 2.12: N ( B ) = λ m (cid:48) N ( A ) − ( λ n − ) N ( B ) = , • 3.15: N ( A ) = ( λ m (cid:48) − λ m + ) N ( A ) − ( λ n − ) N ( B ) = , • 3.16: N ( B ) = λ m (cid:48) N ( A ) − ( λ n − ) N ( B ) = , • 3.17: N ( C ) = ( λ m − ) N ( B ) + N ( B ) = . We have relabeled subfamilies 2.15-2.17 in Table 3 of [9] as 3.15-3.17 here; thegroupings naturally correspond to concentric hexagons about the center.
In [9], a full parametrization for zeros of type I − V I are given. For expositorypurposes, we include an algorithm for generating all ( m , n , k ) satisfying I : N ( A ) = I I : N ( A ) = . Types
I I I − V I admit similar parameterizations; each case requires solving a Dio-phantine equation of the form x y = u v , where x , y , u , v are linear expressions in m , n , and k . Proposition 20
With k > and even, all solutions to m , n even , m (cid:48) = m + n − k , λ m (cid:48) = λ m + λ n (51) are given by m = N , n = Q − P − , k = N − P (52) for some integers N , P , Q with1. N ≥ , PQ = N ( N + ) for ≤ P < Q and P < N , and3. P and Q (resp. P and N ) have opposite (resp. same) parity. entral Values for Clebsch-Gordan coefficients 21 Proof
Consider the equation A + = B + C (53)with all A , B , C > A − C A + C = B − B + . (54)Thus solutions are given precisely when A = P + Q , B = N + , C = Q − P . (55)Now completing the square in (51) yields ( m (cid:48) + ) + = ( m + ) + ( n + ) , (56)and the proposition follows. (cid:3) For example, when N = , P = Q =
12, we have ( m , n , k ) = ( , , ) . SeeFig. 3.
15 15 15 15 15 15 15 15 15 -45 -30 -15 0 15 30 45 60 75 90 45 0 -30 -45 -45 -30 0 45 105 180 270 45 45 15 -30 -75 -105 -105 -60 45 225 495 45 90 105 75 0 -105 -210 -270 -225 0 495 45 135 240 315 315 210 0 -270 -495 -495 45 180 420 735 1050 1260 1260 990 495
Fig. 3 M ( , , ) : the smallest example with type I zeros Consideration of divisibility properties allows one to directly parametrize somesubseries of solutions to (51), as noted in the table below. The first line covers allcases where P divides N . The next series gives the general series where the odd partof P divides N +
1; in this case, with 0 < b , c < a + , bc = ( mod a ) , bc (cid:44) ( mod a + ) . (57)Noting that N and N + I I : N ( A ) =
0, we have
Proposition 21
With k > and odd, all solutions to m , n even , m (cid:48) = m + n − k , λ m (cid:48) + = λ m + λ n (58) Table 2
Some Basic Series of Type I Zeros (Position A1)
P N N + s s ( t + ) s + t ( s + ) ( s + ) ( s + )( t + ) ( s + ) ( s + )( t + ) . . . . . . a ( a + s + b ) ( a + s + b )( a + t + c ) are given by m = N + , n = Q − P − , k = N − P + for some integers N , P , Q with1. N ≥ , PQ = N ( N + ) for ≤ P < Q and P < N , and3. P and Q (resp. P and N ) have opposite (resp. same) parity.Proof Completing the square in (58) yields ( m (cid:48) + ) + = ( m + ) + ( n + ) , (60)and the proof now follows as in Proposition 20. (cid:3) Remark.
For example, we have ( m , n , k ) = ( , , ) when N = , P = Q = l ≥ , solutions of (58) of the form ( m , n , k ) = ( l , , ) correspond to thefourth degenerate case. That is, with respect to M ( m , n , k ) , we obtain a vertical zerotriplet with a single proper zero. m odd, n even The remaining two cases allow for a simultaneous treatment. In both cases, the centeris a square of size 2, and the lower-right entries are given by Propositions 17 and 18.
Proposition 22
Suppose m is odd and n is even, and M ( m , n , k ) has central square (cid:34) X (cid:48) ∗ A X X (cid:35) for nonzero X . Then A = m (cid:48) − mm (cid:48) + if k even ; A = m (cid:48) + m + m (cid:48) + if k odd entral Values for Clebsch-Gordan coefficients 23 where m (cid:48) = m + n − k . Proof
See Proposition 19. In this case, the Weyl group symmetry yields ( m + ) X = (− ) k ( m (cid:48) + ) X (cid:48) . (61) (cid:3) Consider the following fourth-quadrant submatrix, where X represents the lower-right entry of the central square: A X X A X B XA X B X C XA X B X C X D . (62)As before, we have Theorem 6
Let X be the lower-right entry determined by Propositions 17 or 18, anddefine λ s = s ( s + ) . The first two columns of (62) are computed recursively by A = m (cid:48) − mm (cid:48) + if k even ; A = m (cid:48) + m + m (cid:48) + if k odd ; B = , (63) A s + = ( λ m (cid:48) − λ m + λ s + + ) A s + ( λ s − λ n ) B s λ m − λ s + , (64) B s + = ( λ m (cid:48) + ) A s + ( λ s − λ n ) B s λ m − λ s + . (65)In the subcentral triangle, unreduced denominators are equal along rows andincrease by a factor of λ m − λ s + as we pass from the s -th to the ( s + ) -st row. Thatis, with s ≥ , the denominator for index s + d s + = ( m (cid:48) + ) s (cid:214) l = ( λ m − λ l + ) = s + m (cid:48) + m + ( m + s + ) ! ( m − s − ) ! . (66)This implies immediately Corollary 4
For s ≥ , let N ( A s ) and N ( B s ) be the numerators in the unreducedexpressions of A s and B s , respectively. Then N ( A s ) and N ( B s ) are computed recur-sively by N ( A ) = m (cid:48) − m if k even ; N ( A ) = m (cid:48) + m + if k odd ; N ( B ) = m (cid:48) + , (67) (cid:20) N ( A s + ) N ( B s + ) (cid:21) = (cid:20) λ m (cid:48) − λ m + λ s + + λ s − λ n λ m (cid:48) + λ s − λ n (cid:21) (cid:20) N ( A s ) N ( B s ) (cid:21) . (68)As before, numerators correspond to the following positions, up to a C × C symmetry: m odd , n even , k even . . .
14 3 . . . . .
10 3 . .
14 2 . IV V I . . .
12 2 . V I • R . . . . R • V I .
10 3 . .
10 2 . V I IV . . . .
10 2 . . . .
12 3 .
14 3 . . , • R : N ( A ) = m (cid:48) − m = n − k = , • 2.8: N ( A ) = ( m (cid:48) − m )( λ m (cid:48) − λ m + ) − λ n ( m (cid:48) + ) = , • V I : N ( B ) = ( m (cid:48) + )[( m (cid:48) − m )( m (cid:48) + ) − λ n ] = , • 3.8: N ( A ) = ( λ m (cid:48) − λ m + ) N ( A ) − ( λ n − ) N ( B ) = , • 2.10: N ( B ) = ( λ m (cid:48) + ) N ( A ) − ( λ n − ) N ( B ) = , • 3.12: N ( B ) = ( λ m (cid:48) + ) N ( A ) − ( λ n − ) N ( B ) = , • IV: N ( C ) = ( m (cid:48) + )[ λ m − λ n − + ( m (cid:48) − m )( m (cid:48) + )] = , • 2.6: N ( C ) = ( λ m − ) N ( B ) + N ( B ) = , • 3.14: N ( C ) = ( λ m − ) N ( B ) + N ( B ) = , • 2.4: N ( D ) = ( λ m − ) N ( C ) + N ( C ) = , • 3.6: N ( D ) = ( λ m − ) N ( C ) + N ( C ) = , • 3.4: N ( E ) = ( λ m − ) N ( D ) + N ( D ) = . m odd , n even , k odd . . .
13 3 . . . . . . .
13 2 . I I I V . . .
11 2 . V • • . . . . • • V . . . . V I I I . . . . . . . .
11 3 .
13 3 . . , • 2.7: N ( A ) = ( m (cid:48) + m + )( λ m (cid:48) − λ m + ) − λ n ( m (cid:48) + ) = , • V : N ( B ) = ( m (cid:48) + )[( m (cid:48) + m + )( m (cid:48) + ) − λ n ] = , • 3.7: N ( A ) = ( λ m (cid:48) − λ m + ) N ( A ) − ( λ n − ) N ( B ) = , • 2.9: N ( B ) = ( λ m (cid:48) + ) N ( A ) − ( λ n − ) N ( B ) = , • 3.11: N ( B ) = ( λ m (cid:48) + ) N ( A ) − ( λ n − ) N ( B ) = , • I I I : N ( C ) = ( m (cid:48) + )[ λ m − λ n − + ( m (cid:48) + m + )( m (cid:48) + )] = , • 2.5: N ( C ) = ( λ m − ) N ( B ) + N ( B ) = , • 3.13: N ( C ) = ( λ m − ) N ( B ) + N ( B ) = , • 2.3: N ( D ) = ( λ m − ) N ( C ) + N ( C ) = , • 3.5: N ( D ) = ( λ m − ) N ( C ) + N ( C ) = , • 3.3: N ( E ) = ( λ m − ) N ( D ) + N ( D ) = . entral Values for Clebsch-Gordan coefficients 25 For types 3.9 and 3.10, simultaneously consider the entries Z of this type near A . Application of both recurrences yields N ( Z ) = ( λ m (cid:48) + λ m + ) − λ m (cid:48) λ m − − λ n [( m (cid:48) + ) N ( A ) + λ m − ] and Z = A N ( Z )( λ m (cid:48) − ) .
10 Computer Implementation
In [9], an analysis is given for certain vanishing c m , n , k ( i , j ) with J = m + n − k < , . We leave it to the reader to pursue those details there.The algorithms in this work require only rudimentary programming expertise,implemented on conventional hardware (2013 MacBook Pro). The figures wereconstructed using Excel, which was also used to compute all M ( m , n , k ) above.Other algorithms, such the numerator formulas, were stress-tested using MAPLE. References
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