UDC 530.1(018) + 530.113

MATHEMATICAL PHYSICS

S. I. ALISHAUSKAS, Academician of the Academy of Sciences of the Lithuanian SSR A. P. YUTSIS

ON THE CLEBSCH–GORDAN COEFFICIENTS OF SYMMETRIC REPRESENTATIONS OF THE GROUPS \(SU_n\)

The importance of the theory of representations of the groups \(SU_n\) for modern theoretical physics makes its further development necessary. This applies especially to the reduction of the direct product of irreducible representations, expressed by a Clebsch–Gordan series. Questions concerning the properties of the Clebsch–Gordan coefficients (C.–G.) and methods for their rapid determination still await solution.

In this note we shall indicate a method for expressing the isoscalar factors of the C.–G. coefficients of symmetric representations of the groups \(SU_n\) through a product of \(n-2\) factors, each of which depends on the parameters of representations of the groups \(SU_k\) and \(U_1 \times SU_{k-1}\) \((k=3,4,\ldots,n)\). Let us recall that symmetric representations are irreducible representations with a Young diagram consisting of one row, or representations contragredient to them. The Young diagram of the latter consists of \(n-1\) (\(k-1\) for the subgroups \(SU_k\)) equal rows.

In solving the problem posed we shall proceed from the construction of recurrence relations for the C.–G. coefficients under consideration, since this method is the most convenient (cf., for example, \((^1,^2)\)). We begin the exposition by indicating the method of using these relations in the case of the group \(SU_3\).

We shall denote the basis functions of irreducible representations of the group by

\[ \left| \begin{matrix} (\lambda\mu)\\ IMY \end{matrix} \right\rangle , \tag{1} \]

where \(\lambda+\mu\) and \(\mu\) are the lengths of the rows of the Young diagram; \(I\) and \(M\) are the isospin and its projection; \(Y\) is the hypercharge. We shall use the system of phases \((^3)\), since it facilitates the generalization of the results to the case of groups with \(n>3\). In this system of phases the matrix elements of the operators \(E_{k,k-1}\) are positive. We shall denote the C.–G. coefficients by

\[ \left[ \begin{matrix} (\lambda_1\mu_1) & (\lambda_2\mu_2) & (\lambda\mu)_\rho\\ I_1M_1Y_1 & I_2M_2Y_2 & IMY \end{matrix} \right] = \left[ \begin{matrix} (\lambda_1\mu_1) & (\lambda_2\mu_2) & (\lambda\mu)_\rho\\ I_1Y_1 & I_2Y_2 & IY \end{matrix} \right] \left[ \begin{matrix} I_1 & I_2 & I\\ M_1 & M_2 & M \end{matrix} \right]. \tag{2} \]

The first factor on the right-hand side represents the isoscalar factor, and the second the C.–G. coefficients of the group \(SU_2\). Here \(M=M_1+M_2\), \(Y=Y_1+Y_2\). The parameter \(\rho\), distinguishing repeated representations, will not be needed by us. We remove the remaining phase indeterminacy of the C.–G. coefficients by considering their value positive at \(I_2=0\).

For symmetric representations \(\lambda\) or \(\mu\) is equal to zero. The two representations being coupled may be of identical or different contragredientness. The expressions for the isoscalar factors depend on this. Using the relation obtained when the infinitesimal operator \(E_{32}\) acts on a basis function of the coupled representations, we obtain the following recurrence relation:

\[ \left[ \begin{matrix} (\lambda_1 0) & (\lambda_2 0) & (\lambda_1+\lambda_2-2s,\ s)\\ I_1 & I_2 & I+1 \end{matrix} \right] \left[(I_1+I_2+I-s+1)(\lambda_1+\lambda_2-s-I_1-I_2-I)\right]^{1/2} = \]

\[ = \left[ \begin{array}{ccc} (\lambda_1 0) & (\lambda_2 0) & (\lambda_1+\lambda_2-2s,s')\\ I_1-\frac12 & I_2 & I+\frac12 \end{array} \right] \bigl[(I_1-I_2+I)(\lambda_1-2I_1+1)\bigr]^{1/2}+ \]
\[ +\left[ \begin{array}{ccc} (\lambda_1 0) & (\lambda_2 0) & (\lambda_1+\lambda_2-2s,s')\\ I_1 & I_2-\frac12 & I+\frac12 \end{array} \right] \bigl[(I-I_1+I_2)(\lambda_2-2I_2+1)\bigr]^{1/2}. \tag{3} \]

Comparing this formula with formula (12.3) for the Clebsch–Gordan coefficient of \(SU_2\) from (3), we find that they are isomorphic. By changing the constraints on \(I\) and \(M\) we obtain another recurrence relation, isomorphic to which one may be obtained by combining (16.18) and (12.3) from \((^4)\). The corresponding formulas for the isoscalar factors of \(SU_3\) and the Clebsch–Gordan coefficients of \(SU_2\) coincide if one makes the substitution

\[ \left[ \begin{array}{ccc} (\lambda_1 0) & (\lambda_2 0) & (\lambda_1+\lambda_2-2s,s)\\ I_1 & I_2 & I \end{array} \right] = \]
\[ = \left[ \begin{array}{ccc} \dfrac{\lambda_1-I_1-I_2+I}{2} & \dfrac{\lambda_2-I_1-I_2+I}{2} & \dfrac{\lambda_1+\lambda_2}{2}-s\\[6pt] \dfrac{I+3I_1-I_2-\lambda_1}{2} & \dfrac{I-I_1+3I_2-\lambda_2}{2} & I_1+I_2+I-\dfrac{\lambda_1+\lambda_2}{2} \end{array} \right] = \tag{4a} \]
\[ = \left[ \begin{array}{ccc} I & \dfrac{\lambda_1+\lambda_2}{2}-I_1-I_2 & \dfrac{\lambda_1+\lambda_2}{2}-s\\[6pt] I_1-I_2 & \dfrac{\lambda_1-\lambda_2}{2}-I_1+I_2 & \dfrac{\lambda_1-\lambda_2}{2} \end{array} \right]. \tag{4б} \]

Here \(Y_i\) \((i=1,2)\) are omitted, since they are expressed as

\[ Y_i=2I_i-\frac{2}{3}\lambda_i . \tag{4в} \]

(4a) and (4б) are Clebsch–Gordan coefficients of the group \(SU_2\), with the second obtained from the first by means of Regge symmetry properties for Wigner coefficients.

It is easy to verify that (4a) is normalized with respect to summation over \(I_1\) and \(I_2\) for fixed \(I\) and \(Y\). On the other hand, (4б) is normalized with respect to summation over the parameters of the third column, if it is first multiplied also by a Clebsch–Gordan coefficient of \(SU_2\). It should be noted that from (4) one can obtain other types of substitutions by means of the symmetry properties of the Clebsch–Gordan coefficients of the group \(SU_3\) \((^5)\).

In the case of different contragredientnesses, instead of (3) we obtain

\[ \left[ \begin{array}{ccc} (0\mu) & (\lambda 0) & (\lambda-a,\mu-a)\\ I_1 & I_2 & I \end{array} \right] = \]
\[ =(-1)^{I_1+I_2-I} \left[ \begin{array}{ccc} (\lambda 0) & (0\mu) & (\lambda-a,\mu-a)\\ I_2 & I_1 & I \end{array} \right] = \tag{5a} \]
\[ = \left[ \begin{array}{ccc} \dfrac{\mu-I_1+I_2-I}{2} & \dfrac{I_1-I_2+I+\lambda+1}{2} & \dfrac{\lambda+\mu+1}{2}-a\\[6pt] \dfrac{\mu-3I_1-I_2+I}{2} & \dfrac{I_1+3I_2+I-\lambda+1}{2} & \dfrac{\mu-\lambda+1}{2}+I-I_1+I_2 \end{array} \right], \tag{5б} \]

where

\[ Y_1=\frac{2\mu}{3}-2I_1,\qquad Y_2=2I_2-\frac{2}{3}\lambda; \tag{5в} \]

(5a), (5б) are also Clebsch–Gordan coefficients of the group \(SU_2\), whose recurrence relations are known. However, the relation corresponding to formula (3) has a somewhat different form.

The formulas given make it easy to obtain the formulas of Section 5 of \((^6)\), and our expression (4) is considerably simpler than (7.9) of \((^6)\).

In the case of the groups \(SU_n\), we shall denote the basis functions by a set of Gel'fand–Biedenharn parameters \((^7,^3)\). In the case of symmetric representations this set will be the following:

\[ \left( \begin{array}{cccc} h_n & 0 & 0\ldots & 0\\ & h_{n-1} & 0\ldots & \\ & & h_{n-2} & 0\ldots \end{array} \right) \tag{6a} \]

The basis functions connected from two such representations may be denoted by

\[ \left( \begin{array}{cccccc} h_n+h'_n-s_n & & & s_n & 0 & 0\ \ldots\ 0\\ & h_{n-1}+h'_{n-1}-s_{n-1} & & & s_{n-1} & 0\ \ldots\ 0\\ & & h_{n-2}+h'_{n-2}-s_{n-2} & & & s_{n-2}\ \ldots\ 0\\ & & \ldots & \ldots & \ldots & \end{array} \right) \tag{6б} \]

We denote the separate factor in the C.—G. coefficients for the reduction of \(SU_k\) to \(U_1 \times SU_{k-1}\) by

\[ \left[ \left( \begin{array}{ccc} h_k & 0 & 0\ldots\\ h_{k-1} & 0\ldots & \end{array} \right) \left( \begin{array}{cc} h'_k & 0\ldots\\ h'_{k-1} & \ldots \end{array} \right) \left( \begin{array}{ccc} h_k+h'_k-s_k & s_k & 0\ldots\\ & h_{k-1}+h'_{k-1}-s_{k-1} & s_{k-1}\ldots \end{array} \right) \right] = \]

\[ = H_k = \left( \begin{array}{cc|c} h_k & h'_k & s_k\\ h_{k-1} & h'_{k-1} & s_{k-1} \end{array} \right). \tag{7} \]

Acting with the operator \(E_{k,k-1}\) on functions of the type (6a) and (6б), we ascertain that the expressions for the matrix elements of this operator, according to formula (60) from (3), do not depend on the parameters \((k-i)\) \((i=3,4,\ldots)\), nor on \(k\) in explicit form.

In the case of identical contragrediences we obtain recurrence relations as for the representations of the groups \(SU_3\). Without loss of generality, on the basis of the independence, indicated in (3), of the separate factor of the C.—G. coefficients from the parameters \(k-i\) \((i=2,3,\ldots)\) of the rows, we may assume \(s_{k-i}=0\). Then the expression for the matrix elements of the operator \(E_{k,k-1}\) for any \(k\) coincides with the expression for the matrix element \(E_{32}\) of the group \(SU_3\), where the parameters of the \(k\)-th row stand in place of the 3rd row, \(k-1\) in place of the 2nd, etc. If we take the expression for the separate factor of the C.—G. coefficient

\[ H_k \left( \begin{array}{cc|c} h_k & h'_k & s_k\\ h_{k-1} & h'_{k-1} & s_{k-1} \end{array} \right) = \]

\[ = \left[ \begin{array}{ccc} \dfrac{h_k-s_{k-1}}{2} & \dfrac{h'_k-s_{k-1}}{2} & \dfrac{h_k+h'_k}{2}-s_k\\[1.2em] h_{k-1}-\dfrac{h_k+s_{k-1}}{2} & h'_{k-1}-\dfrac{h'_k+s_{k-1}}{2} & h_{k-1}+h'_{k-1}-s_{k-1}-\dfrac{h_k+h'_k}{2} \end{array} \right], \tag{8} \]

which holds for \(k=2\) and \(3\), and assume that it is valid for the subgroup \(SU_{k-i}\) \((i=1,2,\ldots)\), then in the recurrence relation \(H_{k-1}\) will play the role of the C.—G. coefficient of the group \(SU_2\) in the relation for \(SU_3\). In this case the factors \(H_{k-i}\) \((i=2,3,\ldots)\) are simplified, and the relation for \(H_k\) coincides with the same relation for the group \(SU_3\), if only the basis functions of irreducible representations of the latter are denoted by a set of Gelfand—Biedenharn parameters.

In generalizing (5) to the case of the groups \(SU_n\), we denote the basis function contragredient to the function (6a) by

\[ \left( \begin{array}{ccccc} 0\ldots 0 & 0 & & & g_{-n}\\ \ldots & \ldots & 0 & & -g_{n-1}\\ & \ldots & \ldots & \ldots & -g_{n-2}\\ & \ldots & \ldots & \ldots & \end{array} \right), \qquad g_k \geqslant 0 . \tag{9} \]

Relating representations (9) and (6a) (by reducing the direct product), we obtain basis functions of the type

\[ \left( \begin{array}{cccccc} h_n-a_n & 0 & 0 & \ldots & 0 & a_n-g_n\\ & h_{n-1}-a_{n-1} & 0 & \ldots & 0 & a_{n-1}-g_{n-1}\\ & & h_{n-2}-a_{n-2} & \ldots & a_{n-2}-g_{n-2} & \end{array} \right). \tag{10} \]

Obviously, by adding \(g_n\) to all parameters in (9), or \(g_n-a_n\) in (10), we would get rid of negative parameters; however, for our purposes the forms (9) and (10) are more convenient. We denote the corresponding separate factor of the Clebsch–Gordan coefficient by

\[ G_k\left( \begin{array}{cc||c} g_k & h_k & a_k\\ g_{k-1} & h_{k-1} & a_{k-1} \end{array} \right). \tag{11} \]

If the operator \(E_{k,k-1}\) is applied to functions of the type (10) and one takes \(a_{k-2}=g_{k-2}\), it is found that the matrix elements of this operator can be obtained from the matrix elements of the operator \(E_{32}\) of the group \(SU_3\). For this purpose one should make the substitutions

\[ \lambda-a \to h_k-a_k+k-3,\qquad \mu-a \to g_k-a_k, \]

\[ 2I \to h_{k-1}-g_{k-1}-2a_{k-1}+k-3. \tag{12a} \]

Applying the operator \(E_{k,k-1}\) to basis functions of the type (9) and (6a), we are led to the equivalence

\[ \lambda-2I_2 \to h_k-h_{k-1},\qquad \mu-2I_1 \to g_k-g_{k-1}. \tag{12b} \]

If

\[ G_k\left( \begin{array}{cc||c} g_k & h_k & a_k\\ g_{k-1} & h_{k-1} & a_{k-1} \end{array} \right) = \]

\[ = \left[ \begin{array}{ccc} \dfrac{g_k-g_{k-1}+a_{k-1}}{2} & \dfrac{h_k+g_{k-1}+a_{k-1}+k-2}{2} & \dfrac{g_k+h_k+k-2}{2}-a_k \\[1.2em] \dfrac{g_k-g_{k-1}-a_{k-1}}{2} & h_{k-1}+\dfrac{g_{k-1}-h_k-a_{k-1}+k-2}{2} & h_{k-1}-a_{k-1}+\dfrac{g_k-h_k+k-2}{2} \end{array} \right] \tag{13} \]

is valid for \(G_{k-i}\) \((i=1,2,\ldots)\), then by substituting \(G_{k-i}\) into the corresponding recurrence relation we obtain a mutual correspondence with the recurrence relation for the group \(SU_3\), provided only that in (12) one regards \(a\to a_n\). Therefore, if (13) is valid for \(SU_3\), it is also valid for \(SU_k\) for any \(k=3,4,\ldots\). It should be noted that the symmetry of (5) and (13) with respect to the parameters \(g\) and \(h\) (\(\lambda\) and \(\mu\)) (see (5)) is visible when the Clebsch–Gordan coefficients are expressed in terms of Wigner coefficients with Racah parameters.

From what has been set forth it follows that the indicated method makes it possible to express the Clebsch–Gordan coefficients of symmetric representations of the group \(SU_n\) as

\[ \prod_{k=2}^{n} C_k, \]

where \(C_2\) is the Clebsch–Gordan coefficient of the group \(SU_2\), and \(C_k,\ k\geq 3\), are respectively the factors \(H_k\) or \(G_k\), equal to the Clebsch–Gordan coefficients of the group \(SU_2\).

Institute of Physics and Mathematics
Academy of Sciences of the Lithuanian SSR
Vilnius State University
named after V. Kapsukas

Received
27 IV 1966

CITED LITERATURE

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