Seminar Sophus Lie 1 (1991) 93–97

A method for the computation of Clebsch-Gordan coefficients

Hartmut Schlosser

Clebsch-Gordan series, Clebsch-Gordan coefficients

Let G be a semisimple connected Lie group, 𝔤 its Lie algebra, 𝔤_{ℂ} the complexification of 𝔤. A finite dimensional irreducible representation of G (of 𝔤, 𝔤_{ℂ}) with highest weight m will be denoted by [m]. Let be [m′] and [m′′] be two such representations with representation spaces V′ and V′′. In general, the tensor product [m′] ⊗ [m′′] is a direct sum of irreducible representations [m_{i}] with the multiplicities n_{i}:

[m′] ⊗ [m′′] = ∑_{i∈I} n_{i}[m_{i}] (Clebsch-Gordan series).

Furthermore, we will assume that for every irreducible representation [m] in the representation space V is given a basis {g(m, p_{1}, p_{2}, …) : p_{1}, p_{2}, …} described by parameters p_{1}, p_{2}, … which range over some intervals. This basis we shall call a canonical basis for [m] in the following. (We do not give an exact definition of the canonical basis here). In V′ ⊗ V′′ we have two distinguished bases: The product basis

{g(m′, p′_{1}, p′_{2}, …) ⊗ g(m′′, p′′_{1}, p′′_{2}, …) : p′_{1}, p′_{2}, …, p′′_{1}, p′′_{2}, …}

and the basis consisting of the canonical bases of the irreducible constituents [m_{i}]:

{g^{j_{i}}(m_{i}, p_{1}, p_{2}, …) : i ∈ I, j_{i} = 1, …, n_{i}, p_{1}, p_{2}, …}

The connection between these bases is described by means of the so-called Clebsch-Gordan coefficients CGC(…): g^{j_{i}}(m_{i}, p_{1}, p_{2}, ...) =

∑ CGC(m′, m′′, m_{i}, j_{i}; p′_{1}, …, p′′_{1}, …, p_{1}, …)g(m′, p′_{1}, …) ⊗ g(m′′, p′′_{1}, …).

So, in general we have to solve two problems: I. The computation of the Clebsch-Gordan series (CG series). II. The computation of the Clebsch-Gordan coefficients (CGC).

Remarks to the computation of the CG series

In closed form CG series are known only in a few cases. For instance, if G = SU(2) the irreducible representations are described by the numbers l = 0, 1/2, 1, … and the CG series is given by the formula

(1)  [l_{1}] ⊗ [l_{2}] = ∑_{k=0}^{min{2l_{1},2l_{2}}}[l_{1} + l_{2} − k].

The formula of Kostant-Steinberg gives a solution for arbitrary semisimple Lie algebras 𝔤_{ℂ}: It is possible to compute the multiplicity n_{i} of any irreducible representation in a given tensor product by means of the Weyl group of g_{C}. But it is very hard to work with this formula (see [1]).

A well-known method is that of Littlewood-Richardson: The decomposition of the tensor product is obtained by a graphical procedure working with Young frames. For SU(n) and U(n), this procedure can be used to prove a closed formula similar to (1) (see [3]); in these cases it is easy to calculate the CG series with the aid of a computer.

For example, the irreducible representations D(p, q) of SU(3) are described by two natural numbers p and q (the highest weight of D(p, q) is (p + q, q, 0)) and by the CG series which we get by

D(p_{1}, q_{1}) ⊗ D(p_{2}, q_{2}) = ∑_{i=0}^{i_{1}} ∑_{k=0}^{k_{1}} ∑_{l=l_{0}}^{l_{1}} D(p_{1} + p_{2} − i − 2k + l, q_{1} + q_{2} − i + k − 2l)

i_{1} = min{p_{2}, q_{1}}, k_{1} = min{p_{1}, p_{2} + q_{2} − i}, l_{1} = min{q_{1} + k − i, q_{2}},
l_{0} = max{0, k + i − p_{2}}.

A method for the computation of CGC

The classical CGC are the coefficients connected with the representations of SU(2). Here a canonical basis for the irreducible representation [l] (l = 0, 1/2, 1, ...) can be characterized in the following way: Let

H = ( \begin{matrix}1/2 & 0 \\ 0 & −1/2 \end{matrix} )

be a basis of the Cartan subalgebra of 𝔰𝔩(2, ℂ); the normed eigenvectors of [l](H) build a canonical basis for [l] (these vectors are determined by this condition up to a factor of absolute value 1; we choose this factor equal to 1). We denote the basis by {g(l, μ) : μ = 0, …, 2l}. Here g(l, 0) is the highest vector. In many papers the parameter μ ranges over the eigenvalues −l, −l + 1, …, l of [l](H). The CGC are given by the closed formula

CGC(l_{1}, l_{2}, k; μ_{1}, μ_{2}, μ) = \sqrt{\begin{matrix}\binom{2l_{1}}{μ_{1}} & \binom{2l_{2}}{μ_{2}} & \binom{2l_{1}}{k} & \binom{2l_{2}}{k} \\ \binom{2(l_{1}+l_{2}−k)}{μ} & \binom{2(l_{1}+l_{2})−k+1}{k} \end{matrix}} ∑_{h=0}^{k} \frac{(−1)^{h}\binom{k}{h}\binom{μ_{1}}{h}\binom{μ_{2}}{k−h}}{\binom{2l_{1}}{h}\binom{2l_{2}}{k−h}}

(2)  (k = 0, ..., min{2l_{1}, 2l_{2}}).

This or an analogous formula can be proved by various methods. A possible way is the following: We look at the matrices

A12 = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ,

A21 = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )

of 𝔰𝔩(2, C) and identify A12 with [l](A12) (A21 with [l](A21)); by

(3)  g(l, μ + 1) = \sqrt{\begin{matrix}( & {2l \atop μ} & ) \\ ( & {2l \atop μ+1} & ) \end{matrix}}·\frac{1}{μ + 1}A21g(l, μ)

we get a recursive relation; the initial vector is the highest vector g(l, 0) of [l] which can be described by

(4)  A21(g(l, 0)) = 0.

If a highest vector g(l_{1} +l_{2} −k, 0) is computed by (4) as an element of the tensor space we can use (3) to compute a canonical basis in terms of the product vectors. Now (2) is an explicit solution of the recursive relation (3)(see [4]). The method can be generalized to a numerical computation of CGC of other groups. For this purpose we assume that a canonical basis g_{0}, g_{1}, …, g_{s} with highest vector g_{0} is introduced by parameters for every irreducible representation of a given group G and the action of the root vectors A_{i}, B_{i} (i = 1, …, n) is known (for instance by the Gelfand-Zetlin formulas). Let A_{i}(B_{i}) be root vectors corresponding to negative (positive) roots. We get all elements of the canonical basis by a suitable application of the A_{i} on g_{0} . g_{0} is characterized by B_{i}(g_{0}) = 0, i = 1, …, n (see [1]). If the vector g_{0} is given we have to look for an algorithm such that g_{j+1} can be calculated in the form

(5)  g_{j+1} = c(i, j)A_{i}g_{j} + ∑_{h<j} c_{h}g_{h}.

More specifically, suppose that we are given a tensor product [m′] ⊗ [m′′] of representations, and let [m_{i}] be an irreducible constituent with multiplicity n_{i}. The possible highest vectors g_{0} of [m_{i}] are linear combinations of some product vectors; these product vectors are determined by the highest weight of [m_{i}], which is a weight of [m′] ⊗ [m′′]. We apply B_{i} to the linear combinations of these product vectors with unknown coefficients and obtain from B_{i}(g_{0}) = 0 a set of homogeneous linear equations for these coefficients. The space of solutions has dimension n_{i} and so for n_{i} > 1 the vector g_{0} must be choosen so as to satisfy further conditions. One possibility for such a choice is described in the next section for SU(3).

Now we assume that a highest vector g_{0} of [m_{i}] is given as a linear combination of product vectors:

g_{0} = ∑_{h} c_{h}g_{1h} ⊗ g_{2h}

(the c_{h} are CGC by definition). Let A_{i} be a suitable root vector such that

g_{1} = cA_{i}(g_{0}) = c ∑ c_{h}A_{i}(g_{1h} ⊗ g_{2h}) = c ∑ c_{h}((A_{i}g_{1h}) ⊗ g_{2h} + g_{1h} ⊗ (A_{i}g_{2h})).

Because A_{i}g_{1h} and A_{i}g_{2h} are known by assumption we get g_{1} as a linear combination of product vectors. The coefficients are the CGC of g_{1} . Similarly we pass from g_{j} to g_{j+1} by (5) and an analogous calculation. A few years ago the procedure was programmed for the group SU(3) with the aid of the computer algebra system reduce (see [2]).

Computation of highest vectors in the case SU(3)

We have three operators A_{i}: A21, A31, A23 and three operators B_{i}: A12, A13, A32. The operators A21, A12 act analogously to the case of SU(2). So a subgroup SU(2) of SU(3) is selected.

In general, in the CG series [m′] ⊗ [m′′] = ∑ n_{i}[m_{i}] of SU(3) we have n_{i} > 1. It is necessary to formulate additional properties for the selection of highest vectors g_{0} as solutions of B_{i}(g_{0}) = 0. We observe:

(i) The irreducible representation [m′ + m′′] is selected from the irreducible constituents [m_{i}]: It has the multiplicity 1; the set of weights of [m′+m′′] is identical with the set of weights of [m′] ⊗ [m′′].

(ii) The canonical basis of an arbitrary representation space of SU(3) provides a layer structure; the layer structure of [m′+m′′] can be transferred in a natural way to the product basis.

This gives solutions of A12(g_{0}) = 0 which are similar to the SU(2)- solutions (2); these solutions are joined with the layer structure of the product basis and they are compatible with A13(g_{0}) = 0, A32(g_{0}) = 0. So we have a special way to solve the system B_{i}(g_{0}) = 0, and the solutions are related naturally to the subgroup SU(2). The exact description is given in [5].

References

[1] Baruth, A. D. and R. Raczka,“Theory of group representations and applications”, PWN, Warszawa 1977.

[2] Lassner, W. and H. Schlosser, Computation of the Clebsch-Gordan coefficients of SU(3) by a computer program written at reduce, Trudy meshdunar. soweschtsch. po anal. wytschisl. na EWM i ich primeniju w teor. fis., Dubna, 17.–20. 9. 1985, 327–338.

[3] Schlosser, H., A closed formula for the decomposition of the Kronecker product of irreducible representations of SU(n), Math. Nachr. 134 (1987), 237–243.

[4] —, Zur Berechnung der Clebsch-Gordan-Koeffizienten halbeinfacher Liescher Gruppen, I. Ein allgemeines Verfahren; Anwendung auf die Gruppe SU(2), Beiträge zur Algebra und Geometrie 24 (1987), 29–40.

[5] —, Zur Berechnung der Clebsch-Gordan-Koeffizienten halbeinfacher Liescher Gruppen, II. Die Berechnung der Clebsch-Gordan-Koeffizienten der SU(3), Beiträge zur Algebra und Geometrie 25 (1987), 5–23.

Fachrichtungen Mathematik/Informatik Ernst Moritz Arndt-Universität Jahnstrasse 15a O-2200 Greifswald

Received February 18, 1991