SU(N) Clebsch-Gordan coefficients
This CGC generator computes SU(N) Clebsch-Gordan coefficients (CGCs)
for the decomposition of the direct product S x S' of two irreps into
a direct sum of irreps S'',
using a program written by Arne Alex (click to download C++ source code),
based on an algorithm exploiting Gelfand-Tsetlin calculus described here
[equation numbers given below in square brackets refer to this link].
Conventions
- S = irrep label [Eq. (19)]: a sequence of N nonincreasing, nonnegative integers, the last one being 0, e.g. S = (2 2 0) for SU(3)
- dim(S) = dimension of irrep S [Eq. (22)]
- P(S) = irrep index [Eq. (C2)]: assigns to each irrep S a unique nonnegative integer
- M = state label [Eq. (20)]: a Gelfand-Tsetlin pattern,
e.g. the highest-weight state of S=(2 2 0) has the form M =
- Q(M) = state index [Eq. (C7)]: assigns to each state of irrep S a unique integer between 1 and dim(S)
- alpha = outer multiplicity index [Eq. (2)]: a positive integer that distinguishes different occurences of the same irrep S'' in the decomposition of S x S'
The CGCs that express a state |M''> from irrep S'' (with outer multiplicity index alpha)
as a linear combination of states |M> and |M'> from irreps S and S', respectively,
are defined by the formula [Eq. (5)]:
Tasks performed by the CGC-generator (1-4: decomposition of SxS'; 5-10: properties of SU(N) irreps):
- Find complete set of CGCs arising in decomposition of S x S'
- Find all irreps S'' occuring in decomposition of S x S'
- Find CGCs for coupling S and S' to S''
- Find outer multiplicity of S'' in decomposition of S x S'
- Generate list of first n irreps of SU(N)
- Generate list of all states M of irrep S
- Find index P(S) and dimension dim(S) of irrep S
- Find which irrep S is indexed by P
- Find index Q(M) of state M in irrep S
- Find which state M is indexed by Q in irrep S
1. Find complete set of CGCs arising in decomposition of S x S'
2. Find all irreps S'' occuring in decomposition of S x S'
3. Find outer multiplicity of S'' in decomposition of S x S'
4. Find CGCs for coupling S and S' to S''
5. Generate list of first n irreps of SU(N)
6. Generate list of all states M of irrep S
7. Find index P(S) and dimension dim(S) of irrep S
8. Find which irrep S is indexed by P
9. Find index Q(M) of state M in irrep S
10. Find which state M is indexed by Q in irrep S
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