Outer Multiplicity
When non-Abelian symmetry is used, it is possible that there are multiple sectors with the same quantum numbers, while different sectors are indeed associated with different (orthogonal) Clebsch-Gordan coefficients. It is called outer multiplicity. One simple example is:
[S,I] = getLocalSpace( 'Spin',1/2);
A1 = getIdentity(I.E,2,I.E,2);
A2 = getIdentity(A1,3,I.E,2);
A12 = contract(A1, '3',A2, '1')
Q: 3x [1 1 1 1] having 'SU2', { , , , * }
data: 4-D double (352 bytes) 1 x 1 x 1 x 3 => 2 x 2 x 2 x 8
1. 1x1x1x2 | 2x2x2x2 [ 1 ; 1 ; 1 ; 1 ] 16 B
2. 1x1x1x2 | 2x2x2x2 [ 1 ; 1 ; 1 ; 1 ] 16 B
3. 1x1x1x1 | 2x2x2x4 [ 1 ; 1 ; 1 ; 3 ] 2.
The first and second sectors have the same quantum numbers! But, of course, they are orthogonal. Consider their parts:
Here we replaced the reduced matrix elements (.data{..}) with ones, to focus on the Clebsch-Gordan coefficients. Then their overlap, i.e., contraction between A12a and the conjugate of A12b vanishes.
On the other hand, three incoming legs (from the first to the third) look identical, so it seems that the tensor remains the same after permuting the incoming legs. But it's not! The overlap with A12a and the permutation of the first and third legs of A12a is not unity. It means that the Clebsch-Gordan coefficients can change by permuting legs.
Q: [] having 'A,SU2',
data: 0-D double (112 bytes)
1. 1x1 [ ] 0.5
Why? Let's draw the tensor network diagram for the first and second sectors of A12.
Attached to the legs are spin quantum numbers. Fusing two \(S=1/2\) (two incoming legs of A1) lead to two values of spin \(S=0\oplus 1\) (outgoing leg of A1). These two values can result in \(S=1/2\) (outgoing leg of A2) by being fused with one \(S=1/2\) (bottom incoming leg of A2). The outer multiplicity of A12 reflects this two different ways of having four legs of \(S=1/2\). In this diagram, it is clear to see that the left leg of A1 and the bottom leg of A2 are not equivalent.