Consult pt1 here.

After a bit of thought, I came to the conclusion that I needed to study the \(\gamma\) present in the computed group functions (GF). From them, observer if there is any pattern showing how can we exploit the number of zeroes to reduce the computational complexity.

As the main image of this post reveals, is not difficult to see that from the sum of **GFs a lot of them are zero**. For instance, in the \(SU4/SU3\) case, in the original —using the original scattering matrix— we use 15 non zero terms. However, using the coset matrix, only 4 are required.

The situation improve in the \(SU5/SU4\) case.

On another theme, also note the fact that we **only require two different double-coset representatives**.

So, task for the next time I work on this

- Study the possible form of the double coset representatives.
- Create a new Julia method that avoids computing orthogonal irrep elements if the monomial is zero. Then, benchmark it.
- Count the total number of orthogonal entries saved.
- Try to find the form of the \(\gamma\)s. That is, the double coset representatives in the case of using a coset matrix.

On the first point. My guess is that the correct non \(e\) double coset representative is of the form \( (1, n-1, n-2, \dots, 2)\).

Update. The case that I was studying is not the most interesting. I was seeing the case of multiple photons in a channel. However, the savings, though not as great, they are still present. I added two new tasks.