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## Savings due to zeros pt2

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

1. Study the possible form of the double coset representatives.
2. Create a new Julia method that avoids computing orthogonal irrep elements if the monomial is zero. Then, benchmark it.
3. Count the total number of orthogonal entries saved.
4. 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.

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## Savings due to zeros pt1

Consult pt2 here.

In this post I will provide a journal about the savings due to the presence of zeros in the submatrix needed to compute the sum rules.

Day 0

Problems found regarding the savings. The $${\rm det}$$ and a transformation are not always valid. Sometimes, the submatrix is not Hessenberg.

Day 1

$T_{U, V}^{\lambda}= \frac{1}{\sqrt{\Theta_{i}^{f}}} \frac{1}{\sqrt{\Theta_{j}^{g}}} \sum_{\cup_\gamma S_{\alpha} \gamma S_{\beta}=S_{n}}\left(\sum_{\sigma \in S_{\alpha} \gamma S_{\beta}} \omega_{i, j}^{\lambda}(\sigma)\right) X_{f\circ\gamma, g}.$

According to the previous equation, if the submatrix present some zeros then at least one $$X_{f\circ\gamma, g}$$ is zero. This in turn leads to avoiding compute some orthogonal irrep (OI) entries.

This is the unique saving I see at the present.

Thanks to Antti Koskinen Post