# Implementing Clebsch-Gordan Symmetries and Sum Properties

In this first week of the GSoC project, I focused on implementing methods that would simplify terms with Clebsch-Gordan coefficients. This still has a long was to go, but I will outline what I have done so far.

The first step was implementing means of dealing with sums of single coefficients. This would hopefully look something like:

The first implementation of this used an indexing system that was able to index single coefficients, which could then be processed. This allowed the simplification function to act properly in simple numerical cases, so it could do things like:

The problem with this implementation is doing something as simple as having one of the terms have a constant coefficient would break it. In addition, there would be no clear way to extend this to sums involving products of multiple Clebsch-Gordan coefficients.

To deal with this, I started working a solution that could deal with having constant coefficients and products of coefficients. Currently implemented is method which creates list of tuples containing information about the Clebsch-Gordan coefficients and the leading coefficients of the Clebsch-Gordan coefficients. Currently, the only implemented logic is only able to deal with the case in that could be dealt with in the previous implementation, however, this should be able to expand to encompass more exotic cases.

Another thing that was touched on this last week was treating symmetries. These are quite simple to implement, as they need only return new Clebsch-Gordan coefficients in place of old ones, just with the parameters changed in correspondence with the symmetry operation. The key will be using these symmetries to help in simplifying terms. This will be based on the development of better logic in the simplification method and the implementation of some means of determining if these symmetries can be used to apply some property of the Clebsch-Gordan coefficients that can simplify the expression.

I will be out of town this next week on a vacation, and will not be able to get work in, but I will continue working on this when I return, with the intention of getting it to a state that can be pushed within the next couple weeks.