Vision: automatic decomposition of e.g. tensor products

[fingolfin]

The CHEVIE paper lists a nice example where the scalar product of a certain character $\chi_2$ with itself is decomposed into a linear combination of irreducibles.

The first step is easy: compute the scalar product of the tensor with each irreducible.

But then one has to start a case analysis: first even $q$, then odd $q$. In each case, certain exceptions can be ruled out, or can be further studied by specialization (then after specializing, in many cases it becomes clear a case cannot occur, etc.)... Until in the end it actually produces a decomposition -- or rather two, depending on whether $q$ is even or odd.

It would be super nice if some or even all of this could be automated.

[fingolfin]

The book example in PR #104 would be a good starting ground: It says near the end

By working through the other possible exceptions and irreducible character types, and
handling duplicates, one finally obtains

We should do that, i.e., work through all the cases and write the result up. And then try to add helpers to simplify or even fully automate as many steps as we can.