# Penrose notation and finite group representations

I have been using Penrose notation quite a lot recently, for instance trying to make sense of Penrose’s Applications of negative dimensional tensors.

While thinking about group algebras, I wondered how hard it would be to translate the basic main results for representations of finite groups into graphical notation (following the first couple sections of Fulton and Harris). These handwritten notes have one interpretation. It covers idempotents, characters, characters of irreducible representations being an orthogonal basis, and the sum of squares of the dimensions.

It seems like it is not a coincidence that the features of the diagrams are like filters. It is like vectors resonate in the loops, and destructive interference causes only particular “frequencies” to emerge.

I would like to see if there were a way to calculate the modes of coupled oscillators. Tensor products seem to represent a maximally uncoupled system, and the integration operator (convolution) seems to be maximally coupled. The interesting modes of a system of coupled oscillators have frequencies involving the square root of a sum, however, so it would have to be some other kind of operation.