# 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.