The history of a project

Way back in grad school, when I was learning about Clebsch–Gordan coefficients and the finite tools used to study continuous groups, I wondered if there was a symmetry that went the other way: could one find continuous symmetries from mixing the elements of a finite group? And could one use the mixing field theory formalism we were studying to represent particle interactions as a finite group?

I gave it a try, and after an embarrassing false start, found that there could be. More detailed inspection showed that it wasn't a very plausible physics model, but there was something interesting (to me) going on. I managed to publish what I had, but I had enough on my plate to make essentially no progress for years. The question I wanted to answer was: given a finite group, can I predict what its continuous symmetries (of this obscure type) will be?

In one of my spurts of activity, I found that another grad student had a textbook I wanted for the study, and went to buy it. He insisted on selling two books as a bundle, so I wound up with Theory of Group Representations as well, which I hadn't wanted. Worse, the book I did want didn't help me much.

The project lay idle again for a while, until I decided to BFI tackle a simple family of groups in a systematic way--and I got a result. I wrote it up, but wanted to supply some tools for study to go with it before I tried to publish it.

Back burner again.

So I retired, and had some free time. I created the tools for finding the interesting quantities given a finite group, loaded them into GitHub, and did a deep dive literature search one more time--reading years worth of abstracts and skimming a promising paper now and then. I didn't find anything--was this really a new result? That would be cool.

But that day I noticed something for the first time--the symmetry I described was actually much more general than I had been claiming--it was an isomorphism of a group algebra onto itself. No way this was unknown--this is the sort of thing mathematicians are always looking at. And, in an ironic loop back to the begining of it all (Clebsh-Gordon coefficients arise from group representations), I realized I should have been looking at the group representations.

And so, "the stone the builders rejected", Naimark and Stern, Theory of Group Representations page 97, Chapter 2, Section 2.9, Theorem 1, Corollary 1. "The group algebra of a finite group is symmetrically isomorphic to the direct sum of complete matrix algebras."

If I had attacked the problem harder earlier, I'd have learned the answer to my question decades ago. So no, there's nothing new in my work. And no, $SU(3)\times SU(2)\times U(1)$ doesn't pop out.