Issai Schur’s orthogonality relations for characters of group representations are surely a mathematical wonder. Unfortunately, the standard proof, even from great expositors like JP Serre and Michael Artin, typically involves a bunch of index gymnastics that never meant much to me. Back when I was a mathematician, I saw a great proof (for compact Lie groups) in a paper by Raoul Bott. You need to know a little bit about tensor products to understand it, but otherwise it’s fairly simple.
On my exit from the field, I tried to write up a few things I’d learned that seemed like they were not well enough known. This was one of them. The writeup expanded to cover all the necessary background material, and include some other stuff. I will put in a link to it below.
In November, I was visiting my son and his girlfriend in Berkeley, and I ended up speaking to a colleague of hers who does quantum computing. It turned out that he teaches group representation, but hadn’t heard the proof. I dug around and found my paper, but clearly what was really needed was a brief proof for people who already knew their way around some basics of representation theory. So here it is. I couldn’t get it down to one page, so it’s on two pages.
People I’ve shown this proof to have described it as “slick,” which I think is accurate. But I would say it’s more than that. A slick proof does not necessarily cut to the heart of the matter. I think this proof does.
Here is a scanned version of my original writeup, which went for 36 pages. If you know some basic group theory and linear algebra, that should be enough to get you started. After proving the orthogonality relations, it goes on to examine the structure of the group ring, and to prove a general structure theorem for semisimple rings.
Of course this proof is not completely unknown. I learned here that it appears in the book Introduction to Representation Theory by Etingof et al. There is a PDF of what is apparently a related set of lecture notes, where it appears as Theorem 3.8 on p. 37.