Okay, I’m done with measure theory (for now, at least), and not a moment too soon. It’s been good for me to work through all this stuff again, and I hope it’s provided a good resource, but my traffic has really taken a hit, at least as measured by daily page views. I think maybe everybody else hates analysis too?
So let’s go in a completely different direction! I want to talk about the representation theory of permutation groups. Now at least on the surface you might not think there’s a lot to say, but it’s a surprisingly detailed subject. And since every finite group can be embedded in a permutation group — its action on itself by left multiplication permutes its own elements — and many natural symmetries come in the form of permutations, it’s a very useful subject as well.
But as a niche it doesn’t get taught very much. Even the most direct application I know (the representation theory of classical groups) usually avoids getting into the details of this topic. And so it’s likely that many if not most readers haven’t really seen much of it. Along the way we’ll learn a certain amount about the representation theory of more general finite groups, so if you don’t know much about it yet, don’t worry! I’ll try to link back to more general background information where appropriate — and please ask me to fill in points that I seem to skim over — but the coverage should be pretty self-contained. The great thing about this is that you don’t have to have read anything else I’ve written, and you don’t have to be a particularly expert mathematician to follow along! And of course neither do your friends and acquaintances, so this is the perfect chance to pass the word along. Don’t be shy about telling other people to join in!
There’s also going to be a fair amount of focus on the combinatorics involved in the representation theory, and there are some really amazingly beautiful patterns. But we’ll also see some very explicit descriptions of how to actually count these things. If I feel up to it, I may try to actually implement some of the algorithms we see and talk about that over at my new programming weblog The Unapologetic Programmer. Oh yeah: if you haven’t heard yet, I’ve got a new weblog about programming, so all of you into software design should be heading over there and adding it to your RSS reader. And since my work there is a lot more casual, you should tell your other programmer friends about it too!
So enough shameless self-promotion; let’s get down to business.