# The Unapologetic Mathematician

## New Topic: The Representation Theory of the Symmetric Group

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.

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September 7, 2010 -

## 15 Comments »

1. Excellent—I am looking forward to thinking about representations of the symmetric group more.

Thanks for choosing this topic.
Bret

Comment by bretbenesh | September 7, 2010 | Reply

• Great, Bret! Now pass it on to students, colleagues, passers-by on the street… :D

Comment by John Armstrong | September 7, 2010 | Reply

2. Ah, nice choice! I just started following a course in representation theory, so I will keep reading your entries – hopefully with a categorical flavour :)
Although I also liked measure theory.

Comment by wildildildlife | September 7, 2010 | Reply

3. Hello,
if one already knows Young diagrams, characters, subduced representations and connection to SU_n (or at least one _should_ know), what else are they going to learn?

On a different note, where can the representation theory of finite groups be applied? I have as of yet seen only applications in chemistry and crystallography. Me wonders, however, whether there is some connection between representation theory and abstract stuff like subgroup structure, solvability, etc. In any case, looking forward to this series.

Marek

Comment by Marek Bernát | September 7, 2010 | Reply

• Here’s a couple of things. First there’s Burnside’s Theorem, which uses representation theory to prove that a group of order $p^aq^b$ is always solvable. But here’s another nice one, with some shameless self-promotion (for my semi-defunct blog, due to thesis): http://rigtriv.wordpress.com/2007/09/08/group-theory-and-physics/

Now, for John: any chance you’re going to be up to writing a Schubert calculus program? Maybe a package for something that will actually be able to do multiplication of Schubert monomials, instead of just pairwise multiplication (which is all I’ve seen implemented anywhere, which makes the long calculations continue to be awfully tedious).

Comment by Charles Siegel | September 7, 2010 | Reply

• I don’t know, but I’ll take it under consideration…

Comment by John Armstrong | September 8, 2010 | Reply

• The Burnside Theorem application looks interesting and I’ll look into that. As for the blog… well, it looks less interesting as I am a physicist myself, and I’ve seen applications like this (and also to the emission spectra of symmetric molecules using Wigner-Eckart theorem). Thanks anyway though.

Comment by Marek Bernát | September 8, 2010 | Reply

• You say that like everyone — even every mathematics grad student — learns about things like Young diagrams.

Comment by John Armstrong | September 8, 2010 | Reply

• Trying as I may, I don’t see a single sentence where I wrote about everyone learning Young diagrams :-) I was just interested whether there will only be a standard stuff or something fancier. In any case, I always loved group theory and any blog writing about it is great and you got thumbs up from me, so please don’t take my previous comment the wrong way ;-)

Comment by Marek Bernát | September 8, 2010 | Reply

• My point is that what you call “standard stuff” is far from standard to most readers, even with significant mathematical backgrounds.

Comment by John Armstrong | September 8, 2010 | Reply

• Oh, I haven’t realized that. I just assumed that, like everything else they tell us in our physics classes, it must be a very simplified version of what mathematicians do. Of course, we don’t do as many proofs so that might be a part of the reason (we cover a bigger area, but not as deeply). Or maybe group and representation theory aren’t standard courses for every mathematician at all? I always assumed they must be because of the fundamental importance of group theory in almost every mathematical topic I know of.

Anyways, sorry for the confusion. I am sure I will learn something new from you even about the “standard” stuff :-)

Comment by Marek Bernát | September 9, 2010 | Reply

• Group theory, yes; some representation theory, usually. Specializing on the representations of the symmetric group, not so much.

My only formal exposure to any of this was only as much as was needed for use with Schur-Weyl duality and the representation theory of $GL_n(\mathbb{C})$

Comment by John Armstrong | September 9, 2010 | Reply

4. “Is analysis worthwhile? Is the theatre really dead?” song lyrics for reader identification.

I’m ready for your take on the representation theory of permutation groups. And The Unapologetic Programmer.

I have a 5th Math Teacher interview tomorrow, and a 6th on Thursday. These are both in Los Angeles Unified School District, which is cutting things mighty close, ans Monday is the first day of class for students.

Comment by Jonathan Vos Post | September 8, 2010 | Reply

5. […] Before we push on into our new topic, let’s look back at some of the background that we’ve already […]

Pingback by Some Review « The Unapologetic Mathematician | September 8, 2010 | Reply

6. […] since these days we’re concerned with the symmetric group. And it turns out that some nice things happen in […]

Pingback by Conjugates « The Unapologetic Mathematician | September 10, 2010 | Reply