Over at Not Even Wrong, there’s a discussion of David Vogan’s talks at Columbia about the “orbit method” or “orbit philosophy”. This is the view that there is — or at least there should be — a correspondence between unitary irreps of a Lie group and the orbits of a certain action of . As Woit puts it
This is described as a “method” or “philosophy” rather than a theorem because it doesn’t always work, and remains poorly understood in some cases, while at the same time having shown itself to be a powerful source of inspiration in representation theory.
What he doesn’t say in so many words (but which I’m just rude enough to) is that the same statement applies to a lot of theoretical physics. Path integrals are, as they currently stand, prima facie nonsense. In some cases we’ve figured out how to make sense of them, and to give real meaning to the conceptual framework of what should happen. And this isn’t a bad thing. Path integrals have proven to be a powerful source of inspiration, and a lot of actual, solid mathematics and physics has come out of trying to determine what the hell they’re supposed to mean.
Where this becomes a problem is when people take the conceptual framework as literal truth rather than as the inspirational jumping-off point it properly is.
I just got the latest issue of the Notices of the American Mathematical Society today, and was excited to find in it an article by David Vogan on the Atlas Project. In particular, there’s a lot of information on how the programming and computation ran. I found it sort of interesting, but I know there are people out there who just love tweaking programs and squeezing that extra bit of performance out. Most of the time it really doesn’t matter, but the Kazhdan-Lusztig polynomials for split are one of those few calculations that are just so unbelievably massive that you have to use every trick you can just to get the damn thing to fit in the computer.
Oh, and if any of you code junkies wants to Slashdot that article, I’d appreciate a “heard it on The UM” nod ;-)
I’ve posted my notes for Zuckerman’s third and final talk. This should get us up to “what exactly does the Atlas Project do?”
Maybe we have enough pull around here to get Jeff Adams to come up and answer that question in a talk before I’m banished from the academy, but even if not I may be able to get time to sit down with him one-on-one once I head back to the Maryland area. Anyhow, my explanations are behind the jump, as usual.
I’ve posted my notes for the first of Zuckerman’s lectures. Hopefully my handwriting isn’t too awful for you. I’ve never been very good with that pen-and-paper stuff.
I’m trying to explain this pretty comprehensibly, but I do have to use some terms most mathematicians know without defining them. I’ve got plans to get to them eventually in the main stream of my writings, but for now the exegesis sits at a middle level. Anyhow, there’s a lot to unpack here, so I’ll put it behind the jump.
Even though Andrew Wiles was speaking at the Branford College master’s tea today, I didn’t go. Zuckerman was giving the first of two or three lectures on this whole KLV thing. And I actually took notes!
Unfortunately the scanner in the computer lab was being evil, so I can’t post them quite yet. I’ll definitely have them by Monday, though, and after that I’ll try to explain what I wrote. They should be already more than readable to mathematicians, though.
What I do have is pictures of a conceptual diagram we constructed on the blackboard in his office the other day. I managed to get it mostly into three parts: 1 2 3 (~700KB each). I apologize for the quality of the middle one — I couldn’t use a flash without washing out the board entirely. It should still be readable. The third cuts off some lists of names associated with the topics they’re next to. The second list is “Jantzen, Vogan, Speh”, while in the first list “H.C.” is Harish-Chandra and “Z.” is Zuckerman.
As for what all this means, that’s what these lectures are to explain more thoroughly. Here we see the entire subject circling “characters”, which are certain functions on the groups we’re interested in. Properly defining them was the subject of today’s lecture. In the lower left is a list of examples of the sorts of groups we’re interested in — is the now-(in)famous one. To the right of the diagram is the statement that two special classes of representations, the “standard” and “irreducible” ones, are related in a certain way. On the right is the recipe for computing the irreducible representations into which the Atlas project’s computation fits.
I just got home from a long discussion with Dr. Zuckerman about this whole business. I’m not quite ready to say exactly what’s going on, but I want to correct a couple errors that I’ve made. Let it not be said that I don’t admit when I’m wrong.
Firstly, in my little added remarks about the Monster group in my “Why We Care” post, I was oversimplifying. First of all, the lattice is not the Leech lattice. The Leech lattice lives in 24-dimensional space for one thing (doh). Basically, you put together three copies of the lattice and then tweak it a bit.
Putting them together I can explain. The simplest lattice is just the integers sitting inside the real line. If you move to the plane, the points with integer coordinates sit at the corners of the squares in a checkerboard tiling of the plane. This is “adding two copies of the integer lattice”. For three copies of , we want 24-tuples of numbers so the first eight, second eight, and third eight are each the coordinates of a point in the lattice.
When you do this, it turns out there’s just enough room to squeeze in some more points to get a new lattice. That’s the Leech lattice. The Monster also isn’t quite just a group of symmetries of this lattice, so there’s still a few more steps to go, but it’s definitely related. So the connection isn’t quite as close as I’d implied, but it’s there.
The other thing is about real forms. I’d forgotten that not every choice of “realification” of the Killing form gives a Lie group, and further that not every choice that does work gives a unique Lie group.
What is true is that to every real form of a complex Lie group , there’s a largest compact subgroup . This means that its ends curve back in on themselves like the circle or the torus, and don’t run off to infinity like the line or the cylinder. Then we can “complexify” this group to get another complex group that’s really interesting to us. This group is a subgroup of , which will be important. In particular, if we take the compact real form of , its maximal compact subgroup is just itself, so its complexification is just back again.
Dr. Adams just sent me a link to an explanation of the technical details for mathematicians in other fields, but it’s still somewhat readable.
I also have been reading the slides for Dr. Vogan’s talk, The Character Table for E8, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness. There’s also an audio recording available (7MB mp3). Incidentally, I’d have gone for The Split Real Form of E8, or How We Learned to Stop Worrying and Love the Character Table, but it’s all good. This talk actually manages to be very generally accessible, and includes all sorts of pretty pictures. Those of you who wanted more visuals than I provided in my rough overview might like to check that one out.
Together, these two are my core that, together with some input from Dr. Zuckerman I’ll be trying to break down into smaller chunks. I highly advise reading at least Vogan’s slides and preferably also Adams’ notes.
I also want to respond to a comment basically asking, “so why the heck should we care about this?” It’s an excellent question, and yet another one the newspaper reports really glossed over without taking seriously. I’ll admit that I glossed it over at first too, since I think this stuff is just too elegant not to love. Still, I’ve mulled this over not just as applies to these calculations, but with regard to a lot of mathematics at this level (thus qualifying the “why we care” as a rant).
This sort of question from a non-mathematician almost always is looking for an engineering response. “What’s it good for?” means, “what can we build with it?” Honestly I have to say “not much”. Representations of Lie groups do have their uses, though, and I can point out a few things they have already been good for.
As indicated in Dr. Vogan’s slides, representations of the one-dimensional Lie groups are concerned with change through time, particularly periodic changes. This means that they’re exceptionally good at talking about periodic phenomena, like waves. Sound waves, light waves, electrical circuits, vibrating strings — they’re all one-dimensional waves. So what? So every time you use the graphic equalizer on your stereo the electronics are taking the signal and performing a fast Fourier transform on it. This turns a function on the line (Lie group) into a function on the space of all representations of the group; that’s the “unitary dual” that Dr. Adams refers to. Then you can adjust the periodic components and reconstruct a new function with much fatter bass, or whatever your tastes are.
The same sorts of things can be done in higher dimensions. Similar techniques revealed that you can’t hear the shape of a drum — there are differently-shaped membranes that have the same vibrational characteristics. What are “orbitals” of electrons around an atomic nucleus (hazy memories of chemistry)? They’re representations of the Lie group !
So what can we do with ? Nothing right now, but there’s plenty we can do (and have done) with representation theory in general.
There’s another reason (beyond the intrinsic beauty of the ideas) to work out the Atlas: more data means more patterns, and more patterns means more interrelationships between seemingly-distinct fields. Quite a few of the greatest theorems in recent years have been saying that this field of mathematics over here and that one over there are “really” the same thing. Everyone knows that Andrew Wiles solved Fermat’s Last Theorem, but what he really did was show that some things in algebraic geometry (the study of solution sets of polynomials) called “elliptic curves” are deeply related to functions with a certain sort of periodicity called “modular forms”. If, as David Corfield asserts, mathematics proceeds by “telling stories”, then each field’s stories become allegories for the other. Hard questions in one area might be translated into questions we know how to solve in the other.
So how does having a lot of data like the Atlas around help out? Because we discover a lot of these relationships from similar patterns in the data, and in many cases (though I hate to admit it) through the same numbers showing up over and over. As just one example, I present the Monstrous Moonshine conjecture. The Monster is a finite, simple group — no normal subgroups, so it can’t be broken down into even a semidirect product of smaller groups — of order (brace yourself)
That’s elements being juggled around in an intricate symmetry. People sat down and calculated its character table, very much a similar project to the current one about . And then there’s a certain special modular form called that just happens to be related to it. How so? John McKay happened to see the -function written out like this:
So? So he’d also seen the dimensions of representations of the Monster, which start with , , , and continue. Every single coefficient in the function came from dimensions of representations of the Monster! And it was conjectured that the pattern continued. In fact it did. Twenty-some years ago, Frenkel, Lepowsky, and Meurman constructed a representation of the Monster that made it clear, and their results are still echoing. One of my colleagues graduated last year and went on to Harvard by studying exactly the same sorts of connections.
And how did it start? By recognizing patterns in a mountain of raw data about representations. What unsolved problems might be translatable into representation theory by reflections found through the Atlas data? Maybe the Navier-Stokes equations, which would give a better understanding of fluid flows and aerodynamics. Maybe the Riemann hypothesis, which would lead to a better understanding of the distributions of prime numbers, which would have an impact on modern cryptography. Who knows?
Oh, and one more thing. How did someone find the Monster in the first place? Well it turns out to be a group of symmetries of a certain collection of points tiling eight-dimensional space. What collection of points? The “Leech lattice”. And you’ve already seen it: that picture of the root system in all the news reports is the basic cell, just like a square is the basic cell of a checkerboard tiling of the plane. And it all comes back around again.
How the heck can you not care about this stuff?
[EDIT: I've found out I was wrong about how the Monster relates to . More info in the link.]
There’s evidently now a wiki for the Atlas project. On one page I found some very helpful advice:
The atlas project has computed Kazhdan-Lusztig polynomials for E8. (That is, the large block of the split real form of E8). The answer consists of two files, totalling 60 Gigabytes. This is too large to download conveniently over the internet. The files have been put on a portable usb/firewire drive (never underestimate the bandwidth of a truck).
Sage advice, that parenthetical.
Okay, another thing to make clear is that there’s not just one group we could mean by . There’s one complex group, and a bunch of “real forms” of the group.
The difference between a real group and a complex group is pretty simply stated: implicitly what I’ve been talking about are real groups. Complex Lie groups are group structures on complex manifolds. That is, they “locally look like” complex -dimensional space. You may remember that the complex numbers look like a plane with the real numbers sitting inside on a line. A complex -manifold looks like a real -manifold, but there’s some extra structure floating around I’ll try to ignore. Basically it deals with how we can “scale” shapes in the manifold by imaginary amounts — how to “multiply by ” — but that’s really horribly oversimplifying.
If we’ve got the complex plane, how do we find the real numbers? You might think we can just read off which points have zero imaginary part, but this actually sort of begs the question: it assumes you already know what the real line in the complex plane is.
What we can do is think of the complex plane as a -dimensional complex manifold. Now there’s a “reflection” of the plane to itself that plays nice with the complex structure: complex conjugation, . The points that are their own conjugates make up the real line. But there’s another reflection that plays nice: . The fixed points here are the circle of radius one!
Now we can see the nonzero complex numbers as a group with multiplication as its operation. Similarly we can see the nonzero real numbers with multiplication and the circle with addition of angles as groups. These are all one-dimensional Lie groups. Each of the latter two is a real form of the first one, and together they make up all the simple real and complex commutative Lie groups.
In general, real forms work something like this. There’s a “reflection” in the complex -manifold whose fixed points form a real -manifold. The technical details of how to find these things are more than I want to go into right now, but this is the visual geometric intuition I use.
As another more interesting example, consider the group . This consists of all matrices with complex entries:
with the property that . This is a complex Lie group of dimension . It has two real forms. One you might be able to guess is , where all the entries in the matrix are real. The other is , which is a subgroup of satisfying the requirement
Both and are -dimensional real Lie groups.
Another interesting thing about them is looking for the biggest subgroup of either that can be made from the two -dimensional real groups above. You can only fit one copy of the nonzero real numbers into and no copies of the circle. On the other hand, you can fit one copy of the circle into and no copies of the nonzero reals. At the complex level, we see this in the fact that you can only fit one copy of the nonzero complex numbers into . Since these are the biggest commutative Lie groups we can find inside these groups, we say in each case that the group has “rank “. In fact, is the group . The subscript tells the rank of the group — the biggest product of copies of the nonzero complex numbers you can fit inside.
Okay, so what about ? We see that it has rank , so there’s a product of eight copies of the nonzero complex numbers sitting inside. When we break down to a real form, each of these will collapse either into a circle or a copy of the nonzero complex numbers. If each one becomes a circle, the whole real form is called “compact” and things are actually pretty fantastically well-behaved. If we collapse each to a copy of the nonzero real numbers we get the “split” real form of , and things are actually pretty fantastically evil. That’s the real Lie group that Adams’ team was working on.
[EDIT: Okay, as I've found I have to say, I've pretty drastically oversimplified things. More info in the link]