What does symmetry have to do with physics?
As my ultimate goal is talk about physics, it's time for a paragraph on that subject, just so we don't lose sight of where we're going. So imagine a game of pool. Pool is a game of physics, the outcome of any play depends on factors such as friction, collision angles, the coefficient of restitution of the balls (ie. their bounciness) and a whole host of other factors. But despite the complexity of the factors involved, we know that they are invariant under rotation. What I mean is that if we were to rotate the table and everything on it by some random angle (eg. from a North-South orientation to a NW-SE orientation) it makes no difference to the game. (Well, maybe the floor has a slight tilt, so let's rotate the room as well if that's the case.) If everything is rotated together then you end up with a situation that's essentially identical. One of the fundamental principles of physics is that the laws of physics don't care about the absolute orientation of anything. We called the set of rotations in a plane SO(2) so we can summarise this by saying that pool has SO(2) symmetry. But does it have SO(3) symmetry? Obviously rotating the table so that it tilts will change the way the game plays out. But that's because of an 'accident' of history, we just happen to be on the surface of the Earth and there's a gravitational field directed downwards. But from the point of view of fundamental physics there's nothing special about 'down'. If we were to tilt the pool table and then tilt the entire Earth underneath it, we should still expect to see no change to the physics of pool. In fact, we get to try this experiment out every day because the entire pool/Earth system rotates through (approximately) 360° every day and we don't notice any change in the way pool tables work. So pool actually has SO(3) symmetry, but because of stuff going on in our neighbourhood (ie. a having a planet under our feet), it looks like it only has SO(2) symmetry. SO(3) is a fundamental symmetry of physics, but that fact that on Earth gravity messes this up and leaves us with SO(2) symmetry is a phenomenon known as symmetry breaking. More of this later, back to the mathematics.
Lie Group Representations
I've talked about how groups can act on spaces. For example SO(3) acts on 3D space and SO(2) acts on 2D spaces, ie. planes. But there's some flexibility here and we can decouple the group from the space it acts on. The way SO(2) acts on a plane is given by a certain rule. We can make up new rules and study those. For example, we could allow SO(2) to act on 3D space by this rule: rotate your x and y coordinates using the same rule as in the plane, and leave z unchanged. So if a certain 2D rotation (ie. a 90 degree rotation) maps the point (x,y) to (y,-x), then according to this new rule it maps (x,y,z) to (y,-x,z). We could make up other rules. For example we could apply SO(2) to y and z, and leave x untouched. In fact, we can make infinitely many rules like this because there are infinitely many axes (by axis I just mean direction) we could choose to leave untouched. We can define a representation to be a rule that takes an element of a group and turns it into a transformation on a space. We're interested mainly in what are called linear representations (I'll give some justification for this in part 3). These are representations where the transformations map the origin to the origin, map straight lines to straight lines, and map the midpoints between pairs of points to the midpoints between pairs of points. It's not hard to see that rotations do all three of these things. In fact, as I will only talk about linear representations, I'll drop the word linear from now on.
I just showed how SO(2) has multiple representations because of the different ways in can be applied to 3D space. But we can also define alternative rules for how to apply SO(2) to the same space, eg. the plane. Here's a really simple one called the trivial representation: we simply say that elements of SO(2) do nothing. It's not very interesting, but it is a perfectly valid representation. Here's another: apply elements of SO(2) backwards. So if we have an element of SO(2) that says "rotate by 10° clockwise" the backwards rule says "apply a 10 degree rotation anticlockwise". It's a different representation. But if you look at the plane from underneath, it actually looks just like the original representation. So even though these are different representations, there's a sense in which it's equivalent to the original one.
But here's a representation of SO(2) that really is different to the original one: If the element of SO(2) says "rotate by x°" we rotate by 2x° instead. We can think of it as a double-speed rule. We apply the rotations in SO(2) twice. If we run through the elements of SO(2) starting at 0° and working our way up to 360°, then the representation rotates our space twice as fast and ends up rotating the space twice.
What about a half-speed rule? Sounds like it might work. But it fails for a simple reason. A 360 degree rotation is the same as a zero degree rotation. But the half-speed rule says that the former should rotate by 180° and the latter should rotate by 0°. These are distinct rotations, so our rule doesn't make any sense. As a result we're restricted to rules that are n-speed rules, where n is an integer. It's not hard to see that if we choose n to be a negative integer then by looking at the plane from underneath we get the same rule as using -n, looking from above. So we can discard the ones corresponding to negative integers. And we're left with one rule for each non-negative integer. In fact, these are all the representations that are possible on a 2D plane.
[Optional "advanced section": Now go back to 3D again. When we apply SO(2) to 3D space we find that SO(2) always leaves some axis fixed. So given a 3D representation of SO(2), our 3D space always splits into a 1D space that's left unchanged, and a 2D space. You can think of SO(2) as using the trivial rule on the 1D space, and using any of the rules in the previous paragraph on the 2D space. Something similar happens in any dimension of space. The representation of SO(2) will split up into pieces with some axes left untouched, and others, always grouped in pairs, that transform like in the previous paragraph. In other words, the representations of SO(2) can be broken down into fundmental building blocks which we call irreducible representations. In the previous paragraph I actually classified all of the irreducible representations of SO(2).
We can use the properties of SO(2) to understand other Lie groups. The idea is that not only is SO(2) frequently a subgroup of Lie groups, it's often a subgroup in multiple ways, ie. there are multiple ways to find a copy of SO(2) in SO(3). We've already seen that SO(2) can be embedded in SO(3) by interpreting rotations around any fixed axis as elements of a sub-SO(2). But notice how if we find two different embeddings of SO(2), they are forced to "interfere" with each other. For example, let's choose an SO(2) that acts on the x-y plane as described above. Suppose we now pick another SO(2) subgroup. No matter how we choose it, it must act on either x or y. There simply isn't room to find two SO(2)'s that don't at some point overlap with each other. But imagine the group of 4-dimensional rotations, caled SO(4). (It's not that scary, it's just like SO(3) except that we can make rotations that "mix up" any pair of directions, or combinations of such rotations.) We could pick one SO(2) that acts on the x-y plane and another that acts on the z-t plane (assuming the fourth axis is called 't'). But we won't be able to pick more than two for the same reason as before. Suppose we pick as many SO(2)'s as are possible. Then what we have is whats known as a maximal torus.
Here's a feeble attempt at drawing all of this. One of the SO(2)'s inside SO(4) rotates one pair of axes into each other, the other one rotates the other pair:
Now think about representations of a Lie group. These are a rule that tell you how each element in the group transforms our space. As the elements of our SO(2) subspaces obviously live in the group the rule must apply to these also. So a representation on a group also gives a representation of all of the SO(2)'s in it. So each SO(2) making up our maximal torus must basicaly act like we described above: by rotating some pairs of axes around at some "speed" and leaving other axes untouched. So if we pick a maximal torus of a Lie group, any representation splits up into a bunch of pairs of axes and each pair has an integer "speed" associated to each SO(2) in the maximal torus. The tuple of "speeds" for each pair of axes is known as a weight. By interpreting the weights as the coordinates of points, the collection of weights that arise from any particular representation can be drawn in a diagram. The dimension of the diagram is the number of non-interfering SO(2)'s in the maximal torus. And if you want to see what these diagrams look like, Garrett Lisi has some nice ones in his paper. Note that that paper also contains some "root diagrams". I don't have time to talk about those except to say that (1) they are weight diagrams for one particular special representation and (2) they tell you a lot about the geometry of all possible sets of weights for a particular Lie group.
One last thing for the "advanced" section: a similar analysis can be carried out for Lie algebras as opposed to Lie groups. When physicists draw diagrams of weights they are often talking about the weights of Lie algebras, but these things are intimately related.
There's probably one important message to take from this section: there are a lot of constraints on representations, you can't just make up any old rule. So just knowing that a given Lie group acts on some space you already know a lot of information, even if you don't know what the exact representation is. BTW One of the biggest applications of representation theory is in chemistry where you can read off information about the number of electrons allowed in atomic orbitals directly from representations of SO(3).]
That was pretty tough. Next time I'll talk about physics and it should get a bit easier. I'll explain why much of modern physics is the study of Lie group representations and I'l explain the 'exceptional' and 'simple' in the title of Garrett's paper.
(And I apologise for all of my sins of omission. For example, the analysis above only works for certain types of Lie group and really everything should be done using complex numbers, not real numbers. But I'm trying to compress a few hundred pages of mathematics into a single posting.)