Coxeter planes

Here are a few nice Coxeter projections of representations of finite Lie algebras. If you don't know what a Coxeter projection is, or a finite Lie algebra for that matter, have a look at my PhD thesis (from which I took these pictures). Or, visit John Stembridge's website for more information and pictures.

Update: some of these pictures do not show the full representation, but only a subset.

The orbits of the two highest dominant weights of the 1764 representation of A₅.

The orbits of the two highest dominant weights of the 5985 representation of A₁₇.

The orbits of the two highest dominant weights of the 442 representation of B₆.

The adjoint representation of B₁₉.

The adjoint representation of D₁₀.

Symmetry in and of nature

In part three of "What on earth has Teake been doing for the last four years?" I'll talk a little on how symmetry appears in nature. When considering this subject, it seems natural to think of symmetric things that appear in nature. Snowflakes, like the one above, are good examples, but also flowers, the wings of butterflies, and sea stars (to name a few) all show some form of symmetry. They fall into the category "things in nature that look kind of symmetric", but in fact have little or nothing to do with the physicist's concept of the symmetry of nature.

One definition of symmetry in physics is as follows: the symmetries of nature are those transformations that do not change the laws of physics. To illustrate this concept, have a look at the following four clocks:

The first clock obviously satisfies the laws of physics. We now apply two different transformations, namely a mirror operation and a time reversal operation. Both give a clock that is distinct from the original one -- you can see the dial moving in the opposite direction. So the clock is not invariant under these transformations, and thus they are not symmetries of the clock. However, there's nothing physically wrong with a clock running in the opposite direction, apart from the fact that it's broken. Therefore parity (a fancy word for a mirror operation) and time reversal are symmetries of the laws of physics of this particular system. In general parity and time reversal are broken, but that's another story.

As a side note, you can see that the combined actions of parity (C) and time reversal (T) do leave the clock invariant, so CT is a symmetry of the clock.

This concept of symmetry is quite powerful. For instance, Special Relativity can be derived by demanding that the laws of physics are invariant under translations, rotations, and boosts (which together make the Poincaré group). Modern physics is to a large extent build on similar considerations of symmetry. So the study of group theory is not just a nice mathematical exercise; it actually has some useful applications throughout the field of physics. In the upcoming blogposts on "What on earth has Teake been doing for the last four years?" I'll try to explain how I applied some fancy group theory to even fancier things like supergravity. Stay tuned!

More symmetry: continuous groups

In my previous post on "What on earth has Teake been doing for the last four years?" I tried to explain the concepts of symmetry and groups. Today we're going on step beyond, and see how they are related to algebras. Hold tight! It's going to be a bumpy ride ...

Let's first start of with the triangle. As we saw in the previous post it only had six distinct symmetries, and these symmetries formed a group. It's a discrete group because there are only a finite number of symmetries, and thus a finite number of elements in that group.
If you want objects with bigger symmetry, all you have to is increase the number of sides of your polygon. Here's for example the pentagon:

From now on we ignore the reflection symmetries and focus only on the rotational ones. It's easy to see the pentagon has six symmetries: rotations over 0˚, 72˚, 144˚, 216˚, and 288˚ all leave it invariant.

When we move up to the hendecagon (the 11-sided regular polygon), it will come as no surprise that the thing has 12 distinct rotational symmetries. But what happens if we crank the number of sides up to infinity? Then our polygon become a circle:

You can rotate it over any angle, and it remains the same. This means it has an infinite amount of symmetry! The mathematical object that describes these symmetries is still a group, but no longer a discrete (finite) one. The symmetry group of the circle is continuous. The reason why we call it continuous is because you can smoothly get from one rotation to another one by continuously applying infinitessimal (i.e. very small) rotations. Another way of phrasing this is to say that every angle between e.g. 72˚ and 144˚ corresponds to a symmetry. This was not so for the pentagon: in that case there are 'gaps' between the rotations. That's why that kind of symmetry is called discrete.

Continuous groups are known as Lie groups (pronounced as "lee"; they're named after Sophus Lie). They contain an infinite amount of elements. But because they're continuous we can parametrize the elements in one or more parameters. For the circle we can write any rotation over an angle θ as R(θ) as

This is just the rotation matrix in two dimensions. If R(θ) is still a group element, it should satisfy the group multiplication rule: R(θ₁) • R(θ₂) = R(θ₁ + θ₂). Or in plain English: the result of two succesive rotations over angles θ₁ and θ₂ should give a new rotation over an angle θ₁ + θ₂. Sure enough, if we brush up on our linear algebra and trigonometry, we find that

So group multiplication is indeed satisfied.
The above parametrization makes for easier bookkeeping of the infinite amount of group elements. But things can be simplified even further! Because the parametrization is continuous, we can take the derivative of R(θ) with respect to θ:

The magic happens when you consider the value of dR(θ)/dθ at zero angle, θ = 0 :

which we call T, for short. This thing is independent of the angle θ. What's more, you can recover all the rotations by simply exponentiating T:

We say that T generates the symmetry group of the circle. In proper mathematical lingo, it is called a generator. This single object captures all the important properties of the infinite symmetry group (well, almost all, but we'll not go into that right now). The bookkeeping now becomes very simply: we can just focus on the generator T, instead of the infinite amount of group elements.

You can show that for more complicated groups (e.g. the symmetry group of the sphere) the above simplification also holds. All the group elements can be led back to a finite number of generators (in the generic case there is more than one generator). These generators no longer are elements of a group. Instead, they form what is known as Lie algebra. But more on that in one of the upcoming episodes of "What on earth has Teake been doing for the last four years?"!

Symmetry

This is the first post of what eventually should become the "What on earth has Teake been doing for the last four years?"-series. Brace yourself: it's about maths and physics. Run while you still can!

I'll try to keep things simple by starting of with a concept that is as mundane as it is fascinating: symmetry. Not only makes it our world round, but it’s also what makes it go round. From the perfect circular wheels on our bikes and cars that deliver an enjoyable ride, to the error-correction protocols that keep e-mails from turning into junk; it’s literally all around us.

So what is symmetry exactly? A symmetry is an action on an object that, once you’re done performing it, does not change that object. It's a somewhat abstract definition, but take for example the triangle, which has 6 symmetries.  There are two different rotations (over 120˚ and 240˚), three reflections, and finally the action of doing nothing at all (yes, that's also a symmetry). You can try them out in the following applet. Clicking on the arrows causes the triangle to rotate and reflect.

This is all pretty straightforward, right? But things start to get interesting we you keep track of the effect of the different rotations and reflections. Let's paint the corners so we can see where they end up:

One thing you'll notice is that doing twice a clockwise rotation is equal to doing one counter-clockwise rotation. The same is true for any other combination of actions -- it will always yield the net effect of one single reflection or rotation. It might also happen that the triangle ends up in the original configuration, but remember that doing nothing is also a symmetry.

The combined actions describe what mathematicians call a group. A group is a set of elements plus a rule of multiplying those elements. Let's call the set G and the multiplication rule "•". Then the precise definition of a group (in which a, b are elements of the set G) is the validity of the following four statements:

• Closure.
  The result of the operation a • b is also in G.
• Identity element.
  There exists an element 1 in G, such that for all elements a in G, the equation 1 • a = a • 1 = a holds.
• Inverse element.
  There exists an element a^-1 in G such that a • a^-1 = a^-1 • a = 1.
• Associativity.
  The equation (a • b) • c = a • (b • c) holds.
  

These four set of rules are called the group axioms. They might sound a bit abstract, but in fact, they're not. Let's have a look at our triangle again.

The symmetry actions are labeled as follows:

• 1: identity element ("doing nothing at all").
• y: counter-clockwise rotation by 120˚.
• p: clockwise rotation by 120˚.
• R: reflection in the top vertex.
• G: reflection in the lower-right vertex.
• B: reflection in the lower-left vertex.

We already noticed that the closure axiom holds. The existence of the identity element is also pretty obvious. What about the identity axiom? This indeed also holds: every action has an inverse. The reflections are their own inverse, whereas the rotations are each other's. The last thing to check is associativity. It's a bit harder to verify, but believe me, it holds too.

Just for completeness sake, here's one last version of the triangle applet. This one includes the group multiplication table, which keeps track of what happens when you first do the action in the first row, followed by the action in the first column. (If you didn't believe me on the validity of associativity you can use this table to check it.)

What I've shown you so far is that the symmetries of the triangle can be described in terms of the mathematical concept of a group. The importance of group theory lies in the fact that any symmetry you can think of can be described as a group, and that conversely all groups describe a symmetry.

By now the answer to the question "What on earth has Teake been doing for the last four years?" will hardly come as a surprise: it's group theory. More on that in part two of this series!