In Random Points on a Sphere (Part 1), we learned an interesting fact. You can take the unit sphere in , randomly choose two points on it, and compute their distance. This gives a random variable, whose moments you can calculate.
And now the interesting part: when n = 1, 2 or 4, and seemingly in no other cases, all the even moments are integers.
These are the dimensions in which the spheres are groups. We can prove that the even moments are integers because they are differences of dimensions of certain representations of these groups. Rogier Brussee and Allen Knutson pointed out that if we want to broaden our line of investigation, we can look at other groups. So that’s what I’ll do today.
If we take a representation of a compact Lie group we get a map from group into a space of square matrices. Since there is a standard metric on any space of square matrices, this lets us define the distance between two points on the group. This is different than the distance defined using the shortest geodesic in the group: instead, we’re taking a straight-line path in the larger space of matrices.
If we randomly choose two points on the group, we get a random variable, namely the distance between them. We can compute the moments of this random variable, and today I’ll prove that the even moments are all integers.
So, we get a sequence of integers from any representation of any compact Lie group So far we’ve only studied groups that are spheres:
• The defining representation of on the real numbers gives the powers of 2.
• The defining representation of on the complex numbers gives the central binomial coefficients
• The defining representation of on the quaternions gives the Catalan numbers.
It could be fun to work out these sequences for other examples. Our proof that the even moments are integers will give a way to calculate these sequences, not by doing integrals over the group, but by counting certain ‘random walks in the Weyl chamber’ of the group. Unfortunately, we need to count walks in a certain weighted way that makes things a bit tricky for me.
But let’s see why the even moments are integers!
If our group representation is real or quaternionic, we can either turn it into a complex representation or adapt my argument below. So, let’s do the complex case.
Let be a compact Lie group with a unitary representation on This means we have a smooth map
where is the algebra of complex matrices, such that
where is the conjugate transpose of the matrix
To define a distance between points on we’ll give its metric
This clearly makes into a -dimensional Euclidean space. But a better way to think about this metric is that it comes from the norm
where is the trace, or sum of the diagonal entries. We have
I want to think about the distance between two randomly chosen points in the group, where ‘randomly chosen’ means with respect to normalized Haar measure: the unique translation-invariant probability Borel measure on the group. But because this measure and also the distance function are translation-invariant, we can equally well think about the distance between the identity 1 and one randomly chosen point in the group. So let’s work out this distance!
I really mean the distance between and so let’s compute that. Actually its square will be nicer, which is why we only consider even moments. We have
Now, any representation of has a character
and characters have many nice properties. So, we should rewrite the distance between and the identity using characters. We have our representation whose character can be seen lurking in the formula we saw:
But there’s another representation lurking here, the dual
This is a fairly lowbrow way of defining the dual representation, good only for unitary representations on but it works well for us here, because it lets us instantly see
This is useful because it lets us write our distance squared
in terms of characters:
So, the distance squared is an integral linear combination of characters. (The constant function 1 is the character of the 1-dimensional trivial representation.)
And this does the job: it shows that all the even moments of our distance squared function are integers!
Why? Because of these two facts:
1) If you take an integral linear combination of characters, and raise it to a power, you get another integral linear combination of characters.
2) If you take an integral linear combination of characters, and integrate it over you get an integer.
I feel like explaining these facts a bit further, because they’re part of a very beautiful branch of math, called character theory, which every mathematician should know. So here’s a quick intro to character theory for beginners. It’s not as elegant as I could make it; it’s not as simple as I could make it: I’ll try to strike a balance here.
There’s an abelian group consisting of formal differences of isomorphism classes of representations of , mod the relation
Elements of are called virtual representations of Unlike actual representations we can subtract them. We can also add them, and the above formula relates addition in to direct sums of representations.
We can also multiply them, by saying
and decreeing that multiplication distributes over addition and subtraction. This makes into a ring, called the representation ring of
There’s a map
where is the ring of continuous complex-valued functions on This map sends each finite-dimensional representation to its character This map is one-to-one because we know a representation up to isomorphism if we know its character. This map is also a ring homomorphism, since
These facts are easy to check directly.
We can integrate continuous complex-valued functions on so we get a map
The first non-obvious fact in character theory is that we can compute inner products of characters as follows:
where the expression at right is the dimension of the space of ‘intertwining operators’, or morphisms of representations, between the representation and the representation
What matters most for us now is that this inner product is an integer. In particular, if is the character of any representation,
is an integer because we can take to be the trivial representation in the previous formula, giving
Thus, the map
actually takes values in
Now, our distance squared function
is actually the image under of an element of the representation ring, namely
So the same is true for any of its powers—and when we integrate any of these powers we get an integer!
This stuff may seem abstract, but if you’re good at tensoring representations of some group, like you should be able to use it to compute the even moments of the distance function on this group more efficiently than using the brute-force direct approach. Instead of complicated integrals we wind up doing combinatorics.
I would like to know what sequence of integers we get for A much easier, less thrilling but still interesting example is This is the 3-dimensional real projective space which we can think of as embedded in the 9-dimensional space of real matrices. It’s sort of cool that I could now work out the even moments of the distance function on this space by hand! But I haven’t done it yet.