Week 6 - Representation Rings

Hello and welcome back to my blog! Last week, I briefly touched on group representations. I omitted the definition of tensor products, as it’s too technical for this blog. However, I’ve realized that at least the multiplication tables of tensor products are essential. Throughout this week, I’ve also realized that these tables are surprisingly interesting to compute, so I’ll focus on them today. I’ll first review and introduce some notions from linear algebra and group theory.

Linear Algebra Concepts

Recall that matrices are arrays of numbers that can be interpreted as “linear maps” of vector spaces.

Furthermore, eigenvalues of a matrix A are scalars λ such that for some nonzero vector v, the output Av is equal to the scalar multiple λv. This essentially means that the matrix A has the same effect as scalar multiplication for any nonzero vector. In fact, another notion arises from this: the multiplicity of an eigenvalue is the dimension of the space of valid vectors v.

I’ve discussed eigenvalues before, in the context of calculating Frobenius-Perron dimension. Here, their use is much simpler and more intuitive: the trace of a matrix is the sum of its eigenvalues, with multiplicity.

One notable property about the trace is that it is preserved by conjugation. Conjugation of some matrix A by some other matrix B outputs the product BAB⁻¹. Through direct computation, one can show that this has the same trace as A. (If you want to try to prove this yourself, it is significantly easier to instead define the trace as the sum of the diagonal entries. Due to this result, the definitions are equivalent.)

Finally, if a matrix A has an eigenvalue λ, then any power Aⁿ has an eigenvalue λⁿ. You can prove this by directly showing that if v is the eigenvector corresponding to λ, then Aⁿv = λⁿv. I recommend this as an exercise if you’re interested.

Group Theory Basics

As a reminder, a group is essentially a multiplicative structure which can capture a wide variety of mathematical concepts. Groups have the exact same notion of conjugation: the conjugation of some element g by h is hgh⁻¹. We can also divide a group into its conjugacy classes: two elements g and i are “equivalent,” and therefore in the same class, if i is the conjugate of g by some h.

Additionally, while groups often represent concrete mathematical ideas, in representation theory it’s usually easier to consider group presentations. These are ways to express the group in terms of elements that generate it, as well as relations between these generators.

That may be confusing, so let’s consider an example. The group Dₙ, also called the dihedral group, is the group of symmetries of a regular n-gon. These symmetries can all be expressed as some combination of rotations and reflections. Let r be a counter-clockwise rotation by 360/n degrees. Then the rest of the rotations can be expressed as r², r³, etc. We also pick some axis of symmetry (arbitrarily), and denote reflection across it by s.

There are two clear ways in which we can get back to where we started. Rotating n times returns us to the original polygon, as does reflecting twice. Therefore, rⁿ and s² should be the identity symmetry, which is just not changing anything. We can find one more relation. Since reflecting flips the orientation of rotations, if we rotate once, reflect, and then rotate again, the two rotations cancel each other out. Therefore, rsr=s.

These are the only three relations, so they fully determine the group.

A brief clarification

When I say that r and s generate the group, you might think I mean that every element in the group is of the form rᵃsᵇ for some a and b. After all, that’s exactly what it would mean in a vector space. However, generate just means that the elements can be expressed as any multiplication of r’s and s’s. Vector addition is commutative, which is why we rearrange them into the conventional format. However, group multiplication usually isn’t commutative, so we can’t reorder it.

Regardless, our relations are enough. For example, if we have an element sr, by the relations we can rewrite it as rⁿ⁻¹s.

Some dihedral group calculations

First, the dihedral group has exactly 2n elements, which are of the form rᵃsᵇ where a can be anything at least 0 but less than n, and b is either 0 or 1.

By playing around with the group relations for some time, we can also derive that the number of separate conjugacy classes is (n+3)/2 if n is odd, and n/2+3 if n is even. These calculations aren’t too interesting and aren’t in the spirit of what I want to discuss today, so I’ll exclude them.

Another fundamental group theory concept is the commutator subgroup. It is basically the set of ghg⁻¹h⁻¹ across all pairs of group elements g and h. We’ll see later why this is important; for now, by a quicker but still not too interesting calculation, it consists of the even powers of r. For odd n, this includes all rotations, so it has size n. For even n, it only includes half of them, so it has size n/2.

Character Theory

Last week, I discussed how to take the direct sum of two representations — by taking the direct sum of the matrices each group element is sent to. Just as with simplicity of objects in fusion categories, a representation is irreducible if it can’t be expressed as the sum of two representations.

Before we transition to character theory, there are two useful facts about irreducible representations that don’t rely on characters.

• The number of irreducible representations equals the number of conjugacy classes of the group.

• The squares of the dimensions of the irreducible representations add to the size of the group.

Character theory, at least as we’ll see it in this project, is about the traces of irreducible representations. In particular, the character of a representation is a map that assigns each element to a complex number: the trace of the matrix from the representation.

Because matrix conjugation preserves trace, group elements in the same conjugacy class correspond to the same complex number. For this reason, instead of saying a character is a function from the group to the complex numbers, we often say it is a function from the conjugacy classes.

There is a specialized dot product between characters. Let z* denote the complex conjugate of some number z. Then, the dot product of characters χ₁ and χ₂ is the average of χ₁(g)*χ₂(g) across all group elements g.

Character tables: some facts

Character tables record the characters of each irreducible representation. Below, I’ll list many important and interesting facts about them, with which we can compute a few examples. I have actually proved all of these as exercises and they’re quite intriguing, but today’s post is already quite long — if you would like to see them or if I have space, I’ll try to include them in later posts.

• The dot product of an irreducible character with itself is 1.

• The dot product of two different irreducible characters is zero. (These conditions state that the rows are orthonormal.)

• Define a similar dot product of columns: for two group elements g and h, it is the sum over all irreducible characters χᵢ of χᵢ(g)*χᵢ(h). If g and h are in the same conjugacy class, this is the size of the group divided by the size of the shared conjugacy class.

• If g and h are in different conjugacy classes above, then the dot product is zero. (This gives an orthogonality condition on the columns as well).

• For a character χ and element g, χ(g)*=χ(g⁻¹).

• For a character χ of dimension n and element g such that gᵐ is the identity, χ(g) decomposes as the sum of n complex numbers of the form z, where zᵐ=1.

Computing the D_3 table

Using these facts, we can compute the character table of many small-sized groups. Let’s do the dihedral group D₃ as an example. It has three conjugacy classes: the identity {1}, as well as {r, r²} and {s, rs, r²s}.

Then there are three irreducible representations. The squares of their dimensions sum to 6, so they must have dimensions 1, 1, and 2 in some order.

Every group has the trivial representation: just assign each element to 1. This character also sends each conjugacy class to 1.

There is one more 1-dimensional representation. Suppose it sends r to some number z. Then it sends r² to z². Because r and r² are in the same conjugacy class, they have the same character. Then z=z², so it’s 1. All 1-dimensional representations also send the group identity to 1.

Now, let’s look at the dot product of this new representation with the trivial one. If the last conjugacy class is sent to z, then the dot product is 1/6 (1+1+1+z+z+z). Since this must be zero, z=-1.

The third representation still must send the group identity to the matrix identity. In dimension 2, this has trace 2 as well. Let the third character send the other conjugacy classes to a and b, respectively. By taking the dot product with each of the one-dimensional representations, we get that 2+2a+3b=0 and 2+2a-3b=0. This is a system of equations, and solving, we find that a=-1 and b=0.

Thus, the character table has rows {1,1,1}, {1,1,-1}, and {2,-1,0}.

A general statement about dihedral groups

In fact, irreducible representations of any dihedral group have dimension one or two.

To prove this, we need another result: the number of dimension one irreducible representations equals the size of the group divided by the size of the commutator subgroup. Referring back to the calculations section, there are 2 1-dimensional irreducible representations if n is odd, and 4 if it is even.

Also using the number of conjugacy classes from the calculations section, there are (n-1)/2 representations left for odd n, and n/2-1 left for even n. The squares of their dimensions are at least four, and they must add to 2n-2 or 2n-4 respectively, as that’s the size of the group minus the contribution from the 1-dimensional representatoins.

This can only happen if all the dimensions are exactly two, which achieves the desired result.

Computing multiplication tables

This post has been quite long, so we seem to be far from the original goal: to compute multiplication tables of the tensor product without actually discussing the tensor product. However, this goal follows quite directly from character tables. I learned about this really clean connection from Section 4.9 of [1], which is also a great source for character theory.

Let Vᵢ denote the representation corresponding to χᵢ. Because these irreducible representations generate the rest, we know that the tensor product Vᵢ ⊗ Vⱼ is a linear combination of them. The characters tell us what the coefficients are. In particular, the coefficient of Vₖ is (χᵢχⱼ)•χₖ. Here, the first part is normal multiplication (i.e. multiply the outputs), and then we take a dot product.

I’ll omit the actual calculation here because it’s just direct arithmetic, but this allows us to exactly grasp what the representation ring’s structure is — at least, when the groups are small enough that the character tables are actually feasible to compute by hand (or with programmed assistance). However, as we’ll soon see, even if we know exactly what the ring looks like, determining if a fusion category is group-theoretical is an extremely technical and difficult task. Stay tuned as I try to navigate this across the next few weeks!

References

[1] Pavel Etingof, Slava Gerovitch, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwender, Dmitry Vaintrob, Elena Yudovina. Introduction to Representation Theory. Student Mathematical Library (2011)