The n-Category Café

I sometimes like to think that vector spaces are “more quantum” than sets. Whether or not that’s a good notion, certainly it does allow more compression in the representation, which makes them better than permutation representations. There are groups for which the linear representations are not as good as they could be.

Let’s say a permutation group is a group equipped with a small faithful permutation representation. “Small” is not a technical term. Any group can be made into a permutation group if you allow the representation to be big. I don’t want to call those “permutation groups”. I might also say that a linear group is a group equipped with a small linear representation. Perhaps “small” should mean that it is efficient to calculate with that representation?

The symmetric and alternating groups are definitely permutation groups. The Mathieu groups are a nice family of exceptional permutation groups. These permutation representations are very small: the group itself has order the factorial (!) of the representation. In other words, the group is of size comparable to the full symmetric group.

Every Lie group comes with a preferred linear representation, namely its adjoint representation. Every nontrivial representation of a simple group is faithful. So every simple Lie group is a linear group, if you admit the adjoint representation. The classical groups, as mentioned already, have a better, smaller linear representation, whose dimension grows as the square root of the adjoint. In other words, the group is of size comparable to the full matrix group.

I tend to count $G_2$ also as a linear group. Whether $F_4, \dots$ count or not is a bit of a grey-area question. Here is another way to measure the size of a (real, say) representation. Look at $V : G \to \mathrm{O}(n)$, and look at the induced map on, say, $\pi_3$, which is a map $\mathbb{Z} \to \mathbb{Z}$ if $G$ is simple and $n \geq 5$. The cokernel of this map measures something about the size. This is called the Dynkin index of the representation. The classical groups and $G_2$ are precisely the groups with a representation $V$ of Dynkin index $1$.

I don’t count $E_8$ as a linear group. Its smallest vector representation is its adjoint, of Dynkin index $30$, which is somewhat large.

The finite simple groups of Lie type have linear representations over finite fields (debatable for exceptional types).

The sporadic finite simple groups come in four families. The Pariahs have no real pattern to them. The First Generation are the Mathieu groups. These, as already mentioned, are permutation. The Second Generation are linear groups. A finite linear group always preserves a lattice. The Second Generation preserves the Leech Lattice.

The Third Generation are not linear groups. The Monster group is the most extreme. It is not of particularly large order (in spite of its reputation): it is smaller than $S_{24}$ or $\mathrm{GL}_{24}(\mathbb{F}_2)$. What makes the Monster large is the relative size of its smallest permutation and linear representations compared to its own order. On the other hand, the Monster does have a “very small” representation: it acts on a VOA of $c=24$.

There is a fully faithful inclusion of groupoids $\{\text{real vector spaces}\} \to \{\text{VOAs}\}$ that sends a vector space $\mathbb{R}^n$ to a certain VOA of $c=n/2$ (called the “free fermion model”); there is also a complex version that sends $\mathbb{C}^n$ to a $\mathbb{Z}$-graded VOA of $c=n$. So the central charge is a measure of “dimension”.

But VOAs allow compression already for Lie groups. The smallest VOA representation of a Lie group is of central charge comparable to the rank of the Lie group (with equality in the simply laced case). For the classical groups, the rank is proportional to the dimension of the defining representation, and indeed this smallest VOA representation is essentially the one coming from the linear representation.

Some VOA representations arise from taking a lattice (i.e. linear) representation and “second quantizing” it. I call this “second quantization” because the lattice itself sometimes comes from quantizing a permutation representation. (I am aware that this is not quite the standard usage of the term “second quantization”, but it is of the same flavour.)

Igor Frenkel dreams that every group has a preferred smallest VOA representation, which is always “small”. His dream moreover goes: the Dynkin-Cartan approach to using the adjoint representation to combinatorially classify Lie groups should extend to more general VOA representations.

Just a comment on Tom’s comment–another striking fact is how close classical physics comes to quantum physics without actually being the same thing. Mathematically, this is reflected in the fact that studying the action of a group on its coadjoint orbits (which are symplectic manifolds) is in some sense a “near miss” to the idea of studying the action of a group on vector spaces. A challenge for the “categorical machine” you are imagining would be to ask if it can also detect the existence of this “near miss.”

Tom wrote:

So I’ve sometimes wondered whether there’s some actual theorem attesting to the special role of the category of vector spaces relative to the category of groups. It might say something like: there is a categorical machine which when fed as input the category of groups produces as output the category of vector spaces, or perhaps the category of pairs consisting of a group and a linear representation of it.

Do you have any thoughts on this? It seems to me that your theorem is somewhat in this direction.

Yes, I have a lot of thoughts on this. As others have pointed out, the fact that the universe is fundamentally quantum-mechanical is a hint that not only human mathematicians, but physical reality itself, is fond of vector spaces. The state of the world is not a point in a mere set: it’s more like a vector in a vector space! There are caveats here, but never mind: I’m talking about the

$$\frac{1}{\sqrt{2}} \Big(\text{live cat} + \text{dead cat}\Big)$$

stuff that freaked out Schrödinger and everyone since.

Mathematically, one reason quantum mechanics works well is that groups act better on vector spaces that on sets. There are lots of manifestations of this, but one is that it’s easier to “continuously turn around” a vector space than a set. This is particularly true of complex vector spaces: the circle is also the group $\mathrm{U}(1)$ of unitary $1 \times 1$ matrices, so you can turn around a 1-dimensional complex vector space with linear transformations. Such rotation symmetries play a fundamental role throughout math and physics.

I spent a lot of time trying to free quantum mechanics from the domination of complex vector spaces, as explained in From finite sets to Feynman diagram with James Dolan and later HDA7: Groupoidification with Alex Hoffnung and Christopher Walker, but I couldn’t figure out a good substitute for $\mathrm{U}(1)$. In Categorified algebra and quantum mechanics, my student Jeffrey Morton bit the bullet and introduced ‘phased sets’ and ‘phased groupoids’ to blend the advantages of $\mathrm{U}(1)$ and groupoids.

I’ve continued thinking about this, and most recently I’ve gotten interested in what the Brauer induction theorem implies about it all. I think there’s a lot left to discover about how groups and vector spaces—particularly complex vector spaces—are connected.