Tannaka Reconstruction and the Monoid of Matrices
Posted by John Baez
You can classify representations of simple Lie groups using Dynkin diagrams, but you can also classify representations of ‘classical’ Lie groups using Young diagrams. Hermann Weyl wrote a whole book on this, The Classical Groups.
This approach is often treated as a bit outdated, since it doesn’t apply to all the simple Lie groups: it leaves out the so-called ‘exceptional’ groups. But what makes a group ‘classical’?
There’s no precise definition, but a classical group always has an obvious representation, you can get other representations by doing obvious things to this obvious one, and it turns out you can get all the representations this way.
For a long time I’ve been hoping to bring these ideas up to date using category theory. I had a bunch of conjectures, but I wasn’t able to prove any of them. Now Todd Trimble and I have made progress:
- John Baez and Todd Trimble, Tannaka reconstruction and the monoid of matrices.
We tackle something even more classical than the classical groups: the monoid of matrices, with matrix multiplication as its monoid operation.
The monoid of matrices has an obvious -dimensional representation, and you can get all its representations from this one by operations that you can apply to any representation. So its category of representations is generated by this one obvious representation, in some sense. And it’s almost freely generated: there’s just one special relation. What’s that, you ask? It’s a relation saying the obvious representation is -dimensional!
That’s the basic idea. We need to make it more precise. We do it using the theory of 2-rigs, where for us a 2-rig is a symmetric monoidal linear category that is Cauchy complete. All the operations you can apply to any representation of a monoid are packed into this jargon.
Let’s write for the monoid of matrices over a field , and for its 2-rig of representations. Then we want to say something like: is the free 2-rig on an object of dimension . That’s the kind of result I’ve been dreaming of.
To get this to be true, though, we need to say what kind of representations we’re talking about! Clearly we want finite-dimensional ones. But we need to be careful: we should only take finite-dimensional algebraic representations. Those are representations where the matrix entries of are polynomials in the matrix entries of . Otherwise, even the monoid of matrices gets lots of 1-dimensional representations coming from automorphisms of the field . Classifying those is a job for Galois theorists, not representation theorists.
So, we define to be the category of algebraic representations of the monoid , and we want to say is the free 2-rig on an object of dimension . But we need to say what it means for an object of a 2-rig to have dimension .
The definition that works is to demand that the st exterior power of should vanish:
But this is true for any vector space of dimension less than or equal to . So in our paper we say has subdimension when this holds. (There’s another stronger condition for having dimension exactly , but interestingly this is not what we want here. You’ll see why shortly.)
So here’s the theorem we prove, with all the fine print filled in:
Theorem. Suppose is a field of characteristic zero and let be the 2-rig of algebraic representations of the monoid . Then the representation of on by matrix multiplication has subdimension . Moreover, is the free 2-rig on an object of subdimension . In other words, suppose is any 2-rig containing an object of subdimension . Then there is a map of 2-rigs,
unique up to natural isomorphism, such that .
Or, in simple catchy terms: is the walking monoid with a representation of subdimension .
To prove this theorem we need to deploy some concepts.
First, the fact that we’re talking about algebraic representations means that we’re not really treating as a bare monoid (a monoid in the category of sets). Instead, we’re treating it as a monoid in the category of affine schemes. But monoids in affine schemes are equivalent to commutative bialgebras, and this is often a more practical way of working with them.
Second, we need to use Tannaka reconstruction. This tells you how to reconstruct a commutative bialgebra from a 2-rig (which is secretly its 2-rig of representations) together with a faithful 2-rig map to (which secretly sends any representation to its underlying vector space).
We want to apply this to the free 2-rig on an object of subdimension . Luckily because of this universal property it automatically gets a 2-rig map to sending to . So we just have to show this map is faithful, apply Tannaka reconstruction, and get out the commutative bialgebra corresponding to !
Well, I say ‘just’, but it takes some real work. It turns out to be useful to bring in the free 2-rig on one object. The reason is that we studied the free 2-rig on one object in two previous papers, so we know a lot about it:
John Baez, Joe Moeller and Todd Trimble, Schur functors and categorified plethysm.
John Baez, Joe Moeller and Todd Trimble, 2-Rig extensions and the splitting principle.
We can use this knowledge if we think of the free 2-rig on an object of subdimension as a quotient of the free 2-rig on one object by a ‘2-ideal’. To do this, we need to develop the theory of ‘2-ideals’. But that’s good anyway — it will be useful for many other things.
So that’s the basic plan of the paper. It was really great working with Todd on this, taking a rough conjecture and building all the machinery necessary to make it precise and prove it.
What about representations of classical groups like , the orthogonal and symplectic groups, and so on? At the end of the paper we state a bunch of conjectures about these. Here’s the simplest one:
Theorem. Suppose is a field of characteristic zero and let be the 2-rig of algebraic representations of Then the representation of on by matrix multiplication has dimension , meaning its th exterior power has an inverse with respect to tensor product. Moreover, is the free 2-rig on an object of dimension .
This ‘inverse with respect to tensor product’ stuff is an abstract way of saying that the determinant representation of has an inverse, namely the representation .
It will take new techniques to prove this. I look forward to people tackling this and our other conjectures. Categorified rig theory can shed new light on group representation theory, bringing Weyl’s beautiful ideas forward into the 21st century.
Re: Tannaka Reconstruction and the Monoid of Matrices
I like the theorem!
Here’s a thought. It’s a striking fact that to understand groups, studying their actions on vector spaces is by far the most effective strategy, compared to studying their actions on sets, or rings, or measure spaces, or anything else. That’s not to say that you get no information from studying group actions on other objects, but the success of linear representation theory is the star of the show. So much so that we just say “representation theory” without even specifying that we’re representing groups in the category of vector spaces rather than somewhere else. It’s taken for granted.
I call this a “fact”, but of course it’s just a social observation, and who’s to say that things might not be different in a century’s time if human beings are still around. 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.