### Quivering with Excitement

#### Posted by John Baez

Over on Google+, David Roberts just told me the most exciting theorem I’ve heard all week. *Every projective variety is the Grassmannian of a quiver representation!* I suppose it’s just another indication of the ‘wildness’ of quiver representations once we leave the safe waters of Gabriel’s theorem.

Let me explain….

To briefly recall: a **quiver** is a category $Q$ freely generated by a finite directed graph. A **quiver representation** is a functor $F: Q \to FinVect$. In other words, it’s just a finite-dimensional vector space for each vertex and a linear operator for each edge. A **morphism of quiver representations** is a natural transformation between functors of this sort. Again, this amounts to something pathetically simple.

The first surprise:

**Gabriel’s Theorem.** A quiver has finitely many isomorphism classes of indecomposable representations iff its underlying undirected graph is a Dynkin diagram of type A, D, or E.

It’s fun to work out examples of this theorem and delve deeper into the relation between quivers and Lie theory. It’s the tip of an iceberg that mathematicians plan to mine until they hit bottom and sea water rushes in and drowns them all.

But now:

Given a quiver $Q$ and representations $F, G$, define the **Grassmanian** $Gr(Q,F,G)$ to be the set of monomorphisms from $G$ to $F$, modulo the action of automorphisms of G. (We can compose a mono $f: G \to F$ with an automorphism $g: G \to G$ to get a new mono $f g: G \to F$.)

To convince yourself that this deserves to be called a Grassmannian, look at the case where $Q$ comes from the graph with a single vertex and no edges. Then a representation of $Q$ is a vector space, a morphism of representations is a linear map, and a Grassmannian consists of all $n$-dimensional subspaces of an $m$-dimensional vector space.

**Lemma.** For any quiver, the Grassmannian $Gr(Q,F,G)$ can be made into a projective variety in some systematic way.

**Reineke’s Theorem.** Every projective variety arises this way for some quiver with at most 3 vertices.

The title of Reineke’s paper is Every projective variety is a quiver Grassmannian. The abstract says merely “The theorem is proved.” The paper is 2 pages long. This is a man who doesn’t beat around the bush.

For more, try Lieven Lebruyn’s blog article Quiver representations can be anything.

Despite the exciting nature of this result, I have no idea what it might be ‘good for’. Is it fun or helpful to classify projective varieties in terms of the quivers that give them?

## Re: Quivering with Excitement

Perhaps something that might be interesting is relating the stratification of the variety by its singularities to the structure of the quiver. A cryptic comment by (I presume) Thomas Riepe at Lieven’s blog post links to this MO question: Grothendieck’s manuscript on topology. One thing that Grothendieck was interested in in the mid 1980s was finding a new basis for topology that was much more geometrically oriented than point-set topology (he called it ‘tame’ topology, I believe), and for which archetypal examples would be varieties and moduli spaces with their natural stratifications. I wonder if using generalisations of quiver Grassmannians leads to other interesting ‘spaces’.

Another interesting thing would be to relate cohomological invariants of varieties (basically, motives) to those of quivers.

There are a whole stack of interesting papers by Reineke dealing with quivers.