The two subspace quiver

This term I’ve been organizing a small reading group on representations of quivers, aimed at understanding the proof of Gabriel’s Theorem that a quiver has finitely many indecomposable representations if and only if its underlying graph is a tree of Dynkin type. Our main source is these notes.

A standard fact in linear algebra says that if U and W are subspaces of a vector space V over a field K then there is a basis for V that contains bases for each of U and W. An essentially equivalent restatement is that every representation of the quiver Q below

\bullet \longrightarrow \bullet \longleftarrow \bullet

in which the two arrows are representated by injective maps, is isomorphic to a direct sum of the indecomposable representations K \rightarrow K \leftarrow K, K \rightarrow K \leftarrow 0, 0 \rightarrow K \leftarrow K, 0 \rightarrow K \leftarrow 0, where all maps have full rank. For example, a basis vector in the chosen basis for V that lies in U but not in W corresponds to the second representation in the list above.

It’s easy to see that the kernel of any map from a source vertex (such as the left-most and right-most vertices above) can be split off. So if we allow the arrows to be represented by non-injective maps then we only get two extra indecomposable representations, namely K \rightarrow 0 \leftarrow 0 and 0 \rightarrow 0 \leftarrow K.

These representations are in bijection with the positive roots of the root system of type A_3 (shown below) generated by taking the simple roots \alpha_1 = e_1-e_2, \alpha_2 = e_2-e_3 and \alpha_3 = e_3-3_4 in real 4-space, and then repeatedly applying the reflection maps

s_\beta(v) = v - 2\langle \beta, v \rangle \beta

for simple roots \beta to generate new roots. Equivalently, the roots are the elements of the orbit of e_1-e_2 under the action of S_4 in its action as permutation matrices on \mathbf{R}^4, namely \{ \pm (e_i - e_j) : 1 \le i < j \le 4 \}.

The positive roots (with respect to the direction e_1-e_4) are \alpha_1, \alpha_2, \alpha_3, \alpha_1+\alpha_2, \alpha_2+\alpha_3, \alpha_1+\alpha_2+\alpha_3; they corresponding to the representations with dimension vectors (1,0,0), (0,1,0), (0,0,1), (1,1,0), (0,1,1), (1,1,1), respectively.

One of the main ideas in the proof of Gabriel's Theorem we are following is to use reflection functors to show that quivers which have the same underlying graph, but different orientations of the arrows, have isomorphic module categories. There is one reflection functor for each sink vertex (all arrows coming in) or source vertex (all arrows going out) of a quiver.

For example, the reflection functor F corresponding to the middle vertex of Q sends representations of Q to representations of the quiver \tilde{Q} obtained by reversing the two arrows going into the middle vertex:

\bullet \longleftarrow \bullet \longrightarrow \bullet.

The quiver representation U \rightarrow V \leftarrow W is sent to U \leftarrow \tilde{V} \rightarrow W where \tilde{V} is the kernel of the canonical map q : U \oplus W \rightarrow V and the arrows in the new representations are given by the compositions \ker q \hookrightarrow U \oplus W \twoheadrightarrow U and \ker q \hookrightarrow U \oplus W \twoheadrightarrow W.

There is a beautiful connection between the action of reflection functors on representations and the reflection maps in the simple roots. In this case, identifying representations of Q and \tilde{Q} with positive roots of A_3, one finds that F acts as a reflection in the root \alpha_2 = e_2-e_3. For example, the representation K \rightarrow K \leftarrow 0 of Q with dimension vector (1,1,0) labelled by the root \alpha_1 + \alpha_2 = e_1 - e_3 is sent to the representation K \leftarrow 0 \rightarrow 0 of \tilde{Q} with dimension vector (1,0,0) labelled by the root \alpha = e_1-e_2. And correspondingly,

s_{(e_2-e_3)} (e_1-e_3) = e_1-e_2.

The functor F kills the indecomposable representation 0 \rightarrow K \leftarrow 0; correspondingly s_{(e_2-e_3)}(e_2-e_3) = -(e_2-e_3) is a negative root.

Some more quick thoughts:

  • It’s possible to work out what the reflection functor F should do to maps between quiver representations from general nonsense, by thinking of \mathrm{ker} as a functor. (The answer is unsurprising: if f is a map from U \rightarrow V \leftarrow W to U' \rightarrow V' \leftarrow W' then Ff is the map from U \leftarrow \ker p \rightarrow W to U' \leftarrow \ker p' \rightarrow W' given by f on U and W, and by the restriction of f : U \oplus W \rightarrow U' \oplus W' to \ker p in the middle.)
  • The reflection functors turning source vertices into sink vertices can be defined by categorical duality (i.e. reversing all arrows) from the functors of the type considered above. This amounts to replacing kernels with cokernels.
  • Applying reflection functors to the basic linear algebra fact at the start of this post gives three further results about special bases of a vector space. (Two of them differ only in notation.) For example, the result for \tilde{Q} is that if U is a subspace of V and q : V \rightarrow W is a linear map, then there is a basis for V containing bases for W and \ker q.

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