Quivers: a year of linear algebra by drawing arrows | lisyarus blog

Quivers: a year of linear algebra by drawing arrows

2026 Jun 9

Disclaimer: if your first instinct was to think about string diagrams, that's not was this post is about. Cool guess, though.

In algebra there's this idea of representations of some object, especially linear representations. It's so important that even chemists typically have the representation theory of finite groups in their curriculum, because those have a profound influence on the structure of atoms and molecules.

And it just so happens that studying representations of a particular kind of object (namely, quivers) can be seen as a generalization of a typical 1-year university linear algebra course :)

Contents

Representations

First, lets talk about representations a bit more.

The idea is that some objects are easier to study and understand than others. For example, permutations are easier to work with than elements of a general group, and there's a theorem that any group can be represented as a group of some permutations.

One of the things we understand the best in math are matrices — they seem to be in the perfect sweet spot between showing up everywhere and also having a deep, rich, and useful theory. It turns out that we can represent many objects as matrices, in which case we call them linear representations, though often the word representation means a linear representation by default.

As a simple example, consider complex numbers: how can we represent the number \(i\) as a real-valued matrix? Well, \(i\) is defined as being a root of the equation \(i^2=-1\), so we just need to find a matrix that satisfies this equation. Note that we'll need to replace the number \(-1\) with the scalar matrix having \(-1\) on the diagonal (because this matrix effectively acts as the number \(-1\) in the matrix algebra).

It's not hard to find a matrix which squares to \(-1\):

\[ \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}^2 = \begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix} \]

You can notice that it's the matrix of rotation by \(90^\circ\), and that's not a coincidence!

If we also represent real numbers \(a\) as scalar matrices \(\begin{pmatrix}a & 0 \\ 0 & a\end{pmatrix}\), we can represent the complex number \(a+bi\) as a matrix

\[ a+bi \mapsto \begin{pmatrix}a & -b \\ b & a\end{pmatrix} \]

and matrices of this form act exacly like complex numbers — the addition and multiplication formulas are the same, as are all the other properties (the technical term is that the algebra of such matrices is isomorphic to the complex numbers).

In general, representations are easiest to define via category theory. For example, take a group, turn it into a one-object category with elements of this group as morphisms (i.e. categorify the group to obtain a groupoid), and then representations of this group are simply functors from this groupoid into some other category. Functors to \(\operatorname{Set}\) give permutation representations, while functors to \(\operatorname{Vect}_K\) give linear representations.

Matrices vs operators

I want to stress one thing which might sound like obvious truth, a boring technicality, or useless garbage, depending on your background. We commonly represent vectors using coordinates, which is especially convenient for direct computations, but an abstract vector isn't the same as a list of coordinates. We can always choose a different basis, and the coordinates of a specific vector will change, but the vector itself won't change — it represents some specific (albeit abstract) geometric object which doesn't depend on a particular encoding scheme that we've arbitrarily chosen.

The same goes for matrices and operators. A matrix is just a table with numbers; it only means something geometric once we've selected the basis vectors — specifically, it means a linear operator acting on these vectors. If we change the basis, the vectors and operators don't change, but the vector coordinates and matrix entries can change.

How important this is depends entirely on your domain of work. I'm bringing this up because it does matter here: by choosing a basis in a clever way, we can see how some complicated object ends up looking very simple. In fact, keeping track of bases will be very important in what follows.

Quivers

So, what exactly is a quiver? It's a directed graph. That's it, that's the definition.

OK, there are a dozen different things that people call a directed graph. Does it allow for multiple edges? Does it allow loops? For a quiver, the answers are yes and yes. In other words, there are no restrictions on edges: there can be several edges between two vertices, and edges can start and end in the same vertex. This type of a directed graph is often called a multidigraph.

Here are a few examples of quivers:

These were made with an online tool called — you wouldn't believe it — q.uiver.app.

Ok, a quiver is a multidigraph, i.e. a directed graph with multi-edges...