Quivers and Their Representations

Quiver representations provide a combinatorial approach to representation theory, connecting algebra, geometry, and category theory through directed graphs.

DefinitionQuiver

A quiver $Q$ is a directed graph consisting of:

A set of vertices $Q_0 = \{1, 2, \ldots, n\}$
A set of arrows $Q_1$ , where each arrow $\alpha: i \to j$ has a source $s(\alpha) = i$ and target $t(\alpha) = j$

Quivers may have multiple arrows between vertices and loops (arrows from a vertex to itself).

ExampleSimple Quivers

Linear quiver $A_n$ : $1 \to 2 \to 3 \to \cdots \to n$

Cyclic quiver: $1 \to 2 \to 3 \to \cdots \to n \to 1$ (oriented cycle)

Star quiver: Multiple arrows emanating from or pointing to a central vertex

Jordan quiver: A single vertex with a loop: $\circ \circlearrowleft$

DefinitionRepresentation of a Quiver

A representation $V$ of a quiver $Q$ over a field $\mathbb{F}$ assigns:

A vector space $V_i$ to each vertex $i \in Q_0$
A linear map $V_\alpha: V_{s(\alpha)} \to V_{t(\alpha)}$ to each arrow $\alpha \in Q_1$

The dimension vector is $\underline{\dim}(V) = (\dim V_1, \ldots, \dim V_n) \in \mathbb{Z}^n$ .

ExampleRepresentation of $A_2$

For the quiver $1 \to 2$ with arrow $\alpha$ , a representation consists of:

Vector spaces $V_1, V_2$
A linear map $\phi: V_1 \to V_2$

Example: $V_1 = \mathbb{C}^2$ , $V_2 = \mathbb{C}^3$ , and $\phi = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$ .

Dimension vector: $(2, 3)$ .

DefinitionMorphism of Representations

A morphism $f: V \to W$ between quiver representations is a collection of linear maps $\{f_i: V_i \to W_i\}_{i \in Q_0}$ making all diagrams commute: $f_{t(\alpha)} \circ V_\alpha = W_\alpha \circ f_{s(\alpha)}$ for all arrows $\alpha \in Q_1$ .

Quiver representations form a category $\text{Rep}(Q)$ .

Remark

Quiver representations generalize many classical theories:

Modules: Representations of a quiver with one vertex are modules over the algebra of loops
Linear algebra: The Jordan quiver captures Jordan normal form theory
Dynkin quivers: Finite-type quivers correspond to ADE classifications
Hereditary algebras: Path algebras $\mathbb{F}Q$ are hereditary, connecting to homological algebra

Quiver representation theory provides a unified framework for understanding indecomposable representations, Auslander-Reiten theory, and derived categories.