Auslander-Reiten Diagrams Overview

Updated 9 November 2025

Auslander-Reiten diagrams are graphical representations that encode the pattern of irreducible morphisms and almost split sequences within additive and triangulated categories.
They are constructed with vertices for indecomposable objects and arrows for irreducible morphisms, constrained by precise mesh relations.
Their applications range from finite-dimensional algebras to stable ∞-categories, unveiling critical symmetries in representation theory.

Auslander-Reiten diagrams are combinatorial and categorical structures that encode the configuration of irreducible morphisms and almost split (Auslander-Reiten) sequences (or triangles) in a given (typically additive, Krull-Schmidt) category such as a module category, derived category, or certain subcategories thereof. Their development arises from the paper of finite-dimensional algebras, but their applications reach into triangulated and even stable $\infty$ -categories, revealing underlying homological and representation-theoretic symmetries.

1. Structural Definition and Characterization

An Auslander-Reiten (AR) diagram, or AR quiver, is constructed from a category $\mathcal{A}$ (typically preabelian, Krull-Schmidt, possibly quasi-abelian) with the following elements:

Vertices: Each isomorphism class of indecomposable object in $\mathcal{A}$ .
Arrows: For indecomposable objects $X, Y$ , the number of arrows $X \rightarrow Y$ equals $\dim_k(\mathrm{rad}_{\mathcal{A}}(X,Y)/\mathrm{rad}_{\mathcal{A}}^2(X,Y))$ . These represent irreducible morphisms (i.e., non-invertible morphisms that cannot be factored further except through a section or retraction).
Mesh relations: For any almost split sequence

$0 \rightarrow \tau Z \xrightarrow{f} E \xrightarrow{g} Z \rightarrow 0$

decomposed as $f = (\alpha_i)$ , $g = (\beta_i)$ with $E = \bigoplus_i E_i$ , the relation

$\sum_i \beta_i \circ \alpha_i = 0$

is imposed in the path category of the quiver.

Formally, in a quasi-abelian, Krull-Schmidt category over a field, every indecomposable non-projective $Z$ (or non-injective, dually) admits a unique AR-sequence, and the AR quiver (diagram) fully records the pattern of these irreducible morphisms, subject to the mesh relations (Shah, 2018).

2. Construction in Quasi-Abelian and Krull-Schmidt Categories

The rigorous construction of AR diagrams in general additive settings relies on the following notions:

Preabelian categories: Additive categories with kernels and cokernels for every morphism.
Semi-abelian/Quasi-abelian: Additional stability under pullbacks (for cokernels) or pushouts (for kernels).
Krull-Schmidt property: Every object decomposes as a finite direct sum of indecomposable objects with local endomorphism rings and split idempotents.

In this context, one identifies left/right almost split and minimal left/right almost split maps. The equivalence of six conditions (including f/g irreducible, endomorphism rings local, and minimality of almost splitness) for a short exact stable sequence being an AR-sequence is a cornerstone—establishing that the AR structure is fully governed by irreducibility and minimality [(Shah, 2018), Thm. 4.9].

The AR quiver thus encodes all such sequences: vertices for indecomposables, one arrow for each class of irreducible morphism, and mesh relations prescribed by the direct summand structure of almost split sequences.

3. Methodology: From Mesh Categories to Universal Symmetries

The AR quiver formalism extends beyond module categories and abelian setups to stable $\infty$ -categories and mesh categories (Sánchez, 4 Nov 2025). For an acyclic quiver $Q$ and a stable $\infty$ -category $\mathcal{C}$ , the repetitive quiver $\mathbb{Z}Q$ (vertices $(n,v)$ , translation $\tau$ , and "mesh" patterns) is central. The mesh $\infty$ -category $\mathcal{C}^{\mathbb{Z}Q,\,\mathrm{mesh}}$ is defined using homotopy pullbacks, forcing exactness or cofiber relations in each mesh square.

This formalism yields these consequences:

Universal autoequivalences: Translations and automorphisms of $\mathbb{Z}Q$ induce autoequivalences on $\mathcal{C}^Q$ , including classical constructions like reflection functors, the AR-translation $\tau$ , and the Serre functor.
Spectral Picard group: The group $\operatorname{Aut}_{tr,\sigma}(\mathbb{Z}Q)$ injects into the spectral Picard group $\operatorname{Pic}_R(Q)$ , unifying abstract symmetries (automorphisms of the AR diagram) with invertible endofunctors of the category (Sánchez, 4 Nov 2025).

4. Variants and Applications Across Representation Theory

The AR diagrammatic language underpins a large taxonomy of representation-theoretic situations:

Module categories over finite-dimensional algebras: The classical AR quiver constructed via irreducible morphisms and mesh relations encodes almost split sequences, leading to well-known configurations (preprojective, preinjective, regular/tubes).
Stable and triangulated categories: In bounded homotopy categories $K^b(\mathrm{mod}\,A)$ and subcategories like $K^b$ (Gorenstein projective), AR triangles replace AR sequences, but the construction is analogous; irreducible morphisms and AR-translation $\tau$ come from the Serre functor and mesh relations (Zheng et al., 2015).
Symmetric special biserial algebras and Brauer graph algebras: The AR diagram is effectively constructed from combinatorics of Brauer graphs, employing the hook/cohook and Green walk algorithms. Here, the structure of the stable AR quiver (exceptional tubes, $ZA_{p,q}$ components) can be predicted from the topology of the underlying graph (Duffield, 2015).
Locally finite/infinite or continuous settings: For locally finite quivers (Paquette, 2012) or continuous quivers of type $A$ (Rock, 2019), the AR "diagram" may become a continuous space or infinite combinatorial structure, but the language of vertices as indecomposables, arrows/edges as irreducible morphisms, and mesh/rectangle relations remains controlling.

5. Classification and Universal Features

Across all contexts above, key classification results and computational techniques are governed by the AR diagram:

Connected components ("tree class"): For algebras of wild or tame type (e.g., Frobenius-Lusztig kernels), stable AR quiver components are classified by infinite Dynkin diagrams ( $A_\infty$ , $D_\infty$ , $A_\infty^\infty$ ) or Euclidean diagrams in cases of tame/wild dichotomy (Külshammer, 2012).
Universal local structure: Every non-projective indecomposable is the end point of a unique (up to isomorphism) AR sequence; each mesh (i.e., the configuration around such an indecomposable) exhibits a "diamond" with mesh relation enforcing the sum of compositions around the mesh equals zero.
Dimension–arrow correspondences and invariants: The number of arrows encodes irreducibles modulo scalar multiples, and in modular/affine settings, numerical invariants such as the Gabriel–Roiter measure and its decomposition into initial, periodic, multiplicity, and final parts, detect the corresponding region/component in the AR diagram (Schmidmeier et al., 2012).

6. Examples and Diagrammatic Realizations

Indecomposables: $P(1)=S(1)=(k \rightarrow 0)$ , $P(2)=(k \xrightarrow{id} k)$ , $S(2)=(0 \rightarrow k)$ .
AR quiver: three vertices, arrows $P(1) \rightarrow P(2)$ , $P(2) \rightarrow S(2)$ .
Mesh: AR sequence $0 \rightarrow P(1) \xrightarrow{a} P(2) \xrightarrow{b} S(2) \rightarrow 0$ , with relation $b \circ a = 0$ .

The stable AR quiver is a single tube of rank $3 \cdot e(v) = 6$ , constructed by explicitly applying hook/cohook moves and reading Green walks.

Indecomposables form a diamond:

$\overline{P_2} \rightarrow \overline{P_1}, \overline{S_2};\quad \overline{P_1}, \overline{S_2} \rightarrow \overline{I_2}$

Mesh relation: $\overline{b}\,\overline{a} + \overline{d}\,\overline{c} = 0$ .

The AR diagram in each case visually encodes morphism structure, mesh relations, and the translation $\tau$ or shift action.

7. Mathematical and Conceptual Significance

AR diagrams provide a bijective bridge between homological algebra (almost split sequences/triangles), combinatorial data (quivers and path categories), and categorical symmetries (auto-equivalences, reflection functors, spectral Picard groups). Their universal description in mesh categories or $\infty$ -categorical settings demonstrates their relevance for higher representation theory, derived geometry, and topological modular invariants (Sánchez, 4 Nov 2025).

The AR diagrammatic formalism is not only crucial for explicit computations, classification, and visualization, but also provides the structural backbone for various dualities and autoequivalences arising in representation theory, cluster theory, and beyond.

Table: Key components in constructing an Auslander-Reiten diagram

Aspect	Data/Condition	Reference
Vertices	Isomorphism classes of indecomposable objects	(Shah, 2018), etc.
Arrow multiplicity	$\dim_k \left(\mathrm{rad}_{\mathcal{A}}(X,Y)/\mathrm{rad}_{\mathcal{A}}^2(X,Y)\right)$	(Shah, 2018)
Mesh relation	For each non-projective $Z$ : $\sum_i \beta_i \circ \alpha_i = 0$ in $k\Gamma(\mathcal{A})$	(Shah, 2018, Sánchez, 4 Nov 2025)
$\tau$ -translation	Unique assignment $\tau Z$ via AR-sequence $0\to \tau Z \to E \to Z \to 0$	(Shah, 2018)
Universal structure	Diagram arises as mesh category of the repetitive quiver (mesh $\infty$ -category, etc.)	(Sánchez, 4 Nov 2025)

Auslander-Reiten diagrams thus constitute a categorical and combinatorial invariant encoding the complete pattern of irreducible morphisms and almost split sequences in a wide class of additive and triangulated categories, capturing both rigid local data and universal symmetries operant in the representation theory of algebras.