Cluster Modules: Algebra & Categorification

• Cluster modules are algebraic structures that represent structured families of modules in cluster algebra theory, higher Auslander–Reiten theory, and quantum affine algebras.
• They exhibit rigidity and defined homological properties, including maximal orthogonality and controlled projective dimensions in cluster-tilted and mesh algebras.
• Cluster modules underpin categorification frameworks by providing exchange relations that generalize T-systems and facilitate monoidal categorifications in diverse algebraic settings.

Cluster Modules

A cluster module is a representation-theoretic object capturing compatible, highly structured families of modules arising in cluster algebra theory, higher Auslander–Reiten theory, and monoidal categorifications of quantum affine algebras. The term has acquired unified significance through the recent work of des Fossés, Ding, and Sun, where "cluster modules" generalize both Kirillov–Reshetikhin and Hernandez–Leclerc modules as special cases, providing a module-theoretic categorification of cluster variables in monoidal categorifications of cluster algebras (Bing et al., 18 Dec 2025). Cluster modules also appear in classical contexts, such as modules over cluster-tilted algebras and in the structure theory of mesh algebras, and in geometric and combinatorial settings such as the Grassmannian cluster categories. The following systematically presents definitions, foundational results, classification theorems, homological properties, and relations to categorification.

1. Definitions and Foundational Constructions

The formation of cluster modules is context-dependent but governed by categorical and combinatorial correspondences. In the monoidal categorification of cluster algebras for a simply-laced quantum affine algebra, a cluster module is defined as follows (Bing et al., 18 Dec 2025):

• Fix a height function $\xi: I \to \mathbb{Z}$ on the Dynkin diagram of a simple Lie algebra $\mathfrak{g}$, its associated quiver $Q_\xi$, integer $l \ge 1$, and cluster category $\mathcal{C}_{Q_\xi}$.
• For each index $(i,r)$ in a certain range, define $$z_{i,r} = \prod_{k \ge 0, r+2k \le \xi(i)} Y_{i,r+2k}$$, and in $\mathcal{C}^{\le \xi}_l$ consider Kirillov–Reshetikhin modules $W^{(i)}_{k,r} = L(\prod_{j=0}^{k-1} Y_{i,r+2j})$.
• The assignment

$$\Psi^{\le \xi}_l(M) = L((\widetilde{x}^{\le \xi}_l)^{\widetilde{g}(M)})$$

associates to each rigid object $M$ in the cluster category a simple module in the monoidal category, where $\widetilde{x}^{\le \xi}_l$, the extended g-vector $\widetilde{g}(M)$, and other combinatorial data are explicitly defined [(Bing et al., 18 Dec 2025), Def 4.1]. Cluster modules are those simple objects of $\mathcal{C}^{\le \xi}_l$ realized in this way.

In the classical finite type setting, cluster-tilted algebras $A=\operatorname{End}_\mathcal{C}(T)$ are defined using a cluster-tilting object $T$ in a cluster category $\mathcal{C}_Q$. All indecomposable $A$-modules are of the form $\operatorname{Hom}_\mathcal{C}(T, M)$ for $M \notin \operatorname{add} T[1]$ (Beaudet et al., 2011, Assem et al., 2012). These are sometimes termed cluster modules in this context.

2. Homological and Categorical Properties

Cluster modules universally exhibit highly rigid and accessible algebraic structures:

• Rigidity and Orthogonality: For cluster tilting modules $M$ in finite-dimensional algebras $A$, $M$ is $n$-cluster tilting if and only if

$$\operatorname{add}(M) = \{X \mid \operatorname{Ext}^i_A(M, X) = 0 \;\forall\; 1 \le i < n\} = \{X \mid \operatorname{Ext}^i_A(X, M) = 0 \;\forall\; 1 \le i < n\}$$

and is "maximal (n–1)-orthogonal" in the sense of Iyama (Marczinzik et al., 19 May 2025, Erdmann, 2021).

• Homological Dimensions: In cluster-tilted algebras, every module has projective (and injective) dimension $0$, $1$, or $\infty$ due to the 1-Gorenstein property (Beaudet et al., 2011, Schiffler et al., 2016). The precise locus of modules with infinite projective dimension is controlled by factorization ideals $I_M \subset \operatorname{End}_\mathcal{C}(T[1])$.
• Canonical Presentations: For induced modules $M \otimes_C B$ over a cluster-tilted algebra $B=C \ltimes E$, there exist explicit (minimal) injective presentations, with modules formed from induced, coinduced, and syzygy functors, reflecting and lifting the homological data from the tilted algebra $C$ (Schiffler et al., 2016).

3. Classification and Reachability

The classification of cluster modules in the monoidal categorification context advances the Hernandez–Leclerc program:

• Extended g-vector and F-polynomial Parametrization: For each rigid object $M$ in the cluster category, the highest $\ell$-weight monomial of the associated cluster module is given by

$$m = (z^{\le \xi}_l)^{\widetilde{g}(M)} = \frac{(z^{\le \xi}_{l-1})^{g(M)}}{F_M|_\mathbb{P}( (y_{i,r})_{(i,r) \in \breve{I}^{\le \xi}_{l-1}} )}$$

where $F_M$ is the cluster $F$-polynomial of $M$ [(Bing et al., 18 Dec 2025), §3.2].

• Multiplicative Reachability: Under the reachability conjecture (Qin), all real simple modules are cluster modules, and the classification map from indecomposable rigid objects in the cluster category to real primes in $\mathcal{C}^{\le\xi}_l$ is a bijection [(Bing et al., 18 Dec 2025), Thm 1.3].
• Classification in Mesh Algebras and Surface Algebras: In mesh algebras of Dynkin or folded type, cluster-tilting modules correspond bijectively to positive roots of the root system, up to the action of automorphisms and orbits, elaborating the combinatorial nature of the correspondence (Erdmann et al., 2019).

4. Exchange, T-systems, and Cluster Relations

• Cluster Exchange Systems: Prime cluster modules (those corresponding to indecomposable, real simple modules) satisfy explicit systems of equations generalizing classical T-systems:

$$[\Psi^{\le\xi}_l(L)][\Psi^{\le\xi}_l(N)] = [\Psi^{\le\xi}_l(M)] \prod_{i\in I} [L(u_i(l))]^{\alpha_i} [L(v_i(l))]^{\alpha_i} + [\Psi^{\le\xi}_l(M')] \prod_{i\in I} [L(u_i(l))]^{\beta_i} [L(v_i(l))]^{\beta_i}$$

for $(L, N)$ an exchange pair in the cluster category and suitable exponents $\alpha_i, \beta_i$ [(Bing et al., 18 Dec 2025), Thm 1.7].

• Specialization to Classical Cases: This framework includes the classical T-systems for Kirillov–Reshetikhin modules and the exchange relations for Hernandez–Leclerc modules as special or limiting cases (Bing et al., 18 Dec 2025).
• Monoidal Categorifications and q-Characters: The $q$-characters of cluster modules, particularly in affine types, are governed by these exchange systems, showing that the category of cluster modules is closed under tensor products up to the combinatorics of cluster monomials (Sakamoto, 10 Dec 2025, Duan et al., 2015).

5. Applications and Explicit Families

• Quantum Affine Algebras (Monoidal Categorifications): Cluster modules produce the exact correspondence between real simple modules and cluster monomials in the cluster algebra, with prime modules matching cluster variables (Sakamoto, 10 Dec 2025, Bing et al., 18 Dec 2025, Duan et al., 2015).
• Grassmannian Cluster Categories: In ${\rm CM}(B_{k,n})$, categories categorifying the coordinate ring of the Grassmannian, the rank-1 cluster modules (rank-1 Cohen–Macaulay) correspond to Plücker coordinates, while higher-rank indecomposable modules with rigid structure generalize the set of cluster variables beyond the initial seed and are uniformly classified via root system combinatorics (Baur et al., 2020, Baur et al., 2020).
• Cluster-Tilted and Mesh Algebras: Every indecomposable module over a cluster-tilted algebra arising from a cluster-tilting object in a cluster category, and every cluster-tilting module in a mesh algebra, is an example of a cluster module in this generalized sense (Beaudet et al., 2011, Erdmann et al., 2019).

6. Open Problems and Research Frontiers

Key directions and unresolved problems:

• Multiplicative Reachability: The full scope of the multiplicative reachability conjecture remains unsettled outside the acyclic, simply-laced cases (Bing et al., 18 Dec 2025).
• Cartan-Determinant and Homological Tests: The connection of cluster tilting modules to classical Cartan-determinant and Tachikawa conjectures in the context of local and self-injective algebras is under active investigation, with evidence only partial (Marczinzik et al., 19 May 2025).
• Beyond Real Simples: Extending classification to modules not strongly real (i.e., not closed under self-tensor) and to general imaginary modules remains a challenge, especially for types beyond ADE and outside of affine type (Sakamoto, 10 Dec 2025).
• Combinatorics and Geometry: The full correspondence between geometric, combinatorial, and representation-theoretic notions of cluster modules in categories categorifying positroid and Grassmannian strata is ongoing with advances in the theory of Postnikov diagrams and dimer models (Çanakçı et al., 2021).

7. Summary Table: Major Types of Cluster Modules

Context	Defining Feature	Principal Reference
Cluster-tilted algs	$\operatorname{Hom}_\mathcal{C}(T,M)$	(Beaudet et al., 2011, Assem et al., 2012)
Quantum affine algs	$\Psi^{\le\xi}_l(M)$ from rigid objects	(Bing et al., 18 Dec 2025)
Kirillov–Reshetikhin	Simple $L(m)$, $m$ highest $\ell$-weight	(Bing et al., 18 Dec 2025, Duan et al., 2015)
Hernandez–Leclerc	Primes via cluster category bijection	(Duan et al., 2015, Bing et al., 18 Dec 2025)
Mesh algebras	Maximal rigid/cluster-tilting modules	(Erdmann et al., 2019)
Grassmannian categories	Rigid indecomposable CM modules	(Baur et al., 2020)

Each incarnation of cluster module encodes the algebraic and categorical realization of the combinatorics of cluster variables, with classification (where possible) achieved via explicit correspondences to rigid/tilting objects, g-vectors, F-polynomials, or quiver representations, depending on context. The ongoing synthesis across these domains continues to illuminate deep connections among cluster algebras, quantum affine algebras, and higher-dimensional homological algebra.