Grassmannian Cluster Categories

• Grassmannian cluster categories are categorical models that lift the combinatorial features of Grassmannian coordinate rings through dimer algebras and Postnikov diagrams.
• They employ quivers with faces and Cohen–Macaulay module theory to establish isomorphisms between dimer algebras and endomorphism algebras, effectively categorifying cluster algebras.
• Their boundary algebras capture invariant frozen data, linking plabic graphs and dimer models, with applications in representation theory, mirror symmetry, and mathematical physics.

Grassmannian cluster categories are categorical models whose structure encodes and lifts the cluster algebraic and combinatorial features of the homogeneous coordinate rings of Grassmannians. Central to this framework is the interplay between Postnikov diagrams, dimer algebras (also called frozen Jacobian algebras), endomorphism algebras of Cohen–Macaulay (CM) modules, and boundary algebras, which together establish both the combinatorial and categorical infrastructure necessary for the categorification of cluster algebras associated to Gr(k, n). Grassmannian cluster categories serve as a geometric and homological realization of the cluster structure, linking dimer models, combinatorics of plabic graphs, Auslander–Reiten theory, and cluster tilting theory.

1. Combinatorial Construction via Postnikov Diagrams and Quivers with Faces

A (k, n)–Postnikov diagram D consists of oriented curves (“strands”) in a disk with n marked boundary points, arranged such that their alternating regions are labeled by k-element subsets of {1, …, n}—these labels correspond to Plücker coordinates, which are the cluster variables of the Grassmannian Gr(k, n). For each Postnikov diagram, one canonically associates a quiver Q(D) with faces: vertices correspond to alternating regions, and directed arrows reflect the oriented crossings of strands with locally consistent relation to the diagram’s orientation.

The key combinatorial data defining the Grassmannian cluster category is encoded in this quiver with faces. The dimer algebra A = A_D is then constructed as the path algebra of Q(D), modulo relations coming from the oriented faces: for each internal arrow α appearing on the boundary of two faces, the relation p₊(α) = p₋(α) enforces that the two cyclic paths (one in each orientation) around the respective faces are identified. This means that the representation theory of the quiver encodes the combinatorics of Plücker coordinates and the mutations of seeds within the cluster algebra structure (Baur et al., 2013).

2. Dimer Algebra, Endomorphism Algebra, and Cohen–Macaulay Modules

Associated to each alternating region (labeled by a k-subset I) is a rank 1 maximal Cohen–Macaulay module M_I over an auxiliary algebra B (a tiled order introduced in prior categorifications of Grassmannians [Jensen–King–Su]). Collecting these, one forms the “cluster–tilting module” T_D = ⨁_{I ∈ Q₀(D)} M_I corresponding to D.

A central technical result is that the dimer algebra A is isomorphic as a graded algebra to the endomorphism algebra End_B(T_D). The isomorphism is constructed so that idempotents act by identity on the corresponding summands, and each arrow α : I → J is sent to a minimal monomial morphism g_{JI} : M_I → M_J. In particular, A and End_B(T_D) carry gradings compatible with the combinatorics of the diagram.

The algebra B, equipped with its category of Cohen–Macaulay modules, thus “categorifies” the cluster algebra: its cluster–tilting objects model seeds/clusters, and its rigid indecomposables correspond to cluster variables.

Structure	Combinatorial Origin	Category-Theoretic Realization
Postnikov diagram D	Regions labeled by k-subsets	Vertices of quiver Q(D) and CM-modules
Quiver with faces	Oriented crossings	Dimer algebra (frozen Jacobian algebra)
Cluster tilting object	Seed of cluster algebra	Direct sum T_D = ⨁ M_I

3. The Boundary Algebra and Frozen Variables

The subalgebra eAe of the dimer algebra, where e is the sum of idempotents corresponding to regions touching the boundary of the disk (i.e., “frozen” in cluster algebra parlance), is isomorphic—up to passage to the opposite algebra—to B: eAe ≅ B^opp. This identification is highly significant:

• The boundary algebra (realized categorically) encodes the “frozen” part of the cluster structure, i.e., the coefficients/Plücker coordinates that persist across all clusters.
• The categorical boundary algebra is robust under changes in the diagram and reflects the invariance of the underlying algebraic structure.
• The endomorphism algebra of the cluster–tilting module captures the dynamics of mutations, but the boundary algebra encodes the ambient combinatorics (such as faces/coefficients that never participate in mutation) (Baur et al., 2013).

This concept extends beyond the disk to more general marked surfaces, including the annulus, where the boundary algebra can often be given by an explicit quiver with relations, independent (up to isomorphism) under allowable diagrammatic moves.

4. Dimer Model Interpretation and Links to Physics

Each Postnikov diagram yields a dimer model with boundary, in which the associated quiver Q(D) and its potential (constructed from the cycles around faces) define a Jacobian or “frozen Jacobian” algebra. The dimer models, along with dual constructions using plabic graphs, enable:

• Passage to a superpotential description with F-term relations (as in physics/brane tiling literature).
• Natural gradings and weightings on arrows, facilitating connections to higher Teichmüller theory and mirror symmetry.
• Embedding into frameworks, such as brane tilings and bipartite field theories, where the combinatorics directly translates to algebraic and geometric features.

The duality between the plabic graph (a planar bicolored graph encoding trip permutations) and the quiver provides a geometric and combinatorial foundation for mutation, cluster tilting, and the organization of rigid/indecomposable modules.

5. Extension to General Surfaces and Invariance

The construction generalizes to other marked surfaces with boundary, notably the annulus. For weak Postnikov diagrams on such a surface, one follows the same local crossing rules as in the disk (allowing for more general configurations, sometimes with weaker global consistency). The boundary algebra associated to such diagrams can be described by an explicit quiver (e.g., Λ_{n,m} in the annulus case), with relations encoding “bridging” arrows between the two boundaries and local commutation/loop relations.

In the annulus, it is shown that the boundary algebra is invariant—up to isomorphism that preserves the labeling of boundary idempotents—under local moves and exchanges, including geometric exchange (akin to quiver mutation). This implies:

• The categorical model persists across surface types, not only simply connected ones.
• The structure of the cluster category and the correspondence with the boundary algebra are robust under diagrammatic and geometric moves.

6. Synthesis: Bridging Combinatorics and Categorification

The Grassmannian cluster category, as constructed from dimer models, provides a powerful framework:

• Postnikov diagrams encode combinatorial seeds for the cluster algebra structure on Gr(k, n).
• The dimer algebra A(D) models the combinatorial ingredients as a Jacobian algebra, realizing the endomorphism algebra of a cluster–tilting object in the CM category.
• The boundary algebra embodies the frozen data, both combinatorially (boundary minors/faces) and categorically (projective/injective modules).
• The duality with plabic graphs and the interpretation in terms of dimer models provide direct geometric and combinatorial mechanisms for computing clusters, mutations, and cluster variables.

Through this machinery, a full bridge between the combinatorial cluster algebra and its categorical realization is achieved, with robustness to diagrammatic moves and surface topology, and with implications for connections to algebraic geometry, representation theory, and mathematical physics (Baur et al., 2013).

7. Further Developments and Applications

The techniques for associating dimer algebras and boundary algebras to Grassmannian cluster categories pave the way for extension to more general cluster varieties (e.g., positroid moduli, double Bruhat cells), as well as for computational applications involving cluster variables and mutations. These constructions provide tools for categorification of cluster algebras arising in diverse geometric and physical contexts, and serve as a foundation for advances in the understanding of mirror symmetry, quantum cluster algebras, and the representation theory of preprojective and CM-algebras related to Grassmannians.