Cyclic Quiver Grassmannians: Geometry & Applications

• Cyclic quiver Grassmannians are moduli spaces of subrepresentations of nilpotent cyclic quiver modules, generalizing classical Schubert varieties.
• They feature cellular decompositions via torus actions with combinatorial indexing through Grassmann necklaces, juggling patterns, and Dellac configurations.
• They serve as geometric models in cluster algebra theory and affine flag varieties, linking representation theory with total positivity and degenerations.

Cyclic quiver Grassmannians are moduli spaces of subrepresentations of nilpotent representations of the equioriented cycle quiver. They generalize Schubert varieties, provide geometric realizations for cluster algebra theory, and serve as degenerations or local models for various types of affine flag and Grassmannian varieties, including symplectic and totally nonnegative versions. Their geometry is governed by a rich combinatorial and representation-theoretic structure, notably including cellular decompositions, GKM theory, and connections with cluster algebras, affine Coxeter groups, and combinatorics such as juggling patterns and Dellac configurations.

1. Definitions and Foundational Constructions

Let $Q = \widetilde{A}_{n-1}$ denote the cyclic (equioriented) quiver with vertex set $Q_0 = \mathbb{Z}/n\mathbb{Z}$ and arrows $a_i : i \to i+1$ (indices mod $n$). A finite-dimensional representation $M$ of $Q$ is specified by a collection of vector spaces $M(i)$ at each vertex and linear maps $M_{a_i}: M(i) \to M(i+1)$. A subrepresentation $U$ comprises subspaces $U(i) \subset M(i)$ satisfying $M_{a_i}(U(i)) \subset U(i+1)$. Fix $e = (e_i)_{i \in Q_0}$ with $0 \leq e_i \leq \dim M(i)$; then the cyclic quiver Grassmannian is the projective variety

$$\mathrm{Gr}_e(M) = \{\, U \text{ subrepresentation of } M \mid \dim U(i) = e_i \, \forall i \, \}.$$

In the context of nilpotent (finite cyclic length) representations, the geometric features depend strongly on the combinatorics of the indecomposable summands and the dimension vector $e$ (Irelli et al., 2018, Feigin et al., 2023).

2. Cellular Decomposition and Combinatorial Indexing

Cyclic quiver Grassmannians associated to (possibly reducible) nilpotent modules admit cellular decompositions via torus actions and Białynicki-Birula (BB) theory. For suitable choices of basis and gradings, the coordinate subrepresentations lying in successor-closed subquivers correspond to fixed points under a $\mathbb{C}^*$ or torus $T$ action, and the BB attracting sets $C_p$ are affine cells (Lanini et al., 2020, Lowiel, 6 Jun 2024).

In the "single-matrix" case ($\mathcal{M}_i = \mathbb{C}^N$ at each vertex, all arrow maps $J$ a nilpotent Jordan matrix), the fixed points and hence BB-cells are in bijection with successor-closed subquivers of the coefficient quiver $Q(\mathcal{M})$ (Lowiel, 6 Jun 2024). In the more general "folded type-A" setting, fixed points correspond to combinatorial data such as Grassmann necklaces, generalized juggling patterns, or affine Dellac configurations, depending on the precise module structure (Feigin et al., 2021, Feigin et al., 2023, Pütz, 2020). The dimensions of the BB-cells are computable in terms of the combinatorics (Coxeter lengths, inversion numbers, or explicit energy functions).

Table: Combinatorial Patterns and Their Roles

Combinatorial Object	Model/Parameterizes	Cell Structure
Successor-closed subquivers	Fixed points of torus actions	BB-cells in general case
Grassmann necklaces	Cells for $X(k, n)$ (type $A$) Grassmannians	Stratification and closures
Juggling patterns	Fixed points for $X(k,n,w)$ (local models)	BB-cells, stratification
Affine Dellac configurations	$M_w = \oplus U(i;w n)^{\oplus 2}$, $e=(wn,...,wn)$	Cellular decomposition
Symplectic juggling patterns	Fixed points for $X(k,2n)^{SP}$	Symplectic cells, orbits

The closure relations of cells correspond to canonical partial orders, e.g., twisted lex order in the single-matrix case (Lowiel, 6 Jun 2024), reverse inclusion for Grassmann necklaces (Feigin et al., 2021), product order for generalized juggling patterns (Feigin et al., 2023), or the Bruhat order from affine Coxeter groups of types $A^{(1)}$, $C^{(1)}$ and their type $C$-symmetric fixed loci (Micheli, 21 Jul 2025, Feigin et al., 30 Jun 2024).

3. Torus Actions, GKM Theory, and Cohomology

For a large family of cyclic quiver Grassmannians, the action of a diagonal algebraic torus $T$ renders the varieties into GKM varieties: finitely many $T$-fixed points, finitely many one-dimensional orbits, and equivariant formality with vanishing of odd cohomology (Lanini et al., 2020, Lanini et al., 2021, Micheli, 21 Jul 2025). The combinatorial data from the fixed point set and the moment graph (edges labeled by characters arising from adjacent BB-cells) encode the equivariant cohomology ring via the GKM localization theorem: $H_T^*(X) \cong \left\{ (f_p)_{p} \mid f_{p} - f_{q} \in (\alpha_{pq}) \text{ for each edge %%%%31%%%%}\right\},$ where $p,q$ are fixed points and $\alpha_{pq}$ is the weight of $T$ on the corresponding $1$-orbit (Pütz et al., 2023, Lowiel, 6 Jun 2024).

The canonical bases for $H_T^*(X)$ (Knutson-Tao, or upper-triangular in ordering by cell closures) satisfy orthogonality properties and lead to explicit computations of multiplicative structure constants via ABBV localization or recursive relations (Lowiel, 6 Jun 2024, Pütz et al., 2023). In the symplectic and Lagrangian settings, torus actions respect the isotropic locus, and the moment graph structure remains combinatorial but with symmetry constraints (Feigin et al., 30 Jun 2024, Micheli, 21 Jul 2025).

4. Examples: Single-Matrix, Symplectic, and Cluster-Theoretic Grassmannians

In the single-matrix nilpotent representation ($\mathcal{M}$ defined by $(J, N)$ with block sizes $j_l$), for $e = (1,...,1)$, all closed BB-cells are smooth and the entire Grassmannian decomposes into products of type $A$ Grassmannians, yielding nonsingular stratification (Lowiel, 6 Jun 2024). The Knutson-Tao basis yields a cell structure analog to Schubert cell decompositions.

For symplectic cyclic Grassmannians, as constructed in $X(k,2n)^{SP}$, the isotropic condition yields varieties that are linear degenerations of the classical symplectic Grassmannians, but with combinatorial cell stratification indexed by symplectic juggling patterns, closure order matching Bruhat order in affine type $C$ (Feigin et al., 30 Jun 2024, Micheli, 21 Jul 2025). All cells are $G^{SP}$-orbits, isomorphic to affine spaces, and their numbers and dimensions match those in algebraic geometry (Euler characteristic of $\mathrm{Gr}(k,2n)^{SP}$, coincidences of dimension and component formulas).

In type $A$ and their folded/cluster-theoretic degenerations, e.g., totally nonnegative Grassmannians and special fibers of local models, cyclic quiver Grassmannians encode positroid cell stratifications and their closure posets, in direct correspondence with combinatorial data such as Grassmann necklaces and generalized juggling patterns (Feigin et al., 2021, Feigin et al., 2023). The closures of top-dimensional cells produce irreducible components parametrized by $k$-element subsets, and explicit desingularizations are constructed via quiver-theoretic extensions, leading to towers of Grassmann bundle fibrations (Pütz et al., 2023, Feigin et al., 2023).

5. Geometry of Irreducible Components and Singularities

The geometry varies with the choice of dimension vector and underlying representation. For $e$ the constant vector, irreducible components can be indexed bijectively by corresponding combinatorial objects discussed above. In special cases (such as single-matrix, type $A$-folded, or symplectic), all irreducible components are normal, Cohen-Macaulay, with rational singularities, and their closures admit desingularizations via explicit modifications of the quiver (adding flags, extending to "cylinder" quivers) (Pütz et al., 2023, Pütz, 2020, Feigin et al., 2023, Feigin et al., 30 Jun 2024). The largest cells are isomorphic to affine spaces of maximal dimension, matching dimensions with classical or degenerate flag varieties.

Singularities become more intricate for higher multiplicities or larger dimension vectors, where some cell closures may be singular or not reduced. However, explicit constructions from flag theory and BB-decompositions provide tools for analyzing their geometry systematically.

6. Connections with Affine Flag Varieties, Shimura Local Models, and Cluster Algebras

Cyclic quiver Grassmannians provide explicit geometric models for (degenerate) affine flag varieties, local models of Shimura varieties of type $A$, and compactifications in total positivity and cluster algebra theory.

• Affine Flags: There exist explicit closed embeddings $$X(k,n,w) \hookrightarrow \mathcal{F}\ell = SL_n(\mathbb{C}((t)))/\mathcal{B}$$ as unions of Iwahori-Schubert varieties, with combinatorial parametrization via bounded affine permutations (Feigin et al., 2023, Pütz, 2020, Feigin et al., 30 Jun 2024).
• Shimura Local Models: The special fibers of PEL local models for $GL_n$ with minuscule cocharacter correspond to cyclic quiver Grassmannians $X(k,n,w)$, and their geometric and cohomological invariants follow from the quiver model (Feigin et al., 2023).
• Cluster Algebras: Cellular decomposition and explicit Chow ring generators facilitate the construction of cluster multiplication formulas (e.g., Caldero-Keller), and cyclic Grassmannians provide geometric incarnations of categorified cluster algebra constructions (Irelli et al., 2018, Lowiel, 6 Jun 2024).

Table: Key Geometric/Representation-Theoretic Applications

Application Domain	Cyclic Quiver Grassmannian Role	Reference Papers
Affine flag degeneration	Finite-dimensional approximation	(Pütz, 2020, Feigin et al., 2023)
Shimura local models	Special fiber, local model of type $A$	(Feigin et al., 2023, Feigin et al., 2023)
Cluster algebra theory	Geometric model for cluster multiplication	(Irelli et al., 2018, Lowiel, 6 Jun 2024)
Total positivity/positroids	Positroid stratification, Grassmann necklace	(Feigin et al., 2021)

7. Further Developments: GKM-Equivariant Cohomology and Symplectic Degenerations

Recent work extends the theory to cyclic quiver Grassmannians with symplectic structure and their Lagrangian isotropic loci $X(n,2n)^{SP}$, revealing connections to affine Coxeter groups of type $C^{(1)}$, the combinatorics of symplectic juggling patterns, and GKM-theoretic descriptions of the cohomology (Micheli, 21 Jul 2025, Feigin et al., 30 Jun 2024). These varieties are GKM, their cells are $G^{SP}$-orbits, and their dimension and enumeration coincide with classical symplectic Grassmannians. Embeddings into affine symplectic flag varieties as unions of Schubert varieties are constructed, with morphisms compatible with the orthogonality structure.

The equivariant cohomology is thus accessible via explicit combinatorial and representation-theoretic data from the underlying quiver, extending the reach of GKM and moment graph techniques. Multiplicative and representation-theoretic structures, e.g., from Weyl and Coxeter group actions and symmetric group representations, reflect the deep symmetry and combinatorics present in the geometry of cyclic quiver Grassmannians.

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Key references and foundational papers in this field include:

• “Quiver Grassmannians associated to nilpotent cyclic representations defined by single matrix” (Lowiel, 6 Jun 2024)
• “Symplectic Grassmannians and Cyclic Quivers” (Feigin et al., 30 Jun 2024)
• “GKM-Theory for Torus Actions on Cyclic Quiver Grassmannians” (Lanini et al., 2020)
• “Cell decompositions and algebraicity of cohomology for quiver Grassmannians” (Irelli et al., 2018)
• “Generalized juggling patterns, quiver Grassmannians and affine flag varieties” (Feigin et al., 2023)
• “Degenerate Affine Flag Varieties and Quiver Grassmannians” (Pütz, 2020)
• “Some Lagrangian quiver Grassmannians for the equioriented cycle” (Micheli, 21 Jul 2025)
• “Totally nonnegative Grassmannians, Grassmann necklaces and quiver Grassmannians” (Feigin et al., 2021)
• “Laumon parahoric local models via quiver Grassmannians” (Feigin et al., 2023)
• “Permutation actions on Quiver Grassmannians for the equioriented cycle via GKM-Theory” (Lanini et al., 2021)
• “Desingularizations of Quiver Grassmannians for the Equioriented Cycle Quiver” (Pütz et al., 2023)

These works establish the current landscape, methodologies, and combinatorial-geometric correspondences fundamental to the theory of cyclic quiver Grassmannians.