Quiver Grassmannians

Updated 28 October 2025

Quiver Grassmannians are projective varieties that parametrize subrepresentations of a quiver’s finite-dimensional representation, bridging classical Grassmannians with modern representation theory.
They display diverse geometric properties such as singularities, affine cell decompositions, and Euler characteristic computations via F-polynomials.
Their applications span cluster algebras, flag varieties, and tropical geometry, offering insights into combinatorial, cohomological, and motivic invariants.

A quiver Grassmannian is a projective algebraic variety that parametrizes subrepresentations of prescribed dimension inside a fixed finite-dimensional representation of a quiver. Originating as generalized analogues of classical Grassmannians and flag varieties, quiver Grassmannians are central objects in modern geometric representation theory, with significant roles in cluster algebras, combinatorics, moduli theory, and algebraic geometry. Recent research—spanning topics from cell decompositions and cohomology to connections with flag varieties, total positivity, tropical geometry, and motives—has revealed a rich array of structures and properties connected to these varieties.

1. Definition and Types

Given a quiver $Q$ (a directed graph with vertex set $Q_0$ and arrow set $Q_1$ ) and a finite-dimensional representation $M$ (a collection of $\mathbb{C}$ -vector spaces $M_i$ for $i \in Q_0$ and linear maps $M_\alpha\colon M_{s(\alpha)} \to M_{t(\alpha)}$ for each $\alpha \in Q_1$ ), the quiver Grassmannian $\operatorname{Gr}_\mathbf{e}(M)$ is the projective variety whose points are tuples of subspaces $(N_i)_{i \in Q_0}$ satisfying:

$\dim N_i = e_i$ for each $i \in Q_0$ ,
and for each arrow $\alpha: i \to j$ , $M_\alpha(N_i) \subset N_j$ .

When $Q$ is a path (type $A_n$ ), $\operatorname{Gr}_\mathbf{e}(M)$ generalizes to partial or complete flag manifolds; for more general $Q$ , the structure may be far more complicated with arbitrary singularities and multiple irreducible components. Recent generalizations include symplectic quiver Grassmannians, tropical quiver Grassmannians, quiver Grassmannians for bimodules, and quiver Grassmannians associated to quivers with relations.

2. Homological and Geometric Properties

Quiver Grassmannians can be singular, reducible, or even arbitrarily complicated. Several structural results depend on the representation-theoretic type of $Q$ and the rigidity of $M$ :

If $M$ is rigid (i.e., $\mathrm{Ext}^1(M, M) = 0$ ), $\operatorname{Gr}_\mathbf{e}(M)$ is smooth for all $\mathbf{e}$ (Zelevinsky, 2010).
For representations of Dynkin quivers, quiver Grassmannians admit cell decompositions into affine spaces, and their Euler characteristics are nonnegative (Lorscheid et al., 2017, Lorscheid, 2012, Rupel et al., 2018).
For tame affine quivers, all quiver Grassmannians admit affine cell decompositions, though singularities may occur (Lorscheid et al., 2017).
For wild quivers, $\operatorname{Gr}_\mathbf{e}(M)$ can realize any projective variety; in particular, they can have negative Euler characteristics and arbitrary singularities (Ringel, 2017, Ringel, 2013).
The irreducible components of $\operatorname{Gr}_\mathbf{e}(M)$ for direct sums are described homologically: the sum of irreducible components $X$ and $Y$ in $\operatorname{Gr}_A(M)$ and $\operatorname{Gr}_A(N)$ is an irreducible component in $\operatorname{Gr}_A(M \oplus N)$ if and only if the generic extension groups between the corresponding modules vanish (Hubery, 2013).
The smooth loci of minimal-dimensional components in affine quivers coincide with the locus where $\mathrm{Ext}^1(N, M/N) = 0$ (“transverse quiver Grassmannians”) (Irelli et al., 2010).

A key invariant is the $F$ -polynomial

$F_M(\mathbf{u}) = \sum_\mathbf{e} \chi(\operatorname{Gr}_\mathbf{e}(M)) ~ u_1^{e_1}\cdots u_n^{e_n}$

which encodes Euler characteristics of all Grassmannians associated to $M$ (Zelevinsky, 2010).

3. Combinatorial and Cellular Structures

Several classes of quiver Grassmannians are stratified or decomposed into cells labeled by combinatorial objects:

For exceptional and preprojective representations of (generalized) Kronecker quivers, there is a cell decomposition, with cells labeled by maximal Dyck paths or “2-quivers” and compatible pairs (“Dyck path combinatorics”) (Rupel et al., 2018).
In type $A$ and its cyclic versions, for generic representations (“catenoid” or chain-ordered), quiver Grassmannians admit stratifications by orbits of Borel groups or automorphism groups, with each cell corresponding to a unique permutation or “juggling pattern” (Irelli et al., 2015, Feigin et al., 2021).
For degenerate flag varieties and their quiver analogues, cells are explicitly parametrized by Motzkin paths, Grassmann necklaces, or affine Dellac configurations (Pütz, 2020, Feigin et al., 2023).
In symplectic settings, the cellular decomposition is governed by symplectic juggling patterns, and the number of irreducible components matches the Euler characteristic of the classical symplectic Grassmannian (Feigin et al., 30 Jun 2024).
The paper (Lorscheid, 2012) introduces Schubert decompositions of quiver Grassmannians, relating their cells to those of “simpler” quiver Grassmannians by pushforward under morphisms.

Table: Combinatorial Structures Labeling Cells

Setting	Structure	Reference
Type $A$ (catenoid, chain-ordered $M$ )	Weyl group elements	(Irelli et al., 2015)
Cyclic quiver, cyclic flags	Grassmann necklaces	(Feigin et al., 2021)
Kronecker (and rank 2)	Dyck paths/2-quivers	(Rupel et al., 2018)
Symplectic	Symplectic juggling	(Feigin et al., 30 Jun 2024)
Degenerate flags	Motzkin paths	(Irelli et al., 2011)
Affine flag degenerate	Affine Dellac configs	(Pütz, 2020)

4. Cohomology, Poincaré Polynomials, and Canonical Bases

A deep connection has been established between the topology (especially Betti numbers and Poincaré polynomials) of quiver Grassmannians and dual canonical/PBW bases for quantized enveloping algebras:

For acyclic $Q$ , the Poincaré polynomial of a quiver Grassmannian for a rigid representation appears as a coefficient in the expansion of dual canonical (and, for Dynkin types, dual PBW) basis elements in $U_q^-(\mathfrak{g})$ , the negative half of the quantum group (Bi, 2020).
There is a categorical/geometric realization via Lusztig’s category of equivariant perverse sheaves on the representation varieties (Bi, 2020).
In rigid cases, only even-degree cohomology is present (vanishing theorem), explained by parity features of the associated KLR modules (Bi, 2020).
Poincaré polynomials for Grassmannians associated with Schubert quiver Grassmannians can be given in terms of $q$ -analogues of multinomial coefficients, or via enumerative counts of cells (Irelli et al., 2015, Rupel et al., 2018, Feigin et al., 30 Jun 2024).

5. Connections: Flag Varieties, Schubert Varieties, Degenerations

Quiver Grassmannians serve as bridges between moduli of representations and the geometry of classical varieties:

Certain quiver Grassmannians in type $A$ are isomorphic to degenerate flag varieties; others are unions of Schubert varieties or desingularizations thereof (Bott–Samelson resolutions) (Irelli et al., 2011, Iezzi, 17 Feb 2025).
For smooth Schubert varieties in type $A$ , explicit dimension vectors yield a quiver Grassmannian isomorphic to the Schubert variety itself. For arbitrary $w$ , the associated quiver Grassmannian realizes the Bott–Samelson resolution (Iezzi, 17 Feb 2025).
Embeddings of quiver Grassmannians into (affine) flag varieties—with images a union of Schubert cells—are constructed explicitly, including for affine/symplectic cases (Feigin et al., 2023, Feigin et al., 30 Jun 2024).
Degenerate affine Grassmannians (and local models in the theory of Shimura varieties) are realized as quiver Grassmannians or their unions, often with compatible torus actions (Feigin et al., 2023, Feigin et al., 2023).

6. Torus Actions, GKM Theory, and Combinatorial Cohomology

Quiver Grassmannians for cyclic quivers and related settings support torus actions that allow for applications of GKM (Goresky-Kottwitz-MacPherson) theory:

For quiver Grassmannians associated to nilpotent representations of the equioriented cycle, explicit actions by high-rank tori make the varieties GKM; the corresponding moment graphs are computed in terms of successor-closed subquivers and their mutations (Lanini et al., 2020).
The T-equivariant cohomology ring can be described as a subring of functions on the fixed points, with relations determined by edge labels in the moment graph, paralleling Schubert calculus (Feigin et al., 2021, Lanini et al., 2020).
The cell decomposition and associated combinatorics provide combinatorial bases for cohomology in both ordinary and equivariant settings (Pütz, 2020, Lanini et al., 2020).

7. Tropical and Motive-theoretic Aspects

Newer directions include tropical versions and motivic invariants:

The “tropical quiver Grassmannian” and quiver Dressian parametrize systems of tropical linear spaces associated to quivers; the set of realizable points (arising from genuine subrepresentations) can be strictly smaller than the set of tropical solutions to the Plücker-like relations (Iezzi et al., 2023).
Motives of quiver Grassmannians for bimodules are computed recursively, reflecting their cellular decompositions and relations to framed moduli spaces; in favorable cases, these motives are “trivially motivated” (i.e., explicit polynomials in the Lefschetz motive) (Feigin et al., 24 Oct 2025).

Quiver Grassmannians thus form a nexus connecting representation theory, algebraic and symplectic geometry, Schubert calculus, combinatorics, quantum groups, and tropical geometry. Their geometry encodes both the complexity and the categorical structure of the representations to which they are attached, serves as a testing ground for geometric and homological invariants, and continues to motivate new lines of inquiry across several mathematical disciplines.