Lagrangian Grassmannians

• Lagrangian Grassmannians are algebraic varieties that parametrize maximal isotropic subspaces of a symplectic vector space, forming a fundamental construct in geometry and representation theory.
• They exhibit rich intrinsic structures through Schubert calculus, Plücker embeddings, and quantum cohomology, bridging combinatorial, algebraic, and topological properties.
• Their diverse applications include integrable systems, mirror symmetry, and derived category frameworks, offering deep insights into moduli spaces and symplectic geometry.

The Lagrangian Grassmannian is the algebraic variety (or, in the real or complex analytic contexts, the manifold) parametrizing all maximal isotropic subspaces of a finite-dimensional symplectic vector space. Functioning as a foundational object in algebraic geometry, representation theory, integrable systems, and mathematical physics, it exhibits a rich interaction between algebraic, combinatorial, categorical, and geometric phenomena, including distinctive features in its topology, Schubert calculus, derived category structure, and quantum and mirror invariants.

1. Definition and Fundamental Properties

Let $V$ be a $2n$-dimensional vector space over an algebraically closed field of characteristic zero, or over $\mathbb{R}$ or $\mathbb{C}$, equipped with a nondegenerate skew-symmetric bilinear form $\omega: V \otimes V \to k$. A subspace $U \subset V$ is isotropic if $\omega|_{U \times U} = 0$; maximal isotropic subspaces have dimension $n$, called Lagrangian subspaces. The Lagrangian Grassmannian is defined as

$$\mathrm{LG}(n,2n) = \{ U \subset V \mid \dim U = n,\, \omega|_{U\times U}=0 \}.$$

It is a smooth projective variety of dimension $n(n+1)/2$, and a homogeneous space for $\mathrm{Sp}(2n)$ (or, in type C, for $G^\vee = \mathrm{PSp}_{2n}$): $\mathrm{LG}(n,2n) \cong \mathrm{Sp}(2n)/P,$ where $P$ is a maximal parabolic stabilizing a reference Lagrangian subspace (Fonarev, 2019, Fonarev, 2023, Gutt et al., 2018).

The variety possesses rich intrinsic geometry:

• Picard group: $\mathrm{Pic}(\mathrm{LG}(n,2n)) \simeq \mathbb{Z}$, generated by the Plücker line bundle

$\mathcal{O}(1) := \det U^*,$

where $U$ is the rank $n$ tautological bundle.

• Canonical bundle: $\omega_{\mathrm{LG}(n,2n)} \cong \mathcal{O}(-n-1)$.
• Cohomology ring: Generated by the Chern classes $c_i(U^*)$, subject to symplectic-type relations; Schubert cycles are indexed by strict partitions $\lambda = (\lambda_1 > \cdots > \lambda_t > 0)$ with $t \leq n$.
• Plücker embedding: The image in projective space is cut out by quadratic equations encoding symplectic isotropy, including explicit equations involving principal and almost-principal minors (Kim, 3 Mar 2024, Gutt et al., 2018).

2. Schubert Calculus and Topology

Schubert cycles in $\mathrm{LG}(n,2n)$ are defined via incidence with isotropic flags: $$\Omega_\lambda(F_\bullet) = \left\{ \Sigma \in \mathrm{LG}(n,2n) \mid \dim(\Sigma \cap F_{n+1-\lambda_i}) \geq i,\ i = 1, ..., \ell \right\}.$$ Their codimensions are $|\lambda| = \sum_i \lambda_i$. The integral cohomology and homology are concentrated in even degrees, with Betti numbers indexed by strict partitions fitting inside the staircase $n \times n$ (Kim, 2023).

CW decompositions—both in the complex and real settings—are combinatorially controlled by shifted Young diagrams. For complex Lagrangian Grassmannians, all attaching degrees are zero, and odd-dimensional homology vanishes; the real locus $Lag^R(\mathbb{R}^{2n})$ admits a parallel but more subtle CW structure, with $2$-torsion appearing in odd codimensions, which is contractible within the complex manifold (Kim, 2023).

The quantum cohomology ring is generated by special Schubert classes $\sigma_i$ and the quantum parameter $q$, with relations of Giambelli–Pfaffian type and quantum corrections: $$QH^*(\mathrm{LG}(n,2n)) \cong \mathbb{C}[x_1, ..., x_n, q] / \langle x_i^2 - x_{i-1}x_{i+1} - q : i = 1, ..., n \rangle.$$ This presentation is essential for computing Gromov–Witten invariants and quantum Schubert calculus (Cheong, 2017, Pech et al., 2013, Hiep, 2016, Gu et al., 6 Feb 2025).

3. Projective and Combinatorial Models

The Plücker embedding of $\mathrm{LG}(n,2n)$ realizes it as a subvariety of $\mathbb{P}(V_{\omega_n})$, where $V_{\omega_n}$ is the irreducible $\mathrm{Sp}_{2n}$-module of highest weight $\omega_n$. In local charts, points correspond to graphs of symmetric $n \times n$ matrices, and Plücker coordinates are given by their principal minors (Boralevi et al., 2010, Gutt et al., 2018, Arthamonov et al., 2022, Kim, 3 Mar 2024). The equations defining the Lagrangian Grassmannian in these coordinates are quadratics among principal and almost-principal minors, generalizing the classical Grassmann–Plücker relations with symplectic symmetries (Kim, 3 Mar 2024, Arthamonov et al., 2022).

Tropicalizations and matroidal generalizations lead to the introduction of antisymmetric matroids and tropical Lagrangian Grassmannians, defined by analogues of the classical exchange and circuit axioms adapted to the symplectic context (Kim, 3 Mar 2024).

Cluster algebra structures have been established for $\mathrm{LG}(n,2n)$ via symmetric plabic graphs—most notably, rotationally symmetric plabic graphs parametrize the cells of the totally nonnegative Lagrangian Grassmannian, with closure relations encoded by symmetric affine permutations and a refined notion of positroids (Shevchenko, 28 Nov 2025, Wang, 2021, Chepuri et al., 2021). The exchange relations among cluster variables extend the Plücker relations and respect symplectic isotropy.

Newton–Okounkov bodies and superpotential polytopes associated with the Lagrangian Grassmannian relate cluster, toric, and mirror models, confirming cluster duality with their orthogonal Langlands duals (Wang, 2021, Pech et al., 2013).

4. Derived Categories and Exceptional Collections

The bounded derived category of coherent sheaves $D^b(\mathrm{LG}(n,2n))$ admits full exceptional collections, constructed by Kuznetsov–Polishchuk, based on Schur functors applied to the tautological bundle. These collections are organized into exceptional blocks $Y_{h,w}$ indexed by pairs $(h,w)$ with $h+w \leq n+1$ and are characterized by a semiorthogonal decomposition

$$D^b(\mathrm{LG}(n,2n)) = \langle A_0, A_1(1), ..., A_n(n) \rangle,$$

where $A_i$ is generated by right duals $\{ {}^{\lambda} \mid \lambda \in Y_{i, n-i} \}$ and $(j)$ denotes twisting by $\mathcal{O}(j)$. The fullness of these decompositions is established via Lagrangian staircase complexes, which iteratively resolve dual exceptional objects using fundamental symplectic representations and combinatorial manipulations of Young diagrams (Fonarev, 2019, Fonarev, 2023).

Left dual collections—constructed via graded poset duality—provide alternative (but equivalent) semiorthogonal decompositions, leading to explicit resolutions of equivariant vector bundles. The resolutions are indexed by balanced diagrams, and every irreducible equivariant bundle admits an explicit resolution by bundles in the dual block (Fonarev, 2023).

5. Quantum Cohomology, Mirror Symmetry, and Cluster Duality

The small quantum cohomology ring $QH^*(\mathrm{LG}(n,2n))$ is semisimple, with quantum multiplication operators diagonalizable in an explicit Q–polynomial eigenbasis. The spectrum of quantum multiplication by $c_1$ satisfies Conjecture O (Galkin–Golyshev–Iritani) in type C, with unique maximal eigenvalue, multiplicity one, and cyclic symmetry controlled by the Fano index (Cheong, 2017). The flat sections of the quantum differential equations are realized by oscillatory integrals over the mirror Landau–Ginzburg model (Pech et al., 2013).

On the mirror side, a Langlands dual orthogonal Grassmannian equipped with a superpotential explicitly reflects quantum cohomology through the Jacobi ring of critical points. The quantum connection corresponds to the Gauss–Manin connection for the superpotential, and the isomorphism between the Jacobi ring and $QH^*(\mathrm{LG}(m))$ is supported by the Peterson variety description and explicit calculations in low rank. New quantum relations in Schubert calculus—predicted by the LG superpotential—generalize classical Giambelli–Pfaffian formulas (Pech et al., 2013, Wang, 2021).

Cluster structures, Newton–Okounkov bodies, and totally nonnegative models provide a combinatorial and polyhedral framework for Lagrangian Grassmannians, confirming conjectural aspects of mirror and cluster dualities and linking algebraic, geometric, and representation-theoretic invariants (Wang, 2021, Shevchenko, 28 Nov 2025, Chepuri et al., 2021).

6. Applications, Topological and Homotopical Structure

Lagrangian Grassmannians are central to the study of moduli of special subvarieties, integrable PDEs of Monge–Ampère and Goursat-type (via their correspondence to hypersurfaces in $\mathrm{LG}(n,2n)$), and representation theory of classical groups. In topological settings, the CW structure, contractibility of real torsion classes in the complex variety, and the inclusion of the real locus as a subcomplex provide insights into homotopy extension and cellular structure (Kim, 2023).

The Hermitian $K$-theory of $\mathrm{LG}(n,2n)$ exhibits a complete splitting indexed by combinatorial data of shifted Young diagrams, computed via generalized Lagrangian flag schemes, which remain regular or reducible Gorenstein as needed to admit explicit pushforward and pullback operations (Huang et al., 10 Aug 2025).

Gauge-theoretic and mirror‐theoretic constructions (GLSMs, LG mirrors) provide uniform frameworks for quantum cohomology, $K$-theory, and enumerative invariants, relating classical characteristic polynomials (Schur $Q$-functions) to indices of defects and quantum multiplicative structures (Gu et al., 6 Feb 2025).

The role of the Lagrangian Grassmannian in the theory of antisymmetric matroids, tropical and oriented matroid analogues, and combinatorial representation over hyperfields, further advances the interaction with combinatorial geometry, with Baker–Bowler theory and generalizations to tracts ensuring that classical algebraic relations can be extended to very general base structures (Kim, 3 Mar 2024).

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References:

(Fonarev, 2019, Pech et al., 2013, Fonarev, 2023, Gutt et al., 2018, Kim, 2023, Shevchenko, 28 Nov 2025, Wang, 2021, Boralevi et al., 2010, Kim, 3 Mar 2024, Freire et al., 2018, Cheong, 2017, Gu et al., 6 Feb 2025, Huang et al., 10 Aug 2025, Arthamonov et al., 2022, Chepuri et al., 2021, Hiep, 2016, Kristel et al., 2023)