Real Grassmannians in Symplectic Geometry

• Real Grassmannians are manifolds that parameterize k-dimensional subspaces in symplectic vector spaces, with special emphasis on maximal isotropic (Lagrangian) subspaces.
• The topic covers detailed algebraic descriptions using Plücker coordinates, providing explicit equations and homogeneous space representations for isotropic and Lagrangian conditions.
• Insights include practical applications in symplectic geometry, representation theory, and mathematical physics, enriched by stratification methods and geodesic metric analyses.

A real Grassmannian of a symplectic vector space parametrizes the linear subspaces of a given dimension in a finite-dimensional real vector space equipped with a symplectic (nondegenerate skew-symmetric bilinear) form. Particular attention is devoted to the Lagrangian Grassmannian, consisting of maximal isotropic subspaces, and more generally to the stratification of the real Grassmannian by symplectic type. This structure underlies much of modern symplectic geometry, representation theory, and mathematical physics.

1. Definitions: Symplectic, Isotropic, and Lagrangian Grassmannians

Let $(V, \omega)$ be a real symplectic vector space of dimension $2n$, with $\omega \in \Lambda^2 V^*$ a nondegenerate skew form. The ordinary Grassmannian $Gr(k, 2n)$ parametrizes $k$-dimensional subspaces $W \subset V$.

• Isotropic Grassmannian: Subspaces $W \subset V$ with $\omega|_{W} = 0$ (i.e., $W$ is isotropic) form the isotropic Grassmannian:

$$Gr_{is}(k,2n;\mathbb{R}) = \{ W \subset \mathbb{R}^{2n} : \dim W = k, \, \omega|_W = 0 \}$$

This is a compact smooth manifold of real dimension $k(2n-k) - \frac{1}{2}k(k-1)$, corresponding to the homogeneous space $Sp(2n, \mathbb{R})/P_k$, where $P_k$ is a maximal parabolic subgroup stabilizing an isotropic $k$-plane (Lim et al., 30 Jan 2025, Cortes et al., 6 May 2025).

• Lagrangian Grassmannian: Maximal isotropic subspaces (where $\dim W = n$) form the Lagrangian Grassmannian:

$$L(n, 2n) = \{ W \subset V : \dim W = n,\, \omega|_W = 0 \}$$

It is smooth, compact, and connected, with real dimension $\frac{n(n+1)}{2}$ (Kristel et al., 2023, Carrillo-Pacheco et al., 2016). As a homogeneous space $L(n,2n) \cong Sp(2n, \mathbb{R}) / U(n)$; when a compatible complex structure is fixed, $L(n,2n) \cong U(n)/O(n)$ (Kristel et al., 2023).

• Symplectic Grassmannian: The locus of $k$-planes $W$ on which $\omega|_W$ is nondegenerate (i.e., symplectic) defines the symplectic Grassmannian, which is itself a homogeneous space under $Sp(2n,\mathbb{R})$ (Bendokat et al., 2021).

2. Algebraic and Coordinate Descriptions

Isotropic and Lagrangian Conditions in Plücker Coordinates

The Grassmannian $Gr(k,2n)$ admits a Plücker embedding into $\mathbb{P}^{\binom{2n}{k}-1}$. The isotropic locus $SpGr(k,2n)$ is defined by

$C \Omega C^T = 0_{k \times k}$

where $C$ is a $k \times 2n$ full-rank matrix representing basis vectors for $W$ and $\Omega$ is the matrix of the symplectic form. In Plücker coordinates, this isotropy condition yields additional linear equations—called symplectic Plücker relations—cutting out $SpGr(k,2n)$ as a closed subvariety of $Gr(k,2n)$ (Cortes et al., 6 May 2025, Carrillo-Pacheco et al., 2016).

For the real Lagrangian Grassmannian $L(n,2n)$, Carrillo-Pacheco et al. provide explicit linear equations in Plücker coordinates. For every $(n-2)$-subset $\alpha \subset \{1, \ldots, 2n\}$, the contraction linear form

$$H_\alpha := \sum_{i=1}^n X_{ \{ i, \alpha_1, \ldots, \alpha_{n-2}, 2n-i+1 \} } = 0$$

together with the classical Plücker quadrics, cut out $L(n,2n)$ inside $Gr(n,2n)$ (Carrillo-Pacheco et al., 2016).

Local Charts and the Siegel Disk Model

A standard open chart on $L(n,2n)$, based at the coordinate Lagrangian $\langle e_1, \ldots, e_n \rangle$, identifies each nearby Lagrangian as the graph of a symmetric $n \times n$ matrix $S$. Under this identification, the local patch is isomorphic to the vector space of real symmetric matrices, and the atlas consists of such symmetric patches covering $L(n,2n)$ (Carrillo-Pacheco et al., 2016, Kristel et al., 2023).

Alternatively, positive symplectic polarizations parametrized by the Siegel disk $\mathcal{D} = \{Z: Z = \bar{Z}^T, \|Z\|<1\}$ provide another coordinatization, giving $L(n,2n)$ the structure of a bounded symmetric domain of complex dimension $n(n+1)/2$ (Kristel et al., 2023).

3. Homogeneous Space and Morse–Bott Stratification

Homogeneous Descriptions

The real symplectic group acts transitively on $Gr_{is}(k,2n)$ and $L(n,2n)$: $$Gr_{is}(k,2n) \simeq Sp(2n, \mathbb{R}) / P_k \qquad L(n,2n) \simeq Sp(2n, \mathbb{R}) / U(n)$$ where $P_k$ is a parabolic stabilizer and $U(n)$ the unitary stabilizer. With a compatible complex structure, $L(n,2n) \simeq U(n)/O(n)$.

Morse–Bott Decomposition

Given a compatible triple $(g,J,\omega)$, one constructs a Morse–Bott function on $Gr(k;V)$ whose critical loci are subspaces split as

$W = W_0 \oplus (W \cap J W)$

where $W_0 = W \cap W^\omega$ (isotropic kernel) and $W \cap J W$ (maximal complex summand). This stratifies the Grassmannian into Sp(V)-orbits labeled by symplectic type $(n_0, n_+, n_-)$, with

$n_0 + 2n_+ = k, \quad n_0 + 2n_- = 2n - k$

and yields a disjoint union

$$Gr(k; V) = \bigsqcup_{n_0+2n_+=k} Gr(n_0, n_+, n_-; V)$$

Each stable manifold $Gr(n_0, n_+, n_-;V)$ deformation-retracts onto the $U(n)$-orbit $$U(n)/\left(O(n_0) \times U(n_+) \times U(n_-)\right)$$, giving topological control and allowing computation of Betti numbers in closed form (Kim, 23 Jan 2026, Kim, 2024).

4. Involution Model, Schubert Decomposition, and Topology

The isotropic Grassmannian may be modeled by involutions anti-commuting with the symplectic structure: $$\mathcal{P}_k = \{ P \in \mathrm{Mat}_{2n \times 2n}(\mathbb{R}) : P^2 = I, \, P^T J P = -J,\, \mathrm{tr}\,P = 2k - 2n \}$$ This correspondence $W \mapsto P_W$ provides a concrete matrix model for enumerating or composing isotropic subspaces (Lim et al., 30 Jan 2025).

A Schubert cell decomposition, indexed by partitions respecting the isotropic flag, gives a cell structure with known closure relations (Bruhat order) and homology ring generated by Schubert classes subject to Giambelli–Pieri relations for the symplectic case (Lim et al., 30 Jan 2025, Cortes et al., 6 May 2025).

The topology is stratified, with each orbit $Gr(\vec n;V)$ homotopy equivalent to a compact symmetric space of type $A_{n-1}$, e.g., the Lagrangian Grassmannian’s homotopy type is $U(n)/O(n)$, while more general symplectic and coisotropic types yield symmetric quotients reflecting the decomposition of a subspace relative to its symplectic orthogonal (Kim, 2024).

5. Metrics, Geodesics, and Applications

The real symplectic Grassmannian $\SpGr(2n,2k)$ is a smooth manifold equipped with natural pseudo-Riemannian and right-invariant Riemannian metrics:

• The bi-invariant metric on $\Sp(2n)$ descends to $\SpGr(2n,2k)$, with geodesics given by

$$\gamma(t) = \exp(t[\Gamma, P]) P \exp(-t[\Gamma, P])$$

for $\Gamma \in T_P \SpGr$ (Bendokat et al., 2021).

• Local retractions (Cayley transform) and their inverses are explicitly computable, supporting efficient optimization algorithms on $\SpGr(2n,2k)$ with applications in data analysis, structure-preserving reduction, and the “nearest symplectic matrix” problem.

In mathematical physics, symplectic Grassmannians provide the kinematic space for Coulomb-branch amplitudes in $\mathcal{N}=4$ super Yang-Mills theory, where integration over $\SpGr(k,2n)$ with respect to its canonical measure realizes three- and four-point amplitudes as the localization of these integrals (Cortes et al., 6 May 2025).

6. Cohomology, Characteristic Classes, and Homotopy

The real cohomology ring of $Gr(\vec n;V)$ is

$$H^\bullet\left(Gr(\vec n); \mathbb{R}\right) \cong H^\bullet\left(U(n);\mathbb{R}\right)^{O(n_0)\times U(n_+)\times U(n_-)}$$

with generators corresponding to Pontryagin (real bundles) and Chern (complex bundles) classes of tautological subbundles determined by the stratification’s type $(n_0, n_+, n_-)$, with relations arising from the Whitney sum and isotropy constraints (Kim, 2024, Kristel et al., 2023).

Topologically, the Lagrangian Grassmannian has $\pi_1 \cong \mathbb{Z}$ reflecting the Maslov class; higher cohomology is built from symmetric functions of the corresponding characteristic classes. All of $Gr(\vec n;V)$ are connected and admit a strong deformation retraction onto the compact symmetric base $U(n)/(O(n_0) \times U(n_+) \times U(n_-))$.

7. Fiber Bundles, Reductions, and Generalizations

Fiber bundle structures arise from the symplectic reduction perspective: projection from a subspace $W$ to its isotropic kernel $W_0 = W \cap W^\omega$ induces fibrations

$Gr(\vec n; V) \longrightarrow Gr(n_0, 0, n-n_0; V)$

with contractible Siegel-type fibers parameterizing the reduced symplectic data. The same framework governs the stratification of the complex Lagrangian Grassmannian, with orbits classified by signature of the Hermitian form associated to $\omega$ and the complex structure (Kim, 2024).

The singularity and fiber structures are crucial in infinite-dimensional settings (restricted Grassmannians), moduli of Fock representations, loop groups, and in applications to the representation theory of canonical commutation relations (Kristel et al., 2023).

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References:

(Carrillo-Pacheco et al., 2016, Bendokat et al., 2021, Kristel et al., 2023, Kim, 2024, Lim et al., 30 Jan 2025, Cortes et al., 6 May 2025, Kim, 23 Jan 2026).