Symplectic Grassmannians

• Symplectic Grassmannians are varieties of k-dimensional isotropic subspaces in symplectic vector spaces, defined by the vanishing of the symplectic form.
• They arise as rational homogeneous spaces under the symplectic group and are central to studies in Schubert calculus and quantum cohomology.
• Their applications span integrable systems, representation theory, and mathematical physics, influencing developments in quantum K-theory and tropical geometry.

A symplectic Grassmannian is a parameter space of linear subspaces of a symplectic vector space with specific isotropy or nondegeneracy constraints relative to the symplectic form. These spaces play fundamental roles in algebraic geometry, representation theory, integrable systems, and mathematical physics due to their rich geometry, distinctive cohomological invariants, and connections with classical Lie groups and moduli problems. The theory incorporates characterizations via orbit structure, homogenous models, Schubert calculus, quantum (and quantum K-) theory, and tropical, combinatorial, and representation-theoretic degenerations.

1. Definitions, Models, and Basic Structure

Let $V$ be a finite-dimensional vector space (usually over $\mathbb{C}$ or $\mathbb{R}$) equipped with a nondegenerate symplectic (alternating, skew-symmetric) bilinear form $\omega$. For $k \leq n = \tfrac{1}{2}\dim V$, the symplectic Grassmannian $\mathrm{SpGr}(k,2n)$ is the algebraic variety of all $k$-dimensional isotropic subspaces $W \subset V$, i.e.,

$$\mathrm{SpGr}(k,2n) = \{\, W \subset V \mid \dim W = k,\; \omega|_{W \times W} = 0 \,\}$$

["GLSMs for exotic Grassmannians" (Gu et al., 2020)], [“A Characterization of Symplectic Grassmannians” (Occhetta et al., 2016)].

For maximal isotropic subspaces (i.e., $k = n$), $\mathbb{C}$0 is the Lagrangian Grassmannian.

An alternative, non-isotropic version, the symplectic Grassmannian of nondegenerate subspaces, parametrizes $\mathbb{C}$1-planes $\mathbb{C}$2 such that $\mathbb{C}$3 is nondegenerate (symplectic). These varieties, also called symplectic or nonisotropic Grassmannians, can be realized as adjoint orbits of involutions: $\mathbb{C}$4 (Lim et al., 30 Jan 2025).

In both cases, $\mathbb{C}$5 is a rational homogeneous space for the symplectic group $\mathbb{C}$6, typically presented as $\mathbb{C}$7 for the parabolic $\mathbb{C}$8 stabilizing a reference isotropic $\mathbb{C}$9-plane (Gu et al., 2020, Bode, 2012).

The (complex or real) dimension is

$\mathbb{R}$0

["Symplectic Grassmannians and Cyclic Quivers" (Feigin et al., 2024)].

2. Homogeneous Geometry, Orbit Structure, and Topology

Symplectic Grassmannians are homogeneous under $\mathbb{R}$1, with isotropy subgroups described as follows:

• For the symplectic Grassmannian of nondegenerate $\mathbb{R}$2-planes:

$\mathbb{R}$3

The action is transitive, and the spaces are compact connected homogeneous manifolds (Kim, 23 Jan 2026, Lim et al., 30 Jan 2025).

• The Lagrangian Grassmannian $\mathbb{R}$4 is $\mathbb{R}$5, a compact Kähler symmetric space of real dimension $\mathbb{R}$6 and complex dimension $\mathbb{R}$7 (Kristel et al., 2023). The Siegel domain (symmetric complex matrices with positive imaginary part) provides coordinates for an open cell.

In Morse–Bott theoretic terms, the Grassmannian of all subspaces decomposes into finitely many $\mathbb{R}$8-orbits, labeled by triples $\mathbb{R}$9 encoding maximal isotropic, complex, and symplectic content. Each orbit retracts (via negative gradient flow for a quadratic energy function) onto a compact quotient of the unitary group: $\omega$0 with the symplectic Grassmannian appearing as the case $\omega$1 (Kim, 23 Jan 2026, Kim, 2024).

Topological invariants (e.g., fundamental group, cohomology) are computed via this homogeneous bundle structure. For instance, $\omega$2 for the symplectic orbit, $\omega$3 for Lagrangian.

3. Schubert Calculus, Cohomology, and Combinatorial Geometry

Schubert calculus on symplectic Grassmannians generalizes type $\omega$4 theory, with k-strict partitions and isotropic flags governing Schubert data. The classical cohomology ring $\omega$5 is generated by the Chern classes of the tautological and quotient bundles, subject to symplectic Giambelli-type relations. Pfaffian sum formulas (built from equivariant theta and double Schubert polynomials) express Schubert classes in terms of special ones (Ikeda et al., 2014, Hudson et al., 2018).

The table below summarizes generating presentations in cohomology and K-theory:

Structure	Generators	Relations
Cohomology	$\omega$6	Giambelli/Pfaffian
$\omega$7-theory	$\omega$8	$\omega$9

Combinatorial objects controlling cell decompositions include:

• Weyl group: Schubert cells indexed by signed permutations (type $k \leq n = \tfrac{1}{2}\dim V$0) or, for 2-planes, admissible pairs avoiding conjugate indices (Angel et al., 2023).
• Affinization and patterns: Cellular decompositions in quiver degenerations are parameterized by symplectic juggling patterns or combinatorial necklaces, with top-dimensional cells matching classical Euler characteristics (Feigin et al., 2024).
• Toric/Matroidal geometry: $k \leq n = \tfrac{1}{2}\dim V$1-invariant subvarieties correspond to representable symplectic Coxeter matroids, controlling thin Schubert stratifications and moment polytopes (Angel et al., 2023).

The restriction of type $k \leq n = \tfrac{1}{2}\dim V$2 Schubert classes to the symplectic Grassmannian admits positive puzzle formulas ("self-dual puzzles") compatible with mirror symmetry and quantum integrable system structures (Halacheva et al., 2018).

4. Quantum Cohomology and Quantum K Theory

Quantum cohomology $k \leq n = \tfrac{1}{2}\dim V$3 is determined either by recursion on Gromov–Witten invariants (three-point correlators) or via Coulomb branch analysis of gauged linear sigma models (GLSMs) (Gu et al., 2020, Gu et al., 2020). The small quantum cohomology ring is presented by quantum-deformations of the Giambelli–Pierce relations, explicitly: $k \leq n = \tfrac{1}{2}\dim V$4 for the Chern roots $k \leq n = \tfrac{1}{2}\dim V$5, with Klein-type symmetries and quantum corrections entering in the top relations (Gu et al., 2020). In equivariant quantum cohomology, twisted masses $k \leq n = \tfrac{1}{2}\dim V$6 lead to factorial Schur-type relations coupling the equivariant and quantum parameters.

Quantum $k \leq n = \tfrac{1}{2}\dim V$7-theory for $k \leq n = \tfrac{1}{2}\dim V$8 is obtained from GLSM localization with Chern–Simons levels, yielding OPE algebras of half-BPS Wilson loops: $k \leq n = \tfrac{1}{2}\dim V$9 Quantum $\mathrm{SpGr}(k,2n)$0-theory is further organized in "shifted Wilson line" and $\mathrm{SpGr}(k,2n)$1 class bases, reflecting both geometric and physical structures (Gu et al., 2020).

For the odd symplectic Grassmannian $\mathrm{SpGr}(k,2n)$2, the quantum cohomology ring is semisimple—contrasting with the even case—and Dubrovin's conjecture on exceptional collections holds (Pech, 2010).

5. Characterization Theorems and Rigidity

Symplectic Grassmannians are naturally characterized by their varieties of minimal rational tangents (VMRT). If a Fano manifold of Picard number one has VMRTs locally or globally projectively equivalent to those of a symplectic Grassmannian, it is (biregularly) isomorphic to one (Occhetta et al., 2016, Hwang et al., 2019). This fills the "short root" case for rational homogeneous varieties, extending earlier parabolic geometry results.

$\mathrm{SpGr}(k,2n)$3

This rigidity extends under Kähler deformation (Hwang et al., 2019), and reduces the global geometry to local tangent behavior.

6. Quiver, Tropical, and Degeneration Approaches

Degenerate and tropical models reveal further structure:

• Quiver Grassmannians: The flat degenerations of $\mathrm{SpGr}(k,2n)$4 arise as loci of self-orthogonal subrepresentations for cyclic quivers with symplectic structure. Cells correspond to "symplectic juggling patterns" with explicit combinatorial formulas for Euler characteristics and Poincaré polynomials (Feigin et al., 2024).
• Tropicalization: The tropical symplectic Grassmannian is defined by tropicalizations of the Plücker and symplectic relations; these provide matroidal and fan-theoretic perspectives, but several combinatorial pathologies arise for $\mathrm{SpGr}(k,2n)$5 (Balla et al., 2021).
• Linked Grassmannians: In limit linear series theory, symplectic Grassmannians generalize to "linked symplectic Grassmannians" parameterizing chains of isotropic subbundles with expected codimension formulas, crucial in moduli of Brill–Noether loci (Osserman et al., 2011).

7. Applications, Extensions, and Open Directions

Symplectic Grassmannians and their degenerations appear across mathematical physics (mirror symmetry, GLSMs, gauge theory), representation theory (moduli of parabolic sheaves, Beilinson–Bott–Hodge theory), and combinatorics (matroids, puzzles, tableaux). Further developments include:

• Quantum and K-theoretic Schubert calculus (Grothendieck polynomials, double theta functions) (Hudson et al., 2018, Ikeda et al., 2014).
• Symplectic group factorizations into Grassmannians of involutions, with four-factor decompositions paralleling known orthogonal/unitary results (Lim et al., 30 Jan 2025).
• Cell decompositions and $\mathrm{SpGr}(k,2n)$6-fixed point loci controlled by BC-type matroids (Angel et al., 2023).
• Infinite-dimensional and loop Grassmannians (restricted classes, Banach manifold models) (Kristel et al., 2023).
• Open questions include the full characterization of quantum cohomology rings, positive combinatorial formulae for quantum and K-theoretic structure constants, explicit tropical and matroidal classification, and generalizations to other types and to singular moduli problems.

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Key references:

• "GLSMs for exotic Grassmannians" (Gu et al., 2020);
• "A Morse-Bott unification of the Grassmannians of a symplectic vector space" (Kim, 23 Jan 2026);
• "A Characterization of Symplectic Grassmannians" (Occhetta et al., 2016);
• "Quantum K theory of symplectic Grassmannians" (Gu et al., 2020);
• "Characterizing symplectic Grassmannians by varieties of minimal rational tangents" (Hwang et al., 2019);
• "Symplectic Grassmannians and Cyclic Quivers" (Feigin et al., 2024);
• "The tropical symplectic Grassmannian" (Balla et al., 2021);
• "Pfaffian sum formula for the symplectic Grassmannians" (Ikeda et al., 2014);
• "Double Grothendieck Polynomials for Symplectic and Odd Orthogonal Grassmannians" (Hudson et al., 2018);
• "On $\mathrm{SpGr}(k,2n)$7-invariant subvarieties of symplectic Grassmannians..." (Angel et al., 2023).