Grassmannian Formulation Overview

• Grassmannian formulation is a geometric and algebraic framework that encodes k-dimensional subspaces of an n-dimensional vector space, forming a basis for various mathematical theories.
• It leverages explicit models like the Plücker embedding and projection matrices to simplify computations in algebraic geometry, differential geometry, and physics.
• The formulation underpins scattering amplitude computations, optimization algorithms, and combinatorial analyses, enabling efficient and robust numerical methods.

The Grassmannian formulation is a geometric and algebraic framework in which central objects and processes in algebraic geometry, representation theory, mathematical physics, and optimization are encoded in terms of subspaces or "k-planes" inside a vector space. Grassmannian varieties, denoted $\operatorname{Gr}(k,n)$, parametrize $k$-dimensional linear subspaces of an $n$-dimensional (real or complex) vector space and have a rich structure, manifesting in algebraic, differential, combinatorial, and physical contexts. The recent literature develops powerful explicit and computationally efficient models for the Grassmannian, revealing deep connections to scattering amplitudes, supersymmetric sigma models, positive geometry, K-theory, and more (Olson, 2014, Lai et al., 2020, Lim et al., 2024, De et al., 25 Mar 2026, Bendokat et al., 2020, Lisitsyn et al., 31 Dec 2025, Borovik et al., 2023, Arkani-Hamed et al., 2012, Carlsson, 2013, Kreshchuk et al., 2019, Eberhardt et al., 2020, Cortes et al., 6 May 2025).

1. Geometric Models and Algebraic Embeddings

Several concrete models of the Grassmannian exist, each advantageous for particular applications.

• Plücker Embedding: $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$, mapping a $k$-plane to its maximal minors (Plücker coordinates) of a $k \times n$ representative matrix. The image is cut out by quadratic Plücker relations (Borovik et al., 2023, Kreshchuk et al., 2019).
• Projection Matrix Model: The set of symmetric, idempotent $n \times n$ matrices of rank $k$ with $\operatorname{tr}(P)=k$ (Lim et al., 2024, Bendokat et al., 2020). Common in applications for representing subspaces directly.
• Involution Model (Symmetric Orthogonal): The affine variety $$\{ Q \in O(n) : Q^T = Q,\, Q^2 = I,\, \operatorname{tr}(Q)=2k-n \}$$, with $k$0 for $k$1 the projection (Lai et al., 2020, Lim et al., 2024, Lai et al., 2024). Offers numerically stable and computationally efficient formulations for geometric and optimization tasks.

Each model possesses a tractable and explicit description of tangent and normal spaces, metric, exponential and logarithm map, parallel transport, and curvature (Bendokat et al., 2020, Lai et al., 2020, Lai et al., 2024).

2. Differential Geometry and Curvatures

Recent advances provide explicit closed-form expressions for intrinsic and extrinsic differential geometry on the Grassmannian, especially in the involution model (Lai et al., 2024).

• Tangent Space: At $k$2, $k$3.
• Metric: Trace inner product, $k$4.
• Riemann Curvature: For tangent vectors $k$5, $k$6, $k$7, $k$8.
• Sectional, Ricci, Scalar Curvature: Sectional curvature for a $k$9-plane: $n$0; Ricci: $n$1; Scalar: $n$2.
• Second Fundamental Form, Weingarten Map, Mean Curvature: $n$3; Weingarten map $n$4; mean curvature given explicitly in terms of $n$5 and $n$6.
• Higher Tensors: Includes detailed formulas for Schouten, Weyl, Bach, and vanishing of Cotton/non-Riemannian tensors. All admit block-matrix expressions tractable in high-performance linear algebra (Lai et al., 2024).

This detailed catalog enables stable computation of all geometric invariants, facilitating robust optimization and analysis.

3. Grassmannian Formulation in Scattering Amplitudes

Grassmannian integrals unify the computation of tree-level and (in some contexts) loop-level scattering amplitudes in maximally supersymmetric gauge and gravity theories. In $n$7 super Yang-Mills (SYM), the color-ordered $n$8-point N$n$9MHV tree amplitude is given by a contour integral over $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$0:

$$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$1

where $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$2 is a $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$3 matrix (modulo $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$4), $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$5 are cyclic minors, and $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$6 are momentum twistors (Olson, 2014, Arkani-Hamed et al., 2012, Rao, 2014). Positroid cells—loci where certain minors vanish—stratify $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$7 and correspond to physical factorization channels (Arkani-Hamed et al., 2012, Lisitsyn et al., 31 Dec 2025).

Orientation and Locality: Summing residues over these cells requires careful assignment of orientation signs, computable algorithmically via path-dependent products of edge-weights in a poset of cells. The signed boundary operator satisfies $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$8, ensuring exact cancellation of spurious poles and analytic locality of the amplitude (Olson, 2014).

BCFW Bridge Charts: BCFW recursion corresponds to sequences of adjacent transpositions in the permutation labeling of positroid cells, introducing one parameter and $$\operatorname{Gr}(k,n) \to \mathbb{P}^{\binom{n}{k}-1}$$9-form per bridge (Olson, 2014, Arkani-Hamed et al., 2012).

Generalizations: The Grassmannian formalism extends to celestial amplitudes (via Mellin transforms), Wilson loops (amplituhedron interpretation), and to gravity theories, with $k$0 integrals reflecting KLT factorizations and soft theorems (Cachazo et al., 2012, Ferro et al., 2021, De et al., 25 Mar 2026).

4. Combinatorial, Tropical, and Positive Geometries

Positive Grassmannian: The locus where all ordered $k$1 minors are positive, $k$2 embodies a "positive geometry" whose canonical form is the $k$3-fold wedge of $k$4's of positive parameters. Each planar on-shell diagram corresponds to a positroid cell in $k$5 with its canonical measure, underpinning the geometry of scattering amplitudes (Arkani-Hamed et al., 2012).

Non-Planar and Symplectic/Othogonal Grassmannians: Non-planar amplitudes map to unions of positive cells ("oriented regions"), leading to pseudo-positive geometries (Lisitsyn et al., 31 Dec 2025). For Coulomb branch and higher spin applications, symplectic ($k$6) or orthogonal Grassmannians ($k$7) become the integration domains, encoding additional mass or spin structures (Cortes et al., 6 May 2025, De et al., 25 Mar 2026).

Tropical Grassmannian: The polyhedral (fan) structure defined by tropical Plücker relations governs the combinatorics of generalized biadjoint amplitudes, soft theorems, and their factorization, yielding a correspondence between facets of the tropical Grassmannian and leading terms in soft expansions (Sepúlveda et al., 2019).

5. Optimization, Numerical Algorithms, and Applications

Efficient algorithms on Grassmannians underlie problems spanning signal processing, computer vision, neural network training, and physics-influenced variable projection frameworks.

Matrix Models and Riemannian Calculus:

• Involution Model (Lai et al., 2020, Lai et al., 2024). All gradient, Hessian, exponential, and transport operations reduce to explicit block-matrix formulas, usually involving QR decompositions and block-skew exponentials. The model achieves $k$8 arithmetic per iteration, is robust to ill-conditioning, and bypasses eigen/SVD where possible.
• Projector and Stiefel Quotient Models (Bendokat et al., 2020, Bendokat et al., 2020). Projector models favor direct subspace operations but are numerically unstable for high codimension, while the quotient model is stable but operates on equivalence classes.

Optimization Landscape and Variable Projection: The fundamental variable projection (VarPro) method for structured least squares exhibits a Grassmannian landscape with provable absence of spurious local minima, gradient and trust-region methods converge globally with probability 1 in the overparametrized regime. Rank-deficient phases are resolved using smoothly regularized submanifolds (Dus, 30 Jan 2026). Table below summarizes the main computational models (Lai et al., 2020):

Model	Representation	Key Cost	Remarks
Quotient / Stiefel	$k$9, $k \times n$0 mod $k \times n$1	QR + SVD	Stable, equivalence classes
Projection	$k \times n$2, $k \times n$3	Direct projections	Unstable for large $k \times n$4
Involution	$k \times n$5, $k \times n$6	QR + exp(block skew)	Stable, no classes

Curvature Formulas and Computational Geometry: All curvature and geometric quantities—including the full Riemann, Ricci, scalar, Schouten, Weyl, Bach, fundamental forms and delta invariants—proven to admit simple matrix operations (Lai et al., 2024).

6. Algebraic and Topological Invariants

Degree of Affine Grassmannians: Closed-form expressions for the degree of $k \times n$7 in projection and involution matrix models resolve the Devriendt–Friedman–Sturmfels conjecture, with combinatorial and representation-theoretic formulas involving gamma products and Selberg-type integrals (Lim et al., 2024).

Soft Theorems and Dualities: In both physical (amplitudes) and combinatorial contexts, Grassmannians support structural factorization theorems in the soft limit, and exhibit underlying dualities such as $k \times n$8 at the level of canonical forms, delta constraints, and combinatorics (Rao, 2014, Arkani-Hamed et al., 2012, Sepúlveda et al., 2019).

K-theoretic and Infinite-Dimensional Grassmannians: In equivariant K-theory, ind-Grassmannians support projection formulas for virtual classes, generalizing Borel–Weil–Bott and yielding character formulas for infinite-dimensional algebras and Macdonald-type identities (Carlsson, 2013).

7. Extensions: VOAs, Sigma Models, and Physical Theories

Grassmannian coset constructions yield a hierarchy of vertex operator algebras (VOAs) encompassing unitary and Lagrangian types, with explicit central charges, representation theory, and triality/pentality symmetries (Eberhardt et al., 2020). In sigma models, the Grassmannian target space enables the construction of $k \times n$9 supersymmetric models with calculable $n \times n$0-functions, rich large-$n \times n$1 behavior, and exact non-renormalization theorems (Kreshchuk et al., 2019). These structures are central in the algebraic and quantum geometric approaches to moduli, amplitude gluing, and higher symmetry.

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References

• (Olson, 2014) Orientations of BCFW Charts on the Grassmannian
• (Lai et al., 2020) Simpler Grassmannian optimization
• (Lim et al., 2024) Degree of the Grassmannian as an affine variety
• (De et al., 25 Mar 2026) The Vasiliev Grassmannian
• (Bendokat et al., 2020) A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects
• (Lisitsyn et al., 31 Dec 2025) Grassmannian Geometries for Non-Planar On-Shell Diagrams
• (Borovik et al., 2023) Coupled Cluster Degree of the Grassmannian
• (Arkani-Hamed et al., 2012) Scattering Amplitudes and the Positive Grassmannian
• (Carlsson, 2013) A projection formula for the ind-Grassmannian
• (Kreshchuk et al., 2019) On Grassmannian Heterotic Sigma Model
• (Eberhardt et al., 2020) The Grassmannian VOA
• (Cortes et al., 6 May 2025) Symplectic Grassmannian description of the Coulomb branch three and four point amplitudes
• (Lai et al., 2024) Simple matrix expressions for the curvatures of Grassmannian
• (Sepúlveda et al., 2019) A Soft Theorem for the Tropical Grassmannian
• (Rao, 2014) Soft Theorem of N=4 SYM in Grassmannian Formulation
• (Ferro et al., 2021) The Grassmannian for Celestial Superamplitudes
• (Cachazo et al., 2012) Gravity in Twistor Space and its Grassmannian Formulation