Plücker Vector Embeddings in Geometry

• Plücker vector embeddings are a coordinate-free, polynomial construction that maps linear subspaces onto points in a projective space via decomposable multivectors.
• They utilize quadratic and higher-degree Plücker relations to precisely encode incidence, intersection, and isotropy properties in varieties like Grassmannians and Lagrangian spaces.
• They drive applications in algebraic geometry, finite geometry, coding theory, robotics, and computer vision, enabling efficient computation and robust correction algorithms.

Plücker vector embeddings are a foundational construction in projective algebraic geometry, multilinear algebra, and an expanding range of applied domains. They yield a coordinate-free, polynomially structured embedding of linear subspaces—the Grassmannian, Lagrangian, and related moduli spaces—into projective spaces by realizing subspaces as decomposable multivectors. The Plücker embedding encodes incidence and intersection properties via quadratic (and, for generalizations, higher-degree) equations, central in both classical geometry and computational applications.

1. Classical Plücker Embedding and Quadratic Relations

The classical Plücker embedding maps the Grassmannian $\mathrm{Gr}(k, n)$, the variety of $k$-dimensional subspaces in an $n$-dimensional vector space $V$, into the projective space $\mathbb{P}(\bigwedge^k V)$ as follows. If $U\subset V$ with basis $u_1,\ldots,u_k$, the image is the line in $\bigwedge^k V$ spanned by $u_1 \wedge \cdots \wedge u_k$. A point in projective space corresponds to homogeneous coordinates $[p_I]$, where $I$ ranges over all increasing $k$-element subsets of $\{1,\dots,n\}$, and $p_I$ is the coordinate of $e_{i_1}\wedge\dots\wedge e_{i_k}$ in the multivector expansion. These coordinates, called Plücker coordinates, satisfy quadratic polynomial relations known as the Plücker relations, which cut out the image of the Grassmannian inside $\mathbb{P}(\bigwedge^k V)$ (Mandolesi, 13 Oct 2025, Sobczyk, 2018).

For $k=2$, and $V$ of dimension $n+1$, a 2-plane is determined by a bivector $p\wedge q$; the characterization of decomposability is equivalent to the vanishing of certain quadratic expressions in the Plücker coordinates: $$\sum_{m=0}^{n} \lambda_{im}\lambda_{jm} = 0,\quad \forall i<j.$$ For general $k$, the vanishing of $P\wedge P$ leads to a system of multi-index quadratic relations, encoding the fact that decomposable $k$-vectors correspond precisely to $k$-dimensional subspaces (Sobczyk, 2018).

2. Algebraic and Geometric Structure of Plücker Embeddings

The Plücker embedding enables a uniform polynomial description of various moduli spaces. For the Grassmannian $G(n,2n)$, the full image is defined by the quadratic Plücker relations. In the Lagrangian-Grassmannian $L(n,2n)$—the subvariety of maximal isotropic $n$-planes in a $2n$-dimensional symplectic space—additional linear constraints in Plücker coordinates are imposed, arising from the vanishing of a contraction map determined by the symplectic form. These cut out the kernel of maps $f:\bigwedge^n E \to \bigwedge^{n-2} E$, leading to a system of linear forms $H_\alpha$:

$$H_\alpha=\sum_{i=1}^n C_{i,\alpha,2n-i+1}\, p_{i,\alpha_1,\dots,\alpha_{n-2},2n-i+1}=0,$$

where $C_{i,\alpha,2n-i+1}$ encodes the combinatorics of isotropic subspaces (Carrillo-Pacheco et al., 2016).

For fields of arbitrary characteristic, this description remains valid, and the block structure of the associated linear constraint matrix enables efficient algebraic and computational treatment of Lagrangian subvarieties.

3. Extensions: Plücker Embeddings in Finite Fields and Coding Theory

Plücker embeddings have fundamental applications to finite geometry and coding theory. For the finite Grassmannian $G_q(k,n)$, which parameterizes $k$-dimensional subspaces of $\mathbb{F}_q^n$, the Plücker map provides a projective embedding: $$\varphi: G_q(k,n) \to \mathbb{P}^{\binom{n}{k}-1}(\mathbb{F}_q),\qquad U\mapsto [\Delta_I(U)],$$ where $\Delta_I(U)$ is the $k\times k$ minor for columns indexed by $I$ of any generator matrix of $U$. In the context of constant-dimension network codes, orbit codes formed from Galois actions retain their cyclicity via the Plücker embedding. The embedding preserves cardinality, and combinatorial and distance properties (such as Schubert variety stratifications for subspace balls) are naturally encoded in vanishing patterns of Plücker coordinates (Rosenthal et al., 2012).

Explicit worked examples demonstrate how orbit codes' structure and minimum subspace distance can be analyzed directly in Plücker space.

4. Plücker Embedding Beyond the Classical: New Relation Systems and Super/Infinite-Dimensional Cases

The algebraic description of the Grassmannian via the classical quadratic Plücker relations is strongly non-minimal. New generating sets of equations—such as the move–2–indices, or “Plücker-like” relations—cut out the same projective variety but dramatically reduce redundancy. For every pair of multi-indices $J\in\II^n_{p-2}$, $K\in\II^n_{p+2}$: $$\sum_{I\subset K\setminus J,\,|I|=2} (-1)^{\langle J\triangle K,\,I\rangle} \lambda_{J\cup I} \lambda_{K\setminus I} = 0,$$ and for larger $m$ one may enumerate further compact generators. These new presentations, as rigorously established, yield fewer equations, eliminate trivial and duplicate relations, and yield computational advantages—smaller Gröbner bases, more efficient algebraic manipulation—without altering the prime ideal of the Grassmannian (Mandolesi, 13 Oct 2025).

Extensions to the infinite (arc) Grassmannian, as in the semi-infinite Plücker embedding, involve generating functions of Plücker coordinates and yield infinitely many quadratic relations packed into power series identities. The coordinate ring is then identified with dual global Weyl modules for current algebras (Feigin et al., 2017).

5. Super Plücker Embedding and Supersymmetric Structures

The super Plücker embedding incorporates purely odd and mixed symmetry into the geometric framework. For a supervector space $V=V_{\bar 0}\oplus V_{\bar 1}$, the super Grassmannian $\mathrm{Gr}_{r|s}(V)$ is embedded into a weighted projective space involving $\Lambda^{r|s}(V)$ and a parity-reversed component.

The embedding is defined via: $\Phi: \mathrm{Gr}_{r|s}(V) \to \mathbb{P}_{(+1,-1)}(\Lambda^{r|s}(V)\oplus\Lambda^{s|r}(\Pi V)),\quad L\mapsto [\plu_{r|s}(L):\plu^*_{s|r}(L)],$ with coordinates provided by Berezinian minors of a supermatrix. The system of super Plücker relations—quadratic equations among these minors—enforces decomposability in the super sense. There is a direct connection to supercluster algebras: for instance, in $\mathrm{Gr}_{2|0}(\mathbb{R}^{4|1})$, even and odd Plücker variables satisfy relations reflecting cluster algebra exchange patterns and their “super-extensions” (Shemyakova et al., 2019).

In particular, mixed parity, affine charts, and the equivalence between Khudaverdian's relations and the super-Plücker system for certain templates are rigorously established.

6. Computational and Algorithmic Aspects: Plücker Correction and Robust Embedding

In applied domains, especially 3D geometry, robotics, and computer vision, Plücker vector embeddings are critical for robust representation and computation with geometric primitives such as lines. For a 6D vector $(p^\top,m^\top)^\top\in \mathbb{R}^6$ to correspond to a line in Plücker coordinates, the Klein quadric constraint $p^\top m=0$ must be satisfied.

In practical tasks, noisy inputs can violate this constraint. The Plücker correction problem seeks the closest valid Plücker vector. An efficient closed-form algorithm is given: $$x^\star = \frac{a - \alpha b}{1 - \alpha^2},\quad y^\star = \frac{b - \alpha a}{1-\alpha^2}$$ with $\alpha = \frac{2p}{q+\sqrt{q^2-4p^2}}$, $p = a^\top b$, $q = \|a\|^2 + \|b\|^2$. This method outperforms SVD-based corrections in both simplicity and empirical timing, requiring only a fixed number of vector operations and a square root per correction (Cardoso et al., 2016).

This approach enables robust, on-the-fly correction for millions of line elements in applications including camera calibration, structure-from-motion, SLAM, and multi-camera systems.

7. Interconnections and Broader Contexts

Plücker vector embeddings operate as a central unifying tool across multiple areas: algebraic geometry (structure of Grassmann and flag varieties), representation theory (connections to global Weyl modules), finite geometry (coding theory and network coding), geometric computation (computer vision and robotics), and the rapidly growing area of supergeometry and cluster algebra theory.

Advances in the understanding of their underlying algebraic structure—such as minimal or more efficient generators for defining relations—directly enhance the tractability of symbolic computation and open research avenues in representation theory, moduli of special subspaces, and extensions to derived and infinite-dimensional geometry (Mandolesi, 13 Oct 2025, Sobczyk, 2018, Shemyakova et al., 2019, Feigin et al., 2017).

A plausible implication is that continued improvements in the explicit characterization of the Plücker image—via computational, representation-theoretic, and geometric refinement—will further broaden the embedability and analyzability of complex geometric and algebraic data structures within both theoretical mathematics and applied algorithmic systems.