Plücker Embeddings

• Plücker embeddings are canonical projective maps of Grassmannians that use the dth exterior power to represent d-dimensional subspaces as points in projective space.
• They are defined via maximal minors, with quadratic Plücker relations characterizing the embedded image as an algebraic variety.
• They bridge disciplines such as algebraic geometry, combinatorics, tropical geometry, physics, and computer vision through diverse practical applications.

A Plücker embedding is a canonical, projective embedding of a Grassmannian (the parameter space of $d$-dimensional subspaces of an $n$-dimensional vector space) into a projective space, using the exterior power and its associated multilinear coordinates. The theory of Plücker embeddings links multilinear algebra, algebraic geometry, combinatorics, and representation theory through the use of wedge products, determinantal coordinates, and quadratic Plücker relations, which characterize the image as an algebraic variety. This construction is central not just for pure mathematics, but for applications as diverse as mathematical physics, matroid theory, tropical geometry, complexity theory, matrix completion, and controllable generative models in computer vision.

1. Fundamental Definition and Classical Construction

Let $V$ be a vector space of dimension $n$ over a field (typically $\mathbb{C}$ or $\mathbb{R}$). The Grassmannian $\operatorname{Gr}(d, V)$ parametrizes $d$-dimensional linear subspaces of $V$. The Plücker embedding is defined via the $d$th exterior power:

• Each $d$-dimensional subspace $W \subset V$ is represented by a nonzero decomposable $d$-vector $w_1 \wedge \dots \wedge w_d \in \Lambda^d V$, where $\{w_1, \dots, w_d\}$ is a basis for $W$.
• Up to scaling, this point determines a unique element in the projective space $\mathbb{P}(\Lambda^d V)$.

Explicitly, if $\{e_1, \dots, e_n\}$ is a basis for $V$, the “Plücker coordinates” $p_I$ of $W$ (where $I$ ranges over $d$-element subsets of $\{1, \ldots, n\}$) are given as the maximal minors of the $n \times d$ matrix whose columns are the basis vectors of $W$. The map

$$\operatorname{Gr}(d, V) \to \mathbb{P}(\Lambda^d V) : W \mapsto [w_1 \wedge \dots \wedge w_d]$$

is a smooth projective embedding whose image is cut out by quadratic equations known as the Plücker relations.

2. Plücker Relations and Algebraic Structure

The coordinates $p_I$ satisfy a complete set of quadratic relations—Plücker relations—arising from the condition that a decomposable tensor represents a $d$-plane. For classical $d=2$ in $n=4$, these reduce to the identity: $p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0.$ In general, the Plücker relations are given by antisymmetrized quadratic constraints, cutting out the Grassmannian as a subvariety of $\mathbb{P}^{\binom{n}{d}-1}$. These relations are interpreted as characterizing simple $d$-forms—those that can be written as a wedge product of $d$ 1-forms—and their tropical and quantum analogs play central roles in modern generalizations (Giansiracusa et al., 2015, Danilov et al., 2020).

In quantum matrix theory, quantum minors satisfy “quantum Plücker relations,” which also serve to define quantum analogs of the coordinate ring of the Grassmannian and Flag varieties (Danilov et al., 2020).

In tropical geometry and matroid theory, tropical Plücker relations—valuated exchange axioms—define tropical linear spaces and valuated matroids through the analogue of these classical relations, and a new cryptomorphic approach expresses the tropical Plücker property as the freeness of the top tropical wedge power of a certain quotient module (Giansiracusa et al., 2015).

3. Geometric and Combinatorial Properties

The image of the Plücker embedding is a projective subvariety whose points correspond to $d$-planes via their Plücker coordinates. This embedding is equivariant under the natural action of $\mathrm{GL}(V)$. Rational scrolls and other extremal varieties (e.g., those maximizing the number of double points under generic projection) are precisely characterized by their image under the Plücker embedding, as in the Plücker–Clebsch formula and its higher-dimensional analogues (Ciliberto et al., 2010).

For degeneracy loci, dual varieties, and projections, the behavior of Plücker coordinates encodes the enumerative geometry of singularities (nodes, cusps, etc.), as formalized in the classical Plücker formulas and their affine/tropical counterparts (Esterov, 2013, Osserman et al., 2018).

The relation between total positivity (in the sense of Lusztig for real forms of classical Lie types) and Plücker coordinate positivity directly underlies deep interactions between total positivity, matroids, and tropical geometry (Barkley et al., 15 Oct 2024).

4. Extensions and Generalizations

Super and Quantum Plücker Embeddings

In the super context, the Plücker embedding extends to super Grassmannians $G_{r|s}(V)$ by associating to each $(r|s)$-plane in a super vector space $V$ a parity-sensitive pair in a weighted projective superspace $$\mathbb{P}(\Lambda^{r|s}(V) \oplus \Lambda^{s|r}(\Pi V))$$. The super Plücker relations reflect both “even” and “odd” degrees, and their structure is crucial for constructing and analyzing super cluster algebras (Shemyakova et al., 2019).

Quantum Plücker embeddings define the coordinate ring of the quantum Grassmannian, where quantum minors satisfy quantum Plücker-type relations, co-Plücker relations, and Dodgson-type identities. The overall structure and uniqueness of QI-functions on quantum minors rely on these quantum analogues (Danilov et al., 2020).

Tropical and Infinite-Dimensional Plücker Embeddings

Tropical geometry recasts the Plücker embedding in terms of tropical fans and polytope algebras, leading to a framework where affine Thom polynomials translate into tropical Plücker-type formulas for enumerative invariants of affine morphisms, with applications to toric and tropical correspondence theorems (Esterov, 2013).

Further, the paper of the dual infinite wedge and its generalizations exposes uniform boundedness (noetherianity) of the defining equations for “Plücker varieties” even in the infinite-dimensional case, with hyper-Pfaffians furnishing the building blocks for these equations and the associated limiting varieties (Nekrasov, 2020).

5. Applications in Geometry, Physics, and Computation

The Plücker embedding is ubiquitous in enumerative geometry, moduli theory, and representation theory. In mathematical physics, it governs the structure constants of metric 3-Lie algebras through generalized Plücker relations, imposing stringent constraints on allowed gauge symmetries in multiple M2-brane theories (there is no metric 3-Lie algebra associated to $\mathfrak{u}(N)$ for $N > 2$; structure constants must be sums of volume forms on orthogonal 4-planes) (0804.2662).

In discrete and computational geometry, Plücker embeddings and coordinates are foundational for representing lines and higher-dimensional subspaces in computer vision (including multi-view geometry and structure-from-motion), as well as for efficient geometric computation on vector architectures and GPUs. Plücker coordinates enable stable and division-free algorithms for line and plane intersection, robust against floating-point error (Skala, 2022).

In modern machine learning, the Plücker embedding underpins formulations of low-rank matrix completion, where the problem is equivalently cast as intersecting the Plücker-embedded Grassmannian with a family of hyperplanes defined by observed data patterns. Deterministic conditions for unique and finite completion correspond to intersections with prescribed supports, with deep connections to matching fields and matroid theory (Tsakiris, 2020).

Recent advances in score-based generative modeling have used pixelwise Plücker embeddings to condition large video diffusion transformers for fine-grained 3D camera control, providing geometrically faithful representations of rays for each pixel and enabling explicit, differentiable generator control (Bahmani et al., 17 Jul 2024).

6. Broader Impact and Theoretical Consequences

Plücker embeddings represent a central interface between linear algebra, algebraic geometry, and combinatorics. The universality and geometric intuition provided by decomposable tensors, together with the algebraic structure of the Plücker relations, provide the backbone for developments across classical and tropical geometry, quantum group theory, and computational approaches. They consistently serve as a testing ground and unifying theme for generalizations—super, quantum, and tropical—and for the design of extremal and uniformity results (e.g., noetherianity, total positivity).

The explicit characterization of the Plücker embedding’s image, the structure of its defining equations, and their stability under limits have far-reaching consequences in the paper of moduli spaces, tropical and matroid geometry, representation stability, and in the integration of geometric and combinatorial notions into applied contexts such as optimization, error-correcting codes, computer vision, and generative modeling.