Plücker Embeddings: Theory and Applications

• Plücker embeddings are canonical maps that embed Grassmannians into projective space via wedge products, defining quadratic relations known as Plücker relations.
• They provide a projective algebraic structure for decomposable k-vectors, with applications spanning algebraic geometry, representation theory, and integrable systems.
• Generalizations, including super and semi-infinite embeddings, extend Plücker theory to flag varieties and coding theory, driving advances in both pure and applied mathematics.

The Plücker embedding is the canonical projective embedding of a Grassmannian or flag variety, central to projective geometry, representation theory, algebraic geometry, integrable systems, and coding theory. By mapping a $k$-dimensional subspace of a vector space $V$ into projective space via the wedge product of a basis, the embedding realizes the Grassmannian $G(k, n)$ as a projective algebraic variety cut out by quadratic polynomial relations—the Plücker relations. Generalizations include super, relative, skew, and semi-infinite analogs across a spectrum of mathematical disciplines.

1. Classical Plücker Embedding: Construction and Equations

Let $V$ be an $n$-dimensional vector space over a field $k$ (real, complex, or finite). The Grassmannian $G(k, n)$ parametrizes $k$-dimensional subspaces $W \subset V$. For a basis $\{v_1,\dots,v_k\}$ of $W$, the $k$-blade $v_1 \wedge \cdots \wedge v_k$ is well-defined up to scalars and identifies $W$ with a point in projective space $\mathbb{P}(\wedge^k V)$. The Plücker embedding is

$$\varphi: G(k, n) \hookrightarrow \mathbb{P}(\wedge^k V),\qquad W \mapsto [v_1\wedge \cdots\wedge v_k].$$

Expanding $v_1\wedge \cdots \wedge v_k$ in terms of a basis $\{e_{i_1}\wedge \cdots \wedge e_{i_k}\}$, one obtains Plücker coordinates $p_{i_1\cdots i_k}$ as the coefficients.

A point in the image satisfies the quadratic Plücker relations: for any subsets $I$ of size $k-1$ and $J$ of size $k+1$,

$$\sum_{s=1}^{k+1} (-1)^s\, p_{I \cup \{j_s\}}\, p_{J \setminus \{j_s\}} = 0.$$

These relations cut out the Grassmannian as a subvariety of projective space and ensure the decomposability of the $k$-vector in $\wedge^k V$ (Sobczyk, 2018, Christophersen et al., 2014, Aokage et al., 11 Apr 2025).

2. Plücker Relations in Geometric and Algebraic Language

The Plücker relations capture the locus of decomposable $k$-vectors in $\wedge^k V$ and have several equivalent formulations:

• Geometric algebra: In Clifford algebra $G(V)$, the Plücker relations are written as $(A\cdot B)\, B=0$ for all $(k-1)$-blades $A$, where $B$ is a $k$-vector. This condition is necessary and sufficient for decomposability (Sobczyk, 2018).
• Coordinate-free: Decomposable $k$-vectors precisely satisfy these quadratic equations.
• Algorithmic decomposability check: Given coordinates $\{p_I\}$, assemble $B$, compute $(e_K\cdot B) B$ for all $(k-1)$-blades $e_K$, and check if all vanish as multivectors (Sobczyk, 2018, Mandolesi, 13 Oct 2025).

The redundancy of classical Plücker relations has motivated alternative, more efficient quadratic systems, such as the Mandolesi "new Plücker-like equations," which, for $p$-vectors in $\wedge^p V$, yield fewer relations yet still cut out only the decomposable locus (Mandolesi, 13 Oct 2025).

3. Generalizations: Super, Relative, Skew, and Semi-Infinite Plücker Embeddings

Plücker theory extends beyond classical Grassmannians:

• Super Plücker embedding: For the super Grassmannian $G_{r|s}(V)$ of a super-vector space $V=V_0 \oplus V_1$, the map targets a weighted projective space over $\Lambda^{r|s}(V) \oplus \Lambda^{s|r}(\Pi V)$ and is defined via Berezinians and parity-reversal. Super Plücker coordinates satisfy quadratic super-relations, which generalize both the classical and Khudaverdian relations (Shemyakova et al., 2019).
• Relative and skew Plücker relations: These arise from the projective embedding of flag varieties and their Schur determinant realization as Jacobi–Trudi determinants. Relative relations govern minors of mixed size, cutting out full flag varieties in the product of projective spaces; the skew version, defined via skew-Schur determinants and generalized combinatorial data, further extend the Plücker geometry to flags with additional structure (Aokage et al., 11 Apr 2025).
• Semi-infinite Plücker embedding: The Drinfeld–Plücker embedding for semi-infinite flag varieties is governed by "semi-infinite Plücker relations," which involve formal generating series of minors and encode the homogeneous coordinate ring as a direct sum of dual global Weyl modules (Feigin et al., 2017).

4. Representation-Theoretic and Cohomological Aspects

The Plücker embedding admits an equivariant description: the homogeneous coordinate ring of the image (the Plücker algebra) is the ring of $\text{SL}(W)$-invariants in ${\rm Sym}(E^*\otimes W)$, with the ideal generated in degree two by quadratic Plücker relations (Christophersen et al., 2014). Bott's theorem provides a uniform vanishing pattern for the (co)homology of homogeneous vector bundles on $G(r, n)$ and establishes that, outside the simplest cases, higher cotangent cohomology of the Plücker algebra vanishes in large ranges. This vanishing ensures, for example, unobstructedness of deformations of sufficiently high-codimension complete-intersection subvarieties in cones over the Grassmannian (Christophersen et al., 2014).

The algebraic structure is further illuminated by the isomorphism of the coordinate ring modulo relations with duals of irreducible (or global Weyl) modules. For flag or semi-infinite flag varieties, the corresponding homogeneous coordinate ring has basis indexed by Schur polynomials or semi-standard tableaux, further connecting to combinatorics and symmetric function theory (Gatto, 2021, Feigin et al., 2017).

5. Plücker Embedding in Codes, Metrics, and Integrable Systems

Beyond pure algebraic geometry, the Plücker embedding is central in several applied and theoretical settings:

• Coding theory: Orbit codes are families of constant-dimension subspaces used in network coding. Embedding codewords using Plücker coordinates allows efficient storage, explicit coordinate descriptions of balls in the subspace metric, and algebraic decoding schemes leveraging the orbit structure and linearity of Plücker embedding (Trautmann, 2012). The computational complexity is significantly reduced compared to direct minor recomputation.
• Local Plücker formulas: The embedding intertwines with the geometry of holomorphic curves and osculating flags. Local Plücker formulas relate curvatures and metrics of the induced pullback of the Fubini–Study metric to the Cartan matrix of related Lie algebras, with explicit reductions from type $A_n$ (general linear) to orthogonal types $B_n$, $D_n$ via Segre and Veronese factorizations (Degtyarev, 2022).
• Integrable systems: The Hirota bilinear equations for the modified KP hierarchy are equivalent to (skew) Plücker relations among Schur functions; the addition of relative and skew structure matches the appearance of Miura transforms and tau-function addition formulae (Aokage et al., 11 Apr 2025).

6. Plücker Embedding and Schubert Calculus

The Plücker embedding not only realizes the Grassmannian/flag varieties but organizes the geometry of Schubert varieties via explicit coordinate and cohomological data:

• The degree of a Schubert variety under the Plücker embedding equals the number of standard Young tableaux of the associated shape, and this number matches the top-order $x_1$-partial derivative of the corresponding Schur polynomial $S_\lambda(x)$—a consequence of Chern class computations and determinantal formulas (Gatto, 2021).
• Generating functions summing over all Schubert degrees recover explicit exponential forms, reinforcing the transparent combinatorial and geometric structure imposed by Plücker coordinates and relations.

7. Open Problems, Efficiency, and Future Directions

Recent developments include the enumeration and classification of new, less redundant Plücker-like quadratic systems, with open questions regarding minimization of relations, geometric significance of higher-index-level equations, and extension to other homogeneous varieties (symplectic, orthogonal, and super cases) (Mandolesi, 13 Oct 2025). Super cluster algebras, the combinatorics of mutations in super-Grassmannians, and canonical bases compatible with Plücker-type relations remain active research areas (Shemyakova et al., 2019, Aokage et al., 11 Apr 2025).

References:

(Sobczyk, 2018, Christophersen et al., 2014, Trautmann, 2012, Degtyarev, 2022, Aokage et al., 11 Apr 2025, Shemyakova et al., 2019, Mandolesi, 13 Oct 2025, Feigin et al., 2017, Gatto, 2021)