Plücker–Ray Embedding Overview

• Plücker–Ray embedding is an algebraic-geometric construction that maps the Grassmannian to a projective subvariety using determinant-based Plücker coordinates and quadratic relations.
• It provides the underlying framework for network coding applications, particularly in constant-dimension and rank-metric codes, by enabling explicit decoding strategies.
• The construction also connects to modern algebraic geometry and representation theory through noncommutative resolutions and homological projective duality.

The Plücker–Ray embedding is a fundamental algebraic-geometric construction that realizes the Grassmannian $\Gr(k,V)$ of $k$-dimensional subspaces of a finite- or infinite-dimensional vector space $V$ as a projective algebraic subvariety in $\P(\Lambda^k V)$, determined by the Plücker (alternatively, Plücker–Ray) coordinates. This embedding is central in the study of constant-dimension codes for network coding, representation theory, invariant theory, and homological projective duality, and provides the algebraic structure underpinning several decoding and duality frameworks.

1. The Grassmannian and the Plücker Embedding

Let $V$ be an $n$-dimensional vector space over a base field $\Bbbk$ (or $\F_q$ in finite geometry), and $\Gr(k,V)$ the Grassmannian of $k$-planes in $V$. To each subspace $W \subset V$ of dimension $k$, one associates a line in the $k$-th exterior power $\Lambda^k V$ by $w_1 \wedge \cdots \wedge w_k$, where $w_1, \dots, w_k$ form a basis of $W$. The Plücker embedding is the canonical projective morphism

$\iota \colon \Gr(k, V) \to \P(\Lambda^k V)$

given in coordinates by

$$W \mapsto [p_I(W)]_{I\subset\{1,\dots,n\},\,|I|=k},$$

where, relative to a fixed basis $e_1, \dots, e_n$ of $V$, each Plücker coordinate is

$$p_I(W) = \det\left(\text{matrix of } k \text{ basis vectors of } W \text{ in columns of } I\right).$$

Each change of basis on $W$ corresponds to an action by $\GL(k)$, scaling the wedge product by $\det(\text{basis change})$, so these coordinates are projective invariants.

The image of $\Gr(k,V)$ under $\iota$ is a subvariety defined by quadratic Plücker relations inside projective space of dimension $\binom{n}{k} - 1$ (Doyle, 2021).

2. Algebraic Structure: Plücker Coordinates and Relations

The Plücker image consists of all points

$$[p_I] \in \P\left(\Lambda^k V\right) \cong \P^{\binom{n}{k}-1}$$

satisfying the homogeneous Plücker ideal $I_P$, generated by quadratic relations. For disjoint multi-indices $I, J$, with $|I|=k-1$, $|J|=k+1$,

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

where $J = \{j_0 < \dots < j_k\}$, and $I\cap J=\emptyset$. These relations are necessary and sufficient for a point to lie in the Plücker image (Doyle, 2021).

The affine cone $A$ over the Plücker embedding is given by the ring $R = \Sym^\bullet(\Lambda^k V^*) / I_P$.

3. Applications to Coding Theory

Grassmannian codes, specifically constant-dimension codes, are sets of $k$-subspaces of $\F_q^n$ employed in random network coding. The Plücker embedding provides an explicit algebraic representation of codewords and their neighborhoods. For lifted Gabidulin codes $\Clift \subset G_q(k,n)$, matrices of the form $[I_k \mid A]$ represent codewords' row spaces, and certain Plücker coordinates encode the entries of $A$.

If $I = \{1, \dots, k\} \setminus \{s\} \cup \{t\}$, $k < t \le n$, then $p_I$ is (up to sign) the $(s, t-k)$-th entry of the rank-metric code matrix $A$. The set of such $k(n-k)$ Plücker coordinates forms a linear block code $C^p \subset \F_q^{k(n-k)}$, whose parity-check matrix $H^p$ yields linear equations

$\bar p\, H^{pT} = 0$

in the “special” coordinate slots (Trautmann et al., 2013).

The subspace distance

$d_S(U, V) = 2k - 2\dim(U \cap V)$

between subspaces is expressible via the vanishing of selected Plücker coordinates, defining subspace balls in Plücker space by linear equations. Explicitly, a ball $B_{2t}(U_0)$ around the subspace $U_0$ corresponds to the locus where $p_I = 0$ for all $I \not\le (t+1,\dots,k,n-t+1,\dots,n)$, with the “$\le$” relation the Bruhat order (Trautmann et al., 2013, Rosenthal et al., 2012).

4. Cyclic Orbit Codes and Extension-Field Embeddings

When considering orbit codes generated by an irreducible cyclic subgroup of $\GL_n$, the Plücker embedding extends to an explicit description over extension fields. For an irreducible polynomial $p(x) \in \F_q[x]$ of degree $n$, its companion matrix $P \in \GL_n(\F_q)$ defines the group action. The extension-field isomorphism

$\Phi: \F_q^n \to \F_{q^n},\quad (v_1,\dots,v_n) \mapsto \sum_{i=1}^n v_i \alpha^{i-1},$

where $\alpha$ is a root of $p(x)$, induces an embedding on wedge products,

$\phi': \Lambda^k \F_q^n \to \Lambda^k \F_{q^n},$

associating to a row space $\mathcal{U}$ the projective class $[ \phi(u_1) \wedge \cdots \wedge \phi(u_k) ]$ in $P(\Lambda^k \F_{q^n})$ (Rosenthal et al., 2012).

Group equivariance is manifested as follows: For a cyclic code $C = \{ \mathcal{U}\cdot P^i \}$, the Plücker images form a single orbit under the Singer cycle, and the subspace distance as well as code cardinality are preserved in the embedding. These properties facilitate characterization and decoding of such codes via projective algebraic techniques (Rosenthal et al., 2012).

5. The Plücker Embedding in List Decoding

List decoding in Grassmannian codes leverages the Plücker embedding by reducing the decoding problem to solving a system of linear and quadratic equations in Plücker coordinates. For a received subspace $R$, the list decoding up to radius $2t$ entails:

• Linear block code parity-check equations $\bar p\, H^{pT}=0$,
• Linear equations cutting out the subspace ball $B_{2t}(R)$,
• Quadratic shuffle (Plücker) relations,
• Projective normalization (e.g., $p_{1,\dots,k}=1$).

The intersection of these conditions yields all codewords within the prescribed distance. Notably, for lifted Gabidulin codes, only the “special” $k(n-k)$ Plücker coordinates carry non-redundant information, enabling computational reduction (Trautmann et al., 2013).

This approach geometrizes the combinatorial neighborhood structure, reinterpreting list decoding as finding common points of linear and bilinear varieties in projective space.

6. Homological and Categorical Aspects

The Plücker embedding admits deep connections to noncommutative and homological algebraic geometry. The homogeneous coordinate ring of the Plücker variety, modulo the Plücker ideal, forms the affine cone $A$, for which Špenko–Van den Bergh constructed a noncommutative crepant resolution (NCCR). The NCCR $\Lambda = \End_R(M)$, with $M$ the sum of covariant modules tied to irreducible $\SL(k)$-representations indexed by Young diagrams, provides a ring-theoretic resolution of $A$ (Doyle, 2021).

The Kuznetsov component of the derived category of coherent sheaves is realized as a subcategory of matrix factorizations on this NCCR, explicitly linked by Knörrer periodicity. Homological projective duality (HPD) is formulated in this context, with the derived category of linear sections of the Grassmannian semi-orthogonally decomposed in terms of the NCCR and the Fonarev rectangular Lefschetz decomposition.

The HPD theorem articulates, for any linear subspace $L \subset \Lambda^k V^*$, a decomposition relating $D^b(\Gr(k,V)\cap\P(L^\perp))$ and the subcategory $\mathcal{D}|_{\P(L)}$, capturing the deep symmetry between the Plücker embedding and its duals (Doyle, 2021).

7. Summary Table of Key Structures

Structure	Definition / Key Feature	Citation
Plücker coordinates	$k \times k$ minors of a $k \times n$ matrix spanning $W$	(Doyle, 2021)
Plücker relations	Quadratic equations cutting out $\Gr(k, V)$ in projective space	(Doyle, 2021)
Special Plücker coordinates	Coordinates encoding rank-metric Gabidulin code entries	(Trautmann et al., 2013)
NCCR (Špenko–Van den Bergh)	Noncommutative crepant resolution for affine Plücker cone	(Doyle, 2021)
Equivariance in orbit codes	Plücker embedding commutes with cyclic group action via extension field	(Rosenthal et al., 2012)
Subspace (Schubert) ball	Subvariety cut by coordinate vanishing, plus quadratic Plücker relations	(Trautmann et al., 2013, Rosenthal et al., 2012)

The Plücker–Ray embedding thus serves as an algebraic and geometric backbone for the analysis and manipulation of Grassmannian varieties, their applications in coding theory, and their role in modern homological and noncommutative geometry.