Plücker–Ray Embedding in Grassmannians

• The Plücker–Ray embedding is defined by a new system of minimal, non-redundant quadratic (Plücker-like) relations that recover the classical Grassmannian image.
• This approach streamlines computational methods in algebraic geometry, facilitating efficient syzygy analysis, Gröbner basis computations, and applications in network coding.
• It preserves key geometric properties such as smoothness and dimensionality while connecting to broader generalizations in flag varieties and noncommutative geometry.

The Plücker–Ray embedding is a variant of the classical Plücker embedding of Grassmannians, characterized by a new system of quadratic equations—termed "Plücker-like"—that defines the Grassmannian subvariety as a projective algebraic variety. This approach uses a minimal and non-redundant generating set of quadrics, reflecting advances in the theory of defining ideals for Grassmannians and streamlining computational methods in algebraic geometry. The Plücker–Ray embedding recovers the image of the Grassmannian in projective space and its homogeneous ideal, with implications for syzygies, free resolutions, and applications in several branches of mathematics and coding theory.

1. Classical Plücker Embedding and Its Defining Ideal

Given a field $\F$ (either $\R$ or $\C$), let $\Gr(p,n)$ denote the Grassmannian of $p$-dimensional subspaces of $\F^n$. The projective space $\PP(\bigwedge^p\F^n)$ parametrizes lines through the origin in the space of $p$-vectors. The classical Plücker embedding

$\Gr(p,n)\xhookrightarrow{\;\;\;} \PP(\bigwedge^p\F^n), \qquad L=\Span\{v_{i_1},\ldots,v_{i_p}\} \mapsto [v_{i_1}\wedge\cdots\wedge v_{i_p}]$

realizes the Grassmannian as a smooth irreducible subvariety of dimension $p(n-p)$, cut out by the vanishing of a set of quadratic equations—classical Plücker relations.

For a $\R$0-vector $\R$1, where multi-indices are ordered tuples $\R$2, a necessary and sufficient condition for decomposability is that the Plücker coordinates $\R$3 satisfy all relations

$\R$4

where $\R$5 is the combinatorial index pairing. These form $\R$6 quadratic equations, though only $\R$7 independent generators are required to define the ideal. The ideal is prime and homogeneous of degree two (Mandolesi, 13 Oct 2025).

2. The Plücker–Ray Embedding: Definition and New Plücker-like Relations

In contrast to the classical system, the Plücker–Ray embedding is delineated by the so-called "Plücker-like" quadratics, introduced by Mandolesi. These relations operate by moving two indices simultaneously, rather than single-index moves as in the classical setting. Formally, for $\R$8 and $\R$9,

$\C$0

These equations, labeled (PL$\C$1), provide a new generating set for the defining ideal $\C$2 (Mandolesi, 13 Oct 2025).

Key properties of the (PL$\C$3) system:

• All (PL$\C$4) equations are non-trivial and irredundant.
• The total number of equations is $\C$5, a significant reduction compared to the classical system for $\C$6.
• Both the classical (PL) and new (PL$\C$7) systems define the same radical ideal—the locus of decomposable $\C$8-vectors.

Set-theoretically and scheme-theoretically, the vanishing locus of (PL$\C$9) coincides with the classical Plücker image, and Hilbert function arguments show their radicals agree.

3. Explicit Structure, Counting, and Examples

The size and structure of the defining set are readily computable. For the Grassmannian $\Gr(p,n)$0:

• $\Gr(p,n)$1, $\Gr(p,n)$2.
• There are six 3-term (classical) and six 10-term (PL$\Gr(p,n)$3) equations. The affine chart $\Gr(p,n)$4 exhibits nine independent affine coordinates $\Gr(p,n)$5, and the ten-term equations provide explicit quadratic relations.

A correction is noted in the literature, where the correct leading term of the 10-term relation for $\Gr(p,n)$6 is $\Gr(p,n)$7, not $\Gr(p,n)$8.

The total number of (PL$\Gr(p,n)$9) equations is

$p$0

This minimizes redundancy and optimizes the defining system for computational applications such as syzygies and Gröbner bases (Mandolesi, 13 Oct 2025).

4. Hierarchical and Structural Reductions

Each (PL$p$1) equation is a $p$2-sum of specific classical (PL) equations, confirming that $p$3. Conversely, since both systems define precisely the set of decomposable multivectors, $p$4. This structural redundancy elimination yields "smaller" and more efficient generating sets, preserving smoothness and dimension of $p$5, with no effect on singularity properties.

The ideal structure thus enables both theoretical simplification and practical efficiency in algebraic computations related to the Grassmannian and its secant varieties (Mandolesi, 13 Oct 2025).

5. Realization as an Embedding and Scheme-theoretic Features

The Plücker–Ray embedding is explicitly the classical Plücker map $p$6, considered together with the (PL$p$7) ideal generators. The step-by-step procedure is:

1. Introduce homogeneous coordinates $p$8.
2. Impose the (PL$p$9) quadrics.
3. The vanishing locus recovers set-theoretically and scheme-theoretically the Grassmann variety.

The induced projective variety is irreducible, of dimension $\F^n$0, and generically smooth; the radical of the (PL$\F^n$1)-generated ideal is the prime Plücker ideal. This realizes $\F^n$2 as a closed embedding where the minimal generating set is optimized.

6. Connections to Broader Plücker-type Frameworks

This methodology connects to a spectrum of generalizations:

• In infinite-dimensional and current-algebraic contexts, "semi-infinite" Plücker relations govern embeddings of formal or affine flag varieties, as in the Drinfeld–Plücker embedding, which involves an infinite system of relations indexed by formal derivatives and partitions (Feigin et al., 2017).
• The conceptual device—replacing redundant or overspecified systems with minimal generating sets—has analogues in homological algebra and noncommutative geometry, where efficient resolutions and quiver descriptions are sought for moduli and derived categories (Doyle, 2021).

In projective geometry, analogous minimal and transparent embeddings maintain incidence structure, such as transparent embeddings in point-line geometries, further generalizing the approach (Cardinali et al., 2016).

7. Implications and Applications

The Plücker–Ray embedding has notable computational and theoretical implications:

• Reduced generator counts facilitate symbolic computation in syzygy modules and Gröbner basis calculations.
• In coding theory, efficient parametrizations of Grassmannians via wedge powers underlie the structure of constant-dimension and orbit codes, with implications for network coding (Rosenthal et al., 2012, Trautmann et al., 2013).
• In representation theory and algebraic geometry, the defining ideals, especially when minimally generated, are crucial for understanding orbit closures, secant varieties, and homological dualities.

A plausible implication is that the systematic reduction of generator set size for defining ideals will benefit future computational strategies in commutative algebra, as the smoothness and geometric properties of the Grassmannian are unaffected by the particular choice of generating set (Mandolesi, 13 Oct 2025).