Plücker Ray Maps: Grassmannian & Isotropic Geometry

• Plücker ray maps are canonical projective embeddings that associate each N-plane with its decomposable exterior wedge product, defining a unique projective ray.
• They enable a deep connection between Plücker coordinates and Cartan (spinor) maps by linking determinants with Pfaffian line bundles through bilinear identities.
• This framework underpins applications in integrable systems and representation theory, offering explicit coordinate relations and a bridge between algebraic and geometric invariants.

The Plücker map, often referred to in projective-geometric contexts as a "ray map," provides a canonical embedding of a Grassmannian into a projective space by associating to each $N$-plane its exterior wedge product—a decomposable $N$-vector, thus identifying each point with a unique projective ray. In the context of the vector space $W=V\oplus V^*$, where $V$ is an $N$-dimensional complex vector space, this embedding plays a foundational role in relating the geometry of general and isotropic Grassmannians. Its structure also facilitates profound links with the theory of Cartan (spinor) maps, Pfaffian line bundles, and determinant identities of Cauchy-Binet type, particularly on the locus of maximal isotropic subspaces. The key results and constructions in this domain are summarized and rigorously developed in (Balogh et al., 2020).

1. Plücker Embedding and Ray Assignment

Let $V$ be a complex vector space of dimension $N$ and $V^*$ its dual. Consider $W=V\oplus V^*$, and the Grassmannian $Gr_N(W)$ of $N$-dimensional subspaces $w\subset W$. For any basis $w=\{w_1,\dots,w_N\}$ of such a subspace, the exterior product $w_1\wedge\cdots\wedge w_N\in\wedge^N W$ defines a projective equivalence class (a "ray") in $\mathbb P(\wedge^N W)$. The classical Plücker map

$\mathrm{Pl}\;:=\;\Blv:\;Gr_N(W)\;\longrightarrow\;\mathbb P(\wedge^N W),\quad \Blv(\mathrm{span}\{w_1,\dots,w_N\})=[w_1\wedge\cdots\wedge w_N]$

is thus viewed as the canonical "Plücker ray map". Each point of the Grassmannian is associated with a unique line (projective ray) in $\wedge^N W$. The map acts compatibly with projective lines: a one-parameter family $w(t)$ in the Grassmannian traces a projective line in $\mathbb P(\wedge^N W)$ under $\Blv$.

The Plücker coordinates are indexed by partitions $\lambda$ whose Young diagrams fit within an $N\times N$ square. In a combined basis $\{e_1,\dots,e_N,f_1,\dots,f_N\}$ for $W$, the decomposable $N$-vector can be expanded as

$\Blv(w)=\sum_{\lambda\subseteq(N^N)} T_\lambda(w) |\lambda\rangle$

where $T_\lambda(w)=\det W_\lambda(w)$ for each $N\times N$ minor $W_\lambda(w)$ determined by multi-index $\lambda$.

2. Isotropic Grassmannians and the Cartan (Spinor) Map

On $W=V\oplus V^*$, introduce the canonical nondegenerate pairing $Q((X,p),(Y,q))=q(X)+p(Y)$ for $X,Y\in V$, $p,q\in V^*$. The isotropic Grassmannian $\Gr^0(W,Q)$ consists of $N$-planes $w^0$ such that $Q(w_i,w_j)=0$ for all $i,j$.

Cartan’s construction realizes $\wedge V$ as an irreducible Clifford (spinor) module under the action of

$$\Gamma_v = v\wedge + i_v \in \operatorname{End}(\wedge V)$$

where $v\in W$ acts by exterior and interior multiplication. For a maximal isotropic $N$-plane $w^0=\mathrm{span}(w_1,\dots,w_N)$, the operator product $\Gamma_{w_1}\Gamma_{w_2}\cdots\Gamma_{w_N}$ yields a pure spinor in $\wedge V$. The Cartan embedding is then

$\Cay:\;\Gr^0(W,Q) \rightarrow \mathbb P(\wedge V),\quad \Cay(w^0)= [\Gamma_{w_1}\cdots\Gamma_{w_N}]$

The homogeneous coordinates $k_a(w^0)$—Cartan coordinates—are labeled by strict partitions $a=(a_1>\cdots>a_r)$ and are given by

$$k_a(w^0) = (-1)^{\cdots}\operatorname{Pf}(A^{(a)}(w^0))$$

where $A(w^0)$ is the skew-symmetric $N\times N$ affine coordinate matrix, and $A^{(a)}(w^0)$ is the principal $r\times r$ submatrix.

3. Coordinate Description: Plücker and Cartan Systems

The Plücker coordinates $T_\lambda(w)$ arise as determinants of $N\times N$ minors extracted from the $2N\times N$ coordinate matrix $W(w)$ of the $N$-plane $w\subset W$. For isotropic $N$-planes, affine coordinates $A(w^0)$ parameterize the "big cell" in the Grassmannian, and the Cartan coordinates are determined as Pfaffians of principal submatrices.

Object	Coordinates	Explicit Construction
General $N$-plane $w$	Plücker $T_\lambda(w)$	$\det W_\lambda(w)$
Isotropic $N$-plane $w^0$	Cartan $k_a(w^0)$	$(-1)^{\cdots} \operatorname{Pf}A^{(a)}(w^0)$

This coordinate system enables explicit computations and forms the basis for the bilinear relations between these two embeddings on the isotropic locus.

4. Bilinear Identities: Cauchy–Binet–Pfaffian Relation

A fundamental connection arises between the Plücker and Cartan embeddings through a quadratic map $B_N:\wedge V\times\wedge V\rightarrow\wedge^N W$ induced by the spin–pairing. Explicitly, for the isotropic locus, one has

$B_N(\Cay(w^0),\Cay(w^0)) = \Blv(w^0)\quad\text{in}\;\mathbb P(\wedge^N W)$

in particular, the image of the Cartan map under this quadratic spinor pairing is precisely the Plücker image.

In coordinates, this gives a Pfaffian analogue of the Cauchy–Binet formula. Let $A(w^0)$ be the affine coordinate matrix, and $I, J\subset\{1,\dots, N\}$, $|I|=|J|=r$, then for $A(I|J)$ the $r\times r$ submatrix,

$$\det A(I|J) = 2^r\, (-1)^{\frac{r(r-1)}2} \sum_{ \substack{ K,L\;\mathrm{even}\ K\cup L=I\cup J\ K\cap L=I\cap J } } (-1)^{r d+v(I,J,K,L)}\, \operatorname{Pf}A(K|K)\, \operatorname{Pf}A(L|L)$$

where the exponents $d$ and $v(I,J,K,L)$ are combinatorial. Thus, each Plücker coordinate is bilinear in the Cartan coordinates, precisely reflecting the factorization through the Segre embedding.

5. Geometric Interplay: Line Bundles and Embedding Factorization

The Plücker embedding realizes $Gr_N(V\oplus V^*)$ as the variety of decomposable $N$-vectors in $\mathbb P(\wedge^N(V\oplus V^*))$, with each point corresponding to a "ray." Projective lines in the Grassmannian are mapped to projective lines in the image. On $\Gr^0(V\oplus V^*,Q)$, the Cartan map produces a subvariety of pure spinors in $\mathbb P(\wedge V)$, with homogeneous Cartan coordinates realizing holomorphic sections of the dual Pfaffian line bundle $\mathrm{Pf}^*\to\Gr^0$.

A central identity identifies the restriction of the determinantal line bundle $\mathrm{Det}^*\to Gr_N$ to $\Gr^0$ in terms of the square of the Pfaffian line bundle, and expresses the Plücker embedding as the composite of the Segre embedding

$$\mathbb P(\wedge V)\times\mathbb P(\wedge V)\to\mathbb P(\wedge^N(V\oplus V^*))$$

with the Cartan map diagonally. This shows that on isotropic $N$-planes, the determinant-type Plücker coordinates factor through the quadratic pairing of spinor-type (Pfaffian) coordinates.

6. Context and Implications in Representation and Integrable Systems

The framework is motivated by, and directly applies to, the study of $\tau$-functions of the KP and BKP integrable hierarchies, which are interpreted as sections of determinantal and Pfaffian line bundles over infinite-dimensional Grassmannians. In finite-dimensional settings, these maps make explicit the interplay of exterior algebra, Clifford modules, and projective geometry.

This approach establishes important links between classical invariants in linear algebra (determinants, Pfaffians), projective algebraic geometry, and the representation theory of symmetric spaces, with far-reaching consequences for the study of integrable systems, representation theory of Lie algebras, and the geometry of moduli spaces (Balogh et al., 2020).