nLab Plücker embedding

Idea

A Plücker embedding is an embedding of algebraic varieties from the Grassmannian of $k$-planes in a $n$-dimensional vector space $V$ to the projective space $P(\wedge^k V)$ of the $k$-th exterior power of $V$. It sends every $k$-dimensional subspace $W\subset V$ into the ray $[w_1\wedge w_2\wedge\ldots\wedge w_k]\in$ for any ordered basis $w_1,w_2,\ldots,w_k$ of $W$; this clearly does not depend on the basis and its ordering.

The homogeneous coordinate ring of the image is generated by the projective classes of elements of a basis of $\wedge^k V$ given by ordered $k$-th exterior products of elements of a basis of $V$. These are the Plücker coordinates and they satisfy Plücker relations.

There are quantum analogues (deformations).

Related entries

• Hirota equation
• Grassmannian, flag variety
• standard monomial

Literature

• Wikipedia, Plücker embedding

Extensive classical reference is the volume 3 of:

• W. V. D. Hodge, Daniel Pedoe, Methods of algebraic geometry, 3 vols. (reviewed by Coxeter in Bull. Amer. Math. Soc. 55, 3, part 1 (1949) 315–316, euclid)

For quantum analogues see

• E. Taft, Jacob Towber, Quantum deformation of flag schemes and Grassmann schemes $q$-deformation of the shape-algebra for $GL(n)$, J. Algebra 142 (1991), 1-3

A fully noncommutative analogue, the quasi-Plücker coordinates, are found within the Retakh-Gelfand work on quasideterminants and explained in more detail in

• Aaron Lauve, Quantum- and quasi-Plücker coordinates, J. Algebra 296:2 (2006) 440–461 doi
• V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization (2007) [pdf]

Some issues in super-case:

• E. Shemyakova, Theodore Voronov, On super Plücker embedding and possible application to cluster algebras, arXiv:1906.12011

category: algebraic geometry, algebra