Grassmannian

In mathematics, the Grassmannian (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all -dimensional linear subspaces of an -dimensional vector space over a field . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than .[1][2] When is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension .[3] In general they have the structure of a nonsingular projective algebraic variety.

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to , parameterizing them by what are now called Plücker coordinates. (See § Plücker coordinates and Plücker relations below.) Hermann Grassmann later introduced the concept in general.

Notations for Grassmannians vary between authors, and include , ,, to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space .

  1. ^ Lee 2012, p. 22, Example 1.36.
  2. ^ Shafarevich 2013, p. 42, Example 1.24.
  3. ^ Milnor & Stasheff (1974), pp. 57–59.