Grassmannian

Description

Let $F$ be a field and let $0 \le r \le n$ be integers. The Grassmannian $\operatorname{Gr}(r, n)$ is the set of $r$-dimensional linear subspaces of $F^n$.

The Plücker embedding makes it a subset of a projective space. Let $\bigwedge^r F^n$ be the $r$-th exterior power, a part of the exterior algebra, of dimension $\binom{n}{r}$. A subspace $W$ with basis $w_1, \dots, w_r$ is sent to $$W \mapsto [w_1 \wedge \cdots \wedge w_r] \in \mathbb{P}\left(\bigwedge\nolimits^r F^n\right) \cong \mathbb{P}^{\binom{n}{r} - 1}.$$ A change of basis multiplies $w_1 \wedge \cdots \wedge w_r$ by the determinant of the change of basis matrix, so the point is well defined. The map is injective, since $W = \set{v \in F^n : v \wedge w_1 \wedge \cdots \wedge w_r = 0}$. The coordinates of the image are the Plücker coordinates $p_I$, the $r \times r$ minors of the $r \times n$ matrix with rows $w_1, \dots, w_r$.