The Two Lives of the Grassmannian

The real Grassmannian is both a projective variety (via Pl\"ucker coordinates) and an affine variety (via orthogonal projections). We connect these two representations, and we develop the commutative algebra of the latter variety. We introduce the squared Grassmannian, and we study applications to determinantal point processes in statistics.
Comment: 19 pages. Theorem 3.8 and Remark 5.11 are new