Degree of the Grassmannian as an affine variety

The degree of the Grassmannian with respect to the Pl\"ucker embedding is well-known. However, the Pl\"ucker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the Grassmannian is usually embedded as projection matrices Gr (k, Rⁿ) \P R^{n n: P^ = P = P², \; tr (P) = k\} or as involution matrices Gr (k, Rⁿ) \X R^{n n: X^ = X, \; X² = I, \; tr (X) =2k - n\}. We will determine an explicit expression for the degree of the Grassmannian with respect to these embeddings. In so doing, we resolved a conjecture of Devriendt--Friedman--Sturmfels about the degree Gr (2, Rⁿ) and in fact generalized it to Gr (k, Rⁿ). We also proved a set theoretic variant of another conjecture of Devriendt--Friedman--Sturmfels about the limit of Gr (k, Rⁿ) in the sense of Gr\"obner degneration.