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) ∈ Rn × n : P^ = P = P²,\; tr(P) = k\ or as involution matrices Gr(k,Rⁿ) ∈ Rn × 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.