Degree of the Grassmannian as an affine variety

The degree of the Grassmannian with respect to the Plücker embedding is well-known. However, the Plücker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the Grassmannian is usually embedded as projection matrices $\mathrm{Gr}(k,R^n)\cong \{P\in R^{n\times n}:P^T=P=P^2,\mathrm{tr}(P)=k\}$ or as involution matrices $\mathrm{Gr}(k,R^n)\cong \{X\in R^{n\times n}:X^T=X,X^2=I,\mathrm{tr}(X)=2k-n\}$. 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, Reinke, and Sturmfels about the degree of $\mathrm{Gr}(2,R^n)$ and in fact generalized it to $\mathrm{Gr}(k,R^n)$. We also proved a set-theoretic variant of another conjecture of theirs about the limit of $\mathrm{Gr}(k,R^n)$ in the sense of Gröbner degeneration.