Degree of the Grassmannian as an affine variety

TL;DR

This paper computes the degree of Grassmannians under common applied embeddings, resolving a conjecture for -planes and generalizing it to -planes in higher dimensions, and explores limits via Grf6bner degeneration.

Contribution

It provides explicit formulas for the degree of Grassmannians in applied embeddings and proves a conjecture about -planes, extending it to general -planes.

Findings

Derived explicit degree formulas for Grassmannians in projection and involution embeddings.

Resolved a conjecture regarding the degree of -planes in -planes.

Analyzed the limit of Grassmannians using Grf6bner degeneration.

Abstract

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 $\operatorname{Gr}(k,\mathbb{R}^n) \cong \{P \in \mathbb{R}^{n \times n} : P^{\scriptscriptstyle\mathsf{T}} = P = P^2,\; \operatorname{tr}(P) = k\}$ or as involution matrices $\operatorname{Gr}(k,\mathbb{R}^n) \cong \{X \in \mathbb{R}^{n \times n} : X^{\scriptscriptstyle\mathsf{T}} = X,\; X^2 = I,\; \operatorname{tr}(X)=2k - n\}$. 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 $\operatorname{Gr}(2, \mathbb{R}^n)$ and in fact generalized it to $\operatorname{Gr}(k, \mathbb{R}^n)$. We also proved a set theoretic variant of another conjecture of theirs about the limit of $\operatorname{Gr}(k,\mathbb{R}^n)$ in the sense of Gr\"obner degneration.