Degree of the Grassmannian as an affine variety
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916865504182272 |
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| author | Lim, Lek-Heng Ye, Ke |
| author_facet | Lim, Lek-Heng Ye, Ke |
| contents | 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 $\operatorname{Gr}(k,\mathbb{R}^n) \cong \{P \in \mathbb{R}^{n \times n} : P^{\scriptscriptstyle\mathsf{T}} = P = P^2,\; \operatorname{tr}(P) = k\}$ 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 Gröbner degneration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degree of the Grassmannian as an affine variety Lim, Lek-Heng Ye, Ke Algebraic Geometry 14E25, 14F45, 05E14 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 $\operatorname{Gr}(k,\mathbb{R}^n) \cong \{P \in \mathbb{R}^{n \times n} : P^{\scriptscriptstyle\mathsf{T}} = P = P^2,\; \operatorname{tr}(P) = k\}$ 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 Gröbner degneration. |
| title | Degree of the Grassmannian as an affine variety |
| topic | Algebraic Geometry 14E25, 14F45, 05E14 |
| url | https://arxiv.org/abs/2405.05128 |