Minimal equivariant embeddings of the Grassmannian and flag manifold
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914875041644544 |
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| author | Lim, Lek-Heng Ye, Ke |
| author_facet | Lim, Lek-Heng Ye, Ke |
| contents | We show that the flag manifold $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$, two orders of magnitude below the current best known result. We will show that the value $(n-1)(n+2)/2$ is the smallest possible and that any $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding of $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$ in an ambient space of minimal dimension is equivariantly equivalent to the aforementioned one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal equivariant embeddings of the Grassmannian and flag manifold Lim, Lek-Heng Ye, Ke Representation Theory Differential Geometry 14M15, 57R40, 57S25, 14R20, 22E46, 22E70 We show that the flag manifold $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$, two orders of magnitude below the current best known result. We will show that the value $(n-1)(n+2)/2$ is the smallest possible and that any $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding of $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$ in an ambient space of minimal dimension is equivariantly equivalent to the aforementioned one. |
| title | Minimal equivariant embeddings of the Grassmannian and flag manifold |
| topic | Representation Theory Differential Geometry 14M15, 57R40, 57S25, 14R20, 22E46, 22E70 |
| url | https://arxiv.org/abs/2407.12546 |