Morphisms from projective spaces to flags of minimal parabolic subgroups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917342522376192 |
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| author | Bakshi, Sarjick Parameswaran, A J |
| author_facet | Bakshi, Sarjick Parameswaran, A J |
| contents | We classify nonconstant morphisms $\mathbb{P}^m \to G/P$ for $m \le 4$ when $G = SL(n,\mathbb{C})$ (type~$A$) for a minimal parabolic subgroup $P$. Using the Borel presentation of cohomology and explicit Schubert intersection identities, we show that there is no nonconstant morphism $\mathbb{P}^2 \to G/B$; for minimal parabolic subgroup $P_{α_i}$, there are no nonconstant morphisms $\mathbb{P}^3 \to G/P_{α_i}$ when $i \in \{1, n-1\}$, while such morphisms exist for $1 < i < n-1$; and, after correcting an earlier error (pointed out by Yanjie Li), we give an elementary proof that there is no nonconstant morphism $\mathbb{P}^4 \to G/P_{α_i}$ for any minimal parabolic subgroup. The proofs are elementary and cohomological. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00286 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Morphisms from projective spaces to flags of minimal parabolic subgroups Bakshi, Sarjick Parameswaran, A J Algebraic Geometry Combinatorics Representation Theory 14M15, 14M17 We classify nonconstant morphisms $\mathbb{P}^m \to G/P$ for $m \le 4$ when $G = SL(n,\mathbb{C})$ (type~$A$) for a minimal parabolic subgroup $P$. Using the Borel presentation of cohomology and explicit Schubert intersection identities, we show that there is no nonconstant morphism $\mathbb{P}^2 \to G/B$; for minimal parabolic subgroup $P_{α_i}$, there are no nonconstant morphisms $\mathbb{P}^3 \to G/P_{α_i}$ when $i \in \{1, n-1\}$, while such morphisms exist for $1 < i < n-1$; and, after correcting an earlier error (pointed out by Yanjie Li), we give an elementary proof that there is no nonconstant morphism $\mathbb{P}^4 \to G/P_{α_i}$ for any minimal parabolic subgroup. The proofs are elementary and cohomological. |
| title | Morphisms from projective spaces to flags of minimal parabolic subgroups |
| topic | Algebraic Geometry Combinatorics Representation Theory 14M15, 14M17 |
| url | https://arxiv.org/abs/2308.00286 |