Designs on the Tautological bundle
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911285092810752 |
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| author | Yuya, Ikeda |
| author_facet | Yuya, Ikeda |
| contents | In this paper, we introduce the framework of a generalized design, which represents any linear operator as a finite sum of local linear maps attached to finitely many points, thereby abstracting the core of design theory without employing integration. We then construct such a design on the space of sections of the tautological bundle over the complex projective line. By using the irreducible decomposition of this space as an SU(2)-representation, we show that the projection onto its lowest-dimensional summand can be realized as a finite sum of these local maps. Our construction relies on invariant theory for the binary icosahedral group and an analysis of fixed-point subspaces in symmetric tensor representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Designs on the Tautological bundle Yuya, Ikeda Combinatorics Representation Theory Primary 05B30, Secondary 20C15, 22E46, 53C30 In this paper, we introduce the framework of a generalized design, which represents any linear operator as a finite sum of local linear maps attached to finitely many points, thereby abstracting the core of design theory without employing integration. We then construct such a design on the space of sections of the tautological bundle over the complex projective line. By using the irreducible decomposition of this space as an SU(2)-representation, we show that the projection onto its lowest-dimensional summand can be realized as a finite sum of these local maps. Our construction relies on invariant theory for the binary icosahedral group and an analysis of fixed-point subspaces in symmetric tensor representations. |
| title | Designs on the Tautological bundle |
| topic | Combinatorics Representation Theory Primary 05B30, Secondary 20C15, 22E46, 53C30 |
| url | https://arxiv.org/abs/2511.20114 |