The geometry of the six quaternionic equiangular lines in $\mathbb{H}^2$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915033846382592 |
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| author | Waldron, Shayne |
| author_facet | Waldron, Shayne |
| contents | We give a simple presentation of the six quaternionic equiangular lines in $\mathbb{H}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in $\mathbb{H}^2$, with angles { 1/3, 2/3 }, { 1/4, 5/8 } and { 0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt's "finite collineation groups" for lines in $\mathbb{C}^4$, and general methods for finding nice systems of quaternionic lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The geometry of the six quaternionic equiangular lines in $\mathbb{H}^2$ Waldron, Shayne Geometric Topology 05B30, 15B33, 20C25, 20G20, 51M05, 15B57, 51E99, 51M15, 65D30 We give a simple presentation of the six quaternionic equiangular lines in $\mathbb{H}^2$ as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2.A_6 the double cover of $A_6)$. Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in $\mathbb{H}^2$, with angles { 1/3, 2/3 }, { 1/4, 5/8 } and { 0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt's "finite collineation groups" for lines in $\mathbb{C}^4$, and general methods for finding nice systems of quaternionic lines. |
| title | The geometry of the six quaternionic equiangular lines in $\mathbb{H}^2$ |
| topic | Geometric Topology 05B30, 15B33, 20C25, 20G20, 51M05, 15B57, 51E99, 51M15, 65D30 |
| url | https://arxiv.org/abs/2411.16766 |