Tracking the symmetries of $\mathbb Z_3$-orbifold K3s within the Mathieu groups
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Budzik, Kasia Taormina, Anne Ungureanu, Mara Wendland, Katrin Zadeh, Ida G. |
| author_facet | Budzik, Kasia Taormina, Anne Ungureanu, Mara Wendland, Katrin Zadeh, Ida G. |
| contents | For $\mathbb Z_3$-orbifold limits of K3, we provide a counterpart to the extensive studies by Nikulin and others of the geometry and symmetries of classical Kummer surfaces. In particular, we determine the group of holomorphic symplectic automorphisms of $\mathbb Z_3$-orbifold limits of K3. We moreover track this group within two of the Mathieu groups, which involves a variation of Kondo's lattice techniques that Taormina and Wendland introduced earlier in their study of the symmetries of Kummer surfaces and the genesis of their symmetry surfing programme. Specifically, we realise the finite group of symplectic automorphisms of this class of K3 surfaces as a subgroup of the sporadic groups Mathieu 12 and Mathieu 24 in terms of permutations of 12, resp. 24 elements. As a proof of concept, we construct an embedding that yields the largest Mathieu group when the symmetry group of $\mathbb Z_3$-orbifold K3s is combined with all symmetries of Kummer surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_16248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tracking the symmetries of $\mathbb Z_3$-orbifold K3s within the Mathieu groups Budzik, Kasia Taormina, Anne Ungureanu, Mara Wendland, Katrin Zadeh, Ida G. Algebraic Geometry High Energy Physics - Theory Group Theory 14, 06, 20, 81 For $\mathbb Z_3$-orbifold limits of K3, we provide a counterpart to the extensive studies by Nikulin and others of the geometry and symmetries of classical Kummer surfaces. In particular, we determine the group of holomorphic symplectic automorphisms of $\mathbb Z_3$-orbifold limits of K3. We moreover track this group within two of the Mathieu groups, which involves a variation of Kondo's lattice techniques that Taormina and Wendland introduced earlier in their study of the symmetries of Kummer surfaces and the genesis of their symmetry surfing programme. Specifically, we realise the finite group of symplectic automorphisms of this class of K3 surfaces as a subgroup of the sporadic groups Mathieu 12 and Mathieu 24 in terms of permutations of 12, resp. 24 elements. As a proof of concept, we construct an embedding that yields the largest Mathieu group when the symmetry group of $\mathbb Z_3$-orbifold K3s is combined with all symmetries of Kummer surfaces. |
| title | Tracking the symmetries of $\mathbb Z_3$-orbifold K3s within the Mathieu groups |
| topic | Algebraic Geometry High Energy Physics - Theory Group Theory 14, 06, 20, 81 |
| url | https://arxiv.org/abs/2504.16248 |