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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.18765 |
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| _version_ | 1866911224535449600 |
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| author | Baburin, Igor A. |
| author_facet | Baburin, Igor A. |
| contents | Given the integral lattice $Λ^d$ in $d$-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of $Aut(Λ^d)$ have been computed for $d \leq 4$. These partitions can be interpreted as colourings of orbits defined up to permutation of colours. Complete results are obtained for $d=2$ up to 64 orbits, for $d=3$ up to 8 orbits, and for 2 orbits in dimension 4. The automorphism groups of the partitions are also determined. Our results for two orbits in dimension 3 correct the old result of H. Heesch [Z. Kristallogr., (1933), 85, 335--344] who overlooked one partition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On colourings of cubic lattices Baburin, Igor A. Combinatorics Group Theory 20-04 Given the integral lattice $Λ^d$ in $d$-dimensional Euclidean space, partitions of the lattice nodes into orbits of finite-index subgroups of $Aut(Λ^d)$ have been computed for $d \leq 4$. These partitions can be interpreted as colourings of orbits defined up to permutation of colours. Complete results are obtained for $d=2$ up to 64 orbits, for $d=3$ up to 8 orbits, and for 2 orbits in dimension 4. The automorphism groups of the partitions are also determined. Our results for two orbits in dimension 3 correct the old result of H. Heesch [Z. Kristallogr., (1933), 85, 335--344] who overlooked one partition. |
| title | On colourings of cubic lattices |
| topic | Combinatorics Group Theory 20-04 |
| url | https://arxiv.org/abs/2510.18765 |