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Complex coordinates forced by finite orthogonality hypergraphs in dimension three
Abstract
We exhibit two finite orthogonality hypergraphs that admit faithful orthogonal representations by rays in complex three-dimensional space but admit none in real three-dimensional space. Thus, although each orthogonality diagram is unchanged, its coordinatizability in dimension three depends on whether the scalar field is complex or real. The first hypergraph has 20 vertices and 12 contexts. The second hypergraph adds one vertex and two contexts. Phase-adjusted realification also gives explicit faithful orthogonal representations of both hypergraphs in real six-dimensional space.
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BibTeX
@misc{navara2026complex,
title = {{Complex coordinates forced by finite orthogonality hypergraphs in dimension three}},
author = {Mirko Navara and Karl Svozil},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20205},
eprint = {2609.20205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20205},
}
RIS
TY - UNPB
TI - Complex coordinates forced by finite orthogonality hypergraphs in dimension three
AU - Mirko Navara
AU - Karl Svozil
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20205
UR - https://arxiv.org/abs/2609.20205
AB - We exhibit two finite orthogonality hypergraphs that admit faithful orthogonal representations by rays in complex three-dimensional space but admit none in real three-dimensional space. Thus, although each orthogonality diagram is unchanged, its coordinatizability in dimension three depends on whether the scalar field is complex or real. The first hypergraph has 20 vertices and 12 contexts. The second hypergraph adds one vertex and two contexts. Phase-adjusted realification also gives explicit faithful orthogonal representations of both hypergraphs in real six-dimensional space.
ER -