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Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles
Abstract
For an almost simple complex algebraic group G of semisimple rank m, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most m+2 members does. We determine the resulting Helly number (h(G)) for almost simple groups of types A, B, C, G2, F4, E7, and E8, and obtain bounds differing by one in types D and E6. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal G-bundle splits equivariantly. In particular, for d ≥ 2, this splitting property holds on ℙd if and only if d ≥ h(G)
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BibTeX
@misc{tsvelikhovskiy2026helly,
title = {{Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles}},
author = {Boris Tsvelikhovskiy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11340},
eprint = {2610.11340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11340},
}
RIS
TY - UNPB
TI - Helly Numbers for Connected Reductive Groups and Splitting of Toric Principal Bundles
AU - Boris Tsvelikhovskiy
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11340
UR - https://arxiv.org/abs/2610.11340
AB - For an almost simple complex algebraic group G of semisimple rank m, we prove that a finite family of parabolic subgroups contains a common maximal torus whenever every subfamily of at most m+2 members does. We determine the resulting Helly number (h(G)) for almost simple groups of types A, B, C, G_2, F_4, E_7, and E_8, and obtain bounds differing by one in types D and E_6. For general reductive groups of positive semisimple rank, the Helly number is the maximum of those of the almost simple factors. We apply these results to establish a fan-theoretic criterion characterizing smooth toric varieties on which every toric principal G-bundle splits equivariantly. In particular, for d ≥ 2, this splitting property holds on ℙ^d if and only if d ≥ h(G)
ER -