Abstract
We show that for all integers \(m\geqslant 2\), and all integers \(k\geqslant 2\), the orthogonal groups \({{\,\mathrm{O}\,}}^{\pm }(2m,\mathbb {F}_{2^k})\) act on abstract regular polytopes of rank 2m, and the symplectic groups \({{\,\mathrm{Sp}\,}}(2m,\mathbb {F}_{2^k})\) act on abstract regular polytopes of rank \(2m+1\).
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Acknowledgements
The authors are grateful to Bill Kantor for suggesting the use of the Arf invariant, which greatly simplified the calculations of the Witt indices of our orthogonal spaces. They also thank the anonymous referees for providing helpful suggestions.
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This work was partially supported by a Grant from the Simons Foundation (#281435 to Peter Brooksbank).
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Brooksbank, P.A., Ferrara, J.T. & Leemans, D. Orthogonal Groups in Characteristic 2 Acting on Polytopes of High Rank. Discrete Comput Geom 63, 656–669 (2020). https://doi.org/10.1007/s00454-019-00083-0
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DOI: https://doi.org/10.1007/s00454-019-00083-0