Abstract
One of the most fundamental results in the theory of regular near polygons is the result that every regular near 2d-gon, \(d \ge 3\), whose parameters \(s,t,t_i\), \(i \in \{ 0,1,\ldots ,d \}\), satisfy \(s,t_2 \ge 2\) and \(t_3=t_2^2+t_2\) is a dual polar space. The proof of that theorem heavily relies on Tits’ theory of buildings, in particular on Tits’ strong results on covering of chamber systems. In this paper, we give an alternative proof which only employs geometrical and algebraic combinatorial arguments.
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Acknowledgements
The author wants to express his gratitude to Hiroshi Suzuki for communicating his desire to him to have an alternative proof of Theorem 1.2 that does not rely on Tits’ strong results on covering of chamber systems. The author also wants to thank him for his comments on an earlier draft.
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De Bruyn, B. On Near Polygons All Whose Hexes are Dual Polar Spaces. Graphs and Combinatorics 36, 1015–1041 (2020). https://doi.org/10.1007/s00373-020-02166-9
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DOI: https://doi.org/10.1007/s00373-020-02166-9