The valuations of the near hexagons related to the Witt designs S(5,6,12) and S(5,8,24)

✍ Scribed by Bart De Bruyn; Pieter Vandecasteele

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 176 KB
• Volume: 14
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20074

✦ Synopsis

Valuations of dense near polygons were introduced in [16]. In the present paper, we classify all valuations of the near hexagons E 1 and E 2 , which are related to the respective Witt designs Sð5,6,12Þ and Sð5,8,24Þ. Using these classifications, we prove that if a dense near polygon S contains a hex H isomorphic to E 1 or E 2 , then H is classical in S. We will use this result to determine all dense near octagons that contain a hex isomorphic to E 1 or E 2 . As a by-product, we obtain a purely geometrical proof for the nonexistence of regular near 2d-gons, d ! 4, whose parameters s, t, t i (0 i d) satisfy ðs, t 2 , t 3 Þ ¼ ð2, 1, 11Þ or ð2, 2, 14Þ. The nonexistence of these regular near polygons can also be shown with the aid of eigenvalue techniques.