The valuations of the near octagon G(4)

摘要

In De Bruyn (2003) [4] it was shown that the dual polar space DH(2n - 1, 4), n >= 2. has a sub-near 2n-gon G(n) with a large automorphism group. In this paper, we classify the valuations of the near octagon G(4). We show that each such valuation is either classical, the extension of a non-classical valuation of a G(3)-hex or is associated with a valuation of Fano-type of an H-3-hex. In order to describe the latter type of valuation we must study the structure of G(4) with respect to an H-3-hex. This study also allows us to construct new hyperplanes of G(4). We also show that each valuation of G(4) is induced by a (classical) valuation of the dual polar space DH(7, 4).