The valuations of the near octagon \({\mathbb I}_4\)
The electronic journal of combinatorics, Tome 13 (2006)
The maximal and next-to-maximal subspaces of a nonsingular parabolic quadric $Q(2n,2)$, $n \geq 2$, which are not contained in a given hyperbolic quadric $Q^+(2n-1,2) \subset Q(2n,2)$ define a sub near polygon ${\Bbb I}_n$ of the dual polar space $DQ(2n,2)$. It is known that every valuation of $DQ(2n,2)$ induces a valuation of ${\Bbb I}_n$. In this paper, we classify all valuations of the near octagon ${\Bbb I}_4$ and show that they are all induced by a valuation of $DQ(8,2)$. We use this classification to show that there exists up to isomorphism a unique isometric full embedding of ${\Bbb I}_n$ into each of the dual polar spaces $DQ(2n,2)$ and $DH(2n-1,4)$.
@article{10_37236_1102,
author = {Bart De Bruyn and Pieter Vandecasteele},
title = {The valuations of the near octagon \({\mathbb {I}_4\)}},
journal = {The electronic journal of combinatorics},
year = {2006},
volume = {13},
doi = {10.37236/1102},
zbl = {1165.51301},
url = {http://geodesic.mathdoc.fr/articles/10.37236/1102/}
}
Bart De Bruyn; Pieter Vandecasteele. The valuations of the near octagon \({\mathbb I}_4\). The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1102
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