The valuations of the near octagon \(\mathbb G_4\). (English) Zbl 1200.51005
In this paper, the authors classify the valuations of the near octagon, called \(\mathbb{G}_4\). This is a near octagon which is defined as a certain substructure of the dual polar space \(DH(7,4)\), also a near octagon, related to the nondegenerate Hermitian variety in PG\((7,4)\). The near octagon \(\mathbb{G}_4\) is dense, which means that every pair of points at distance 2 in the collineariry graph has at least two common neighbours.
Valuations in near polygons are important tools to classify classes of dense near polygons, to construct and classify geometric hyperplanes of near polygons, and to study isometric full embeddings between near polygons. But none of this is demonstrated in the present paper. The classification is apparently a nontrivial exercise, cleverly accomplished, but it remains a result about a single finite geometric structure, and as such is a rather isolated result.
I note that the first author extends the above classification to all near polygons \(\mathbb{G}_n\), in [Electron. J. Comb. 16, No. 1, Research Paper R137, 29 p. (2009; Zbl 1194.51007)], relying on the paper under review.
Valuations in near polygons are important tools to classify classes of dense near polygons, to construct and classify geometric hyperplanes of near polygons, and to study isometric full embeddings between near polygons. But none of this is demonstrated in the present paper. The classification is apparently a nontrivial exercise, cleverly accomplished, but it remains a result about a single finite geometric structure, and as such is a rather isolated result.
I note that the first author extends the above classification to all near polygons \(\mathbb{G}_n\), in [Electron. J. Comb. 16, No. 1, Research Paper R137, 29 p. (2009; Zbl 1194.51007)], relying on the paper under review.
Reviewer: L. -c. Chen
MSC:
| 51E12 | Generalized quadrangles and generalized polygons in finite geometry |
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