Some Extensions and Embeddings of Near Polygons
Hans Cuypers
and Thomas Meixner
DOI: 10.1023/A:1022471817341
Abstract
Let ( P, L, *) be a near polygon having s + 1 points per line, s > 1, and suppose k is a field. Let V k be the k-vector space with basis { v p | p Ĩ P} \{ v_p |p \in P\} Then the subspace generated by the vectors v 1 = S p*1 v p v_1 = Σ_{p*1} v_p , where l Ĩ \in L, has codimension at least 2 in V k.
This observation is used in two ways. First we derive the existence of certain diagram geometries with flag transitive automorphism group, and secondly, we show that any finite near polygon with 3 points per line can be embedded in an affine GF(3)-space.
Pages: 375–381
Keywords: near polygon; diagram geometry; affine embedding
Full Text: PDF
References
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2. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin, Heidelberg, 1989.
3. H. Cuypers, "A construction of geometries that are almost buildings," preprint.
4. T. Meixner, "Chamber systems with extended diagram," Mitt. Math. Sem. Giessen 165 (1984), 93-104.
5. T. Meixner, "Tits Kammersysteme mit einer transitiven Automorphismengruppe," Mitt. Math. Sem. Giessen 174 (1986).
6. S. Payne and J. Thas, Finite Generalized Quadrangles, Pittman Publishing, Boston, 1984.
7. S. Smith, "A geometric condition for incidence-matrix nullvectors," J. Combin. Theory Series A 51 (1989), 129-134.
8. J. Tits, "Buildings and group amalgamations," in Proc. Groups - St. Andrews 1985, LMS Lecture Notes 121 (1986), 110-127.
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