Combinatorial Construction of Some Near Polygons

✍ Scribed by Bruce.N. Cooperstein; Ernest.E. Shult

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 397 KB
• Volume: 78
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.2768

✦ Synopsis

We give a construction which takes a rank two incidence geometry with three points on a line and returns a geometry of the same type, i.e., with three points on a line. It is also demonstrated that embeddings of the original geometry can be extended to the new geometry. It is shown that the family of dual polar spaces of type Sp(2n, 2) arise recursively from the construction starting with the geometry consisting of one point and no lines. Making use of this construction we inductively construct projective embeddings for these geometries, in particular the embedding in the spin module for the group Sp(2n, 2). We also show that if we apply the construction to a classical near polygon which is isometrically embedded in the near 2n-gon of type Sp(2n, 2) the resulting space is a near polygon. Examples of such classical, isometrically embedding spaces are near 2n-gon of Hamming type on a three letter alphabet and the product of dual polar spaces of types Sp(2k, 2) and Sp(2l, 2) with k+l=n.

1997 Academic Press

1. Introduction

We assume the reader is familar with the concepts of a partial linear rank two incidence geometry 1=(P, L), the collinearity graph of 1, a path of length d, for a positive integer d, subspace of 1, singular subspace of 1, a convex subspace of 1, a projective embedding e: 1 Ä PG(V ), morphism of projective embeddings, for an embedding e^to be universal relative to an embedding e, and for an embedding e to be relatively universal. As a reference consult (Buekenhout [Bu]).