The Universal Embedding of the Near Polygon ${\Bbb G}_n$

Abstract

In an earlier paper, we showed that the dual polar space $DH(2n-1,4)$, $n \geq 2$, has a sub near-$2n$-gon ${\Bbb G}_n$ with a large automorphism group. In this paper, we determine the absolutely universal embedding of this near polygon. We show that the generating and embedding ranks of ${\Bbb G}_n$ are equal to ${2n \choose n}$. We also show that the absolutely universal embedding of ${\Bbb G}_n$ is the unique full polarized embedding of this near polygon.