A decomposition of the universal embedding space for the near polygon $${{\mathbb H}_n}$$

• Authors: Bart De Bruyn
• Venue: Designs, Codes and Cryptography
• Year: 2012

Let $${{\mathbb H}_n, n \geq 1}$$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n + 2 vertices, and let e denote the absolutely universal embedding of $${{\mathbb H}_n}$$ into PG(W), where W is a $${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$$ -dimensional vector space over the field $${{\mathbb F}_2}$$ with two elements. For every point z of $${{\mathbb H}_n}$$ and every $${i \in {\mathbb N}}$$ , let Δ i (z) denote the set of points of $${{\mathbb H}_n}$$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of $${{\mathbb H}_n, W}$$ can be written as a direct sum $${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$$ such that the following four properties hold for every $${i \in \{0,\ldots,n \}}$$ : (1) $${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$$ ; (2) $${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$$ ; (3) $${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$$ ; (4) $${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$$ .