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

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 ${frac{1}{n+2} left(begin{array}{c}2n+2 n+1end{array}right)}$ -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) ${leftlangle e left( bigcup_{j leq i} Delta_j(x) right) rightrangle = {rm PG}(W_0 oplus W_1 oplus cdots oplus W_i)}$ ; (3) ${leftlangle e left( bigcup_{j leq i} Delta_j(y) right) rightrangle = {rm PG}(W_{n-i}oplus W_{n-i+1} oplus cdots oplus W_n)}$ ; (4) ${dim(W_i) = |Delta_i(x) cap Delta_{n-i}(y)| = left(begin{array}{c}n iend{array}right)^2 - left(begin{array}{c}n i-1end{array}right) cdot left(begin{array}{c}n i+1end{array}right)}$ .