A decomposition of the universal embedding space for the near polygon \({{\mathbb H}_n}\)
The book [\textit{B. De Bruyn}, Near polygons. Frontiers in Mathematics. Basel: Birkhäuser (2006; Zbl 1102.51007)] shows that the author of the present article is an expert in the field of near polygons. The author refines a result of \textit{A. Blokhuis} and \textit{A. E. Brouwer} [Des. Codes Cryptography 17, No. 1--3, 299--303 (1999; Zbl 0944.51003)] as follows: Let \(x\) and \(y\) be two opposite points of the near polygon \({\mathbf H}_n\), \(n\geq 1\), denote by \(\Delta_i(x)\) the set of points at distance \(i\) from \(x\), and put \(X_i:=\Delta_i(x)\cap\Delta_{n-i}(y)\), \(i\in\{0,\dots,n\}\). Then (1) \(|X_i|={n\choose{i}}^2-{n\choose{i-1}}.{n\choose{i+1}}\) (2) \(X_0\cup X_1\cup\dots\cup X_n\) is a generating set of \({\mathbf H}_n\). Using the assertion above the author is able to prove the subsequent decomposition theorem: Let \({\mathbf H}_n\), \(n\geq 1\), 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 \({\mathbf H}_n\) into PG\((W)\), where \(W\) is a vector space over the field \({\mathbf F}_2\) with two elements whose dimension equals the Catalan number \(C_n=A.{2n+2\choose{n+1}}\) with \(A:=1/(n+2)\). Then for every pair \(\{x,y\}\) of mutually opposite points of \({\mathbf H}_n\), \(W\) can be written as direct sum \(W_0\oplus W_1\oplus \dots\oplus W_n\) such that the following four properties hold for every \(i\in\{0,\dots,n\}\): (1) \(\langle e\bigl(\Delta_i(x)\cap\Delta_{n-1}(y)\bigr)\rangle=\text{PG}(W_i)\) (2) \(\langle e\bigl(\bigcup_{j\leq i}\Delta_j(x)\bigr)\rangle=\text{PG}(W_0\oplus W_1\oplus\dots\oplus W_i)\) (3) \(\langle e\bigl(\bigcup_{j\leq i}\Delta_j(y)\bigr)\rangle=\text{PG}(W_{n-i}\oplus W_{n-i+1}\oplus\dots\oplus W_n)\) (4) \(\dim(W_i)={n\choose{i}}^2-{n\choose{i-1}}.{n\choose{i+1}}\).
scientific article; zbMATH DE number 6037531