Semi-Pappus configurations; combinatorial generalizations of the Pappus configuration
Consider the \(n\)-dimensional affine space \(AG(n,3)\) on the vector space \(GF(3)^n\) and its canonical basis \(\{{\mathbf e}_i\mid i=1,\dots,n\}\); put \({\mathcal H}_i\) for the hyperplane spanned by the set \(\{{\mathbf e}_j\mid j=1,\dots,n,\;j\neq i\}\). By the semi-Pappus configuration the authors mean the \(({3^n}_{2^{n-1}}{3^{n-1}2^{n-1}}_3)\)-configuration \({\mathcal P}_n\) formed by all points of \(AG(n,3)\) and the affine lines of \(AG(n,3)\) that are not parallel to any of the \({\mathcal H}_i\). The convolution [compare \textit{M. Prażmowska} and \textit{K. Prażmowski}, J. Geom. 85, No.~1--2, 90--109 (2006; Zbl 1114.51002)] \({\mathcal M}_n\) of \({\mathcal P}_n\) and the group \(C_2\) is a \(({2.3^n}_{2^{n}}{2^{n+1}3^{n-1}}_3)\)-configuration. The authors give further representations of \({\mathcal P}_n\) and \({\mathcal M}_n\); \({\mathcal P}_n\) and \({\mathcal M}_n\) are substructures of the generalized Sylvester systems \({\mathbf H}_n(3n)\) resp. \({\mathbf H}_n(4n)\). The configurations \({\mathcal P}_n\) and \({\mathcal M}_n\) can be embedded into the (Fanoian) projective spaces \(PG(3n-1,2)\) resp. \(PG(4n-1,2)\). The configuration \({\mathcal M}_n\) can not be embedded into \(AG(k,3)\) for any \(k\). Moreover the authors determine the automorphisms of \({\mathcal P}_n\) and \({\mathcal M}_n\).
convolution (of a partial Steiner triple system and a group)
scientific article; zbMATH DE number 5935024
scientific article