On configurations of type \(n_ \kappa\) with constant degree of irreducibility
A configuration \(C\) is called of type \(n _ \kappa\) if it is a finite partial projective plane consisting of \(n\) points and \(n\) lines such that every point and every line is incident with \(\kappa\) lines and \(\kappa\) points, respectively. A classification of such configurations was started by T. Reye in 1876. Since then an important contribution was made by V. Martinetti in 1887, who found a rather general procedure transforming a configuration of type \(n_ 3\) into one of type \( (n+1)_ 3\). First of all, the author identifies the configuration of type \(n_ \kappa\) with the regular bipartite graph of order \(2n\) and valency \(\kappa\) such that the girth is at least 6. Then using the graph he generalizes Martinetti's procedure from 3 to general \(\kappa\). The configuration is called (ir)reducible if it is (not) obtained by a (generalized) Martinetti procedure. Next using the incidence matrix he introduces concepts of homogeneous (ir)reducibility of degree 1 \((d)\) and uniform (ir)reducibility to configurations of type \(n_ \kappa\). Projective planes and generalized quadrangles are characterized as homogeneously irreducible and homogeneously reducible configurations. Finally he constructs a homogeneously irreducible configuration of type \((q^ 2)_ q\) and degree \(q-1\), where \(q\) is any prime power greater than 2.
scientific article; zbMATH DE number 554396
scientific article