Association schemes from the action of ${PGL}(2,q)$ fixing a nonsingular conic in ${PG}(2,q)$

Hollmann, Henk D.L. ; Xiang, Qing

Journal of Algebraic Combinatorics, Tome 24 (2006) no. 2, pp. 157-193.

Résumé

Summary: The group $PGL(2, q)$ has an embedding into $PGL(3, q)$ such that it acts as the group fixing a nonsingular conic in $PG(2, q)$. This action affords a coherent configuration  ${\cal R}( q)$ on the set $L {\cal L}( q)$ of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions  ${\cal R} _{+}( q)$ and  ${\cal R} _{ - }( q)$ of  ${\cal R}( q)$ to the set $L {\cal L} _{+}( q)$ of secant (hyperbolic) lines and to the set $L {\cal L} _{ - }( q)$ of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme  ${\cal R} _{ - }( q)$ is pseudocyclic. We further show that the coherent configurations  ${\cal R}( q ^{2})$ with $q$ even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme  ${\cal R} _{+}( q ^{2})$, and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes  ${\cal R} _{+}( q ^{2})$ and  ${\cal R} _{ - }( q ^{2})$. The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.

Détail

Keywords: keywords association scheme, coherent configuration, conic, cross-ratio, exterior line, fusion, pseudocyclic association scheme, secant line, strongly regular graph, tangent line