Generalized quadrangles with a regular point and association schemes

AbstractThere is a new method of constructing generalized quadrangles (GQs) which is based on covering of nets; all GQs with a regular point can be represented in this way. Here we first construct from a generalized quadrangle Q with a regular point a four-class association scheme A(Q) called in brief geometric. It is then natural to call pseudo-geometric any association scheme A with the same parameters as A(Q). We use eigenvalue techniques and the above method of construction to give a characterization of pseudo-geometric association schemes which are geometric

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

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 $R(q)$ on the set $L(q)$ 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 $R_{+}(q)$ and $R_{-}(q)$ to the sets $L_{+}(q)$ of secant lines and to the set $L_{-}(q)$ of exterior lines, respectively, are both association schemes; moreover, we show that the elliptic scheme $R_{-}(q)$ is pseudocyclic. We further show that the coherent configuration $R(q^2)$ with $q$ even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme $R_{+}(q^2)$, and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes $R_{+}(q^2)$ and $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.Comment: 33 page

Fissioned triangular schemes via sharply 3-transitive groups

AbstractIn [D. de Caen, E.R. van Dam, Fissioned triangular schemes via the cross-ratio, European J. Combin. 22 (2001) 297–301], de Caen and van Dam constructed a fission scheme FT(q+1) of the triangular scheme on PG(1,q). This fission scheme comes from the naturally induced action of PGL(2,q) on the 2-element subsets of PG(1,q). The group PGL(2,q) is one of two infinite families of finite sharply 3-transitive groups. The other such family M(q) is a “twisted” version of PGL(2,q), where q is an even power of an odd prime. The group PSL(2,q) is the intersection of PGL(2,q) and Mq(q). In this paper, we investigate the association schemes coming from the actions of PSL(2,q), Mq(q) and PML(2,q), respectively. Through the conic model introduced in [H.D.L. Hollmann, Q. Xiang, Association schemes from the actions of PGL(2,q) fixing a nonsingular conic, J. Algebraic Combin. 24 (2006) 157–193], we introduce an embedding of PML(2,q) into PML(3,q). For each of the three groups mentioned above, this embedding produces two more isomorphic association schemes: one on hyperbolic lines and the other on hyperbolic points (via an orthogonal parity) in a 3-dimensional orthogonal geometry. This embedding enables us to treat these three isomorphic association schemes simultaneously

Fissioned Triangular Schemes via the Cross-ratio

AbstractA construction of association schemes is presented; these are fission schemes of the triangular schemesT (n) where n=q+ 1 with q any prime power. The key observation is quite elementary, being that the natural action of PGL(2, q) on the 2-element subsets of the projective line PG(1, q) is generously transitive. In addition, some observations on the intersection parameters and fusion schemes of these association schemes are made

On a Four-Class Association Scheme

We show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q²), for q >= 4 a power of 2, thereby proving a conjecture of De Caen and Van Dam [2]. This is a fusion of certain relations in the fission scheme FT (q² + 1) obtained from the triangular association scheme (see [2]). Combining three relations in the above four..