A computation of some multiply homogeneous superschemes from transitive permutation groups

• Authors: Izumi Miyamoto
• Venue: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
• Year: 2007

Let G be a doubly transitive permutation group on a set X. A doubly homogeneous superscheme is formed by the orbits on the set of triples of X of G. Let α be a point of aset X and let H be a transitive group on X\{α}. Then from the combinatorial structure of the superscheme formed by the orbits of H on X3,we may construct some doubly homogeneous superschemes on X. We will give a general algorithm to compute such superschemes and show how to implement it practically. In particular if H = Gα, thestabilizer of α in G, then we can construct a superscheme of which automorphism group is G in the cases of moderate size. Furthermore, even if H is not a stabilizer of a doubly transitive group, we can consider some orbit-like sets of a doubly homogeneous superscheme. We see whether such sets form a design in some cases. As a related combinatorial algorithm we have developed a program to compute the automorphism group of a superscheme which is a kind of a labeled hyper graph.