Families of Association Schemes on Triples from Two-Transitive Groups

Jose Maria P. Balmaceda, Dom Vito A. Briones

Published 2021-07-16Version 1

In 1990, Mesner and Bhattacharya extended the notion of association schemes to association schemes on triples (ASTs), whose underlying relations and corresponding adjacency algebras are ternary instead of binary. In particular, they showed that the action of any two-transitive permutation group yields an AST. In this paper, we tabulate the sizes of some ASTs obtained from the two-transitive permutation groups. Following Praeger and Bhattacharya's 2021 preprint, we remark slightly on the circulicity and thinness of some of these ASTs. We also speak briefly on the (partial) associativity of the ternary adjacency algebras of these ASTs, in the senses of Abramov and Lister. Furthermore, we generalize some of the results of Mesner and Bhattacharya regarding the sizes and intersection numbers of ASTs obtained from the projective and affine groups $PSL(2,n)$ and $AGL(1,n)$ to $PSL(k,n)$, $PGL(k,n)$, $P\Gamma L(k,n)$, $AGL(k,n)$ and $A \Gamma L(k,n)$. In particular, one generalization also rectifies an error regarding the intersection numbers of $AGL(1,n)$. Finally, we also determine the sizes of the ASTs obtained from the two-transitive Ree groups $Ree(3^{2k+1})$.