Structure and automorphisms of primitive coherent configurations

Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a “vertex set”; each color represents a “constituent digraph.” CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the graph isomorphism problem; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected. We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai’s 1981 paper in which he showed that only the trivial PCC admits more than exp(Õ(n)) automorphisms. (Here, n is the number of vertices and the Õ hides polylogarithmic factors.) In the present paper we classify all PCCs with more than exp(Õ(n)) automorphisms, making the first progress on Babai’s conjectured classification of all PCCs with more than exp(n) automorphisms. A corollary to Babai’s 1981 result solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an exp(Õ(n)) bound on their order. In a similar vein, our result implies an exp(Õ(n)) upper bound on the order of such groups, with known exceptions. This improvement of Babai’s result was previously known only through the Classification of Finite Simple Groups (Cameron, 1981), while our proof, like Babai’s, is elementary and almost purely combinatorial. Our result also has implications to the complexity of the graph isomorphism problem. PCCs arise naturally as obstacles to combinatorial partitioning approaches to the problem. Our results give an algorithm for deciding isomorphism of PCCs in time exp(Õ(n)), the first improvement over Babai’s exp(Õ(n)) bound. An extended abstract of this paper appeared in the Proceedings of the 47th ACM Symposium on Theory of Computing (STOC’15) under the title Faster canonical forms for primitive coherent configurations. [email protected]. This work was partially supported by a grant from the Simons Foundation (#320173 to Xiaorui Sun). [email protected]. Research supported in part by NSF grant DGE-1144082.