Structure and automorphisms of primitive coherent configurations

Abstract

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 analysis of algorithms; 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 $\exp(\tilde{O}(n^{1/2}))$ automorphisms. (Here, $n$ is the number of vertices and the $\tilde{O}$ hides polylogarithmic factors.) In the present paper we classify all PCCs with more than $\exp(\tilde{O}(n^{1/3}))$ automorphisms, making the first progress on Babai's conjectured classification of all PCCs with more than $\exp(n^ε)$ 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 $\exp(\tilde{O}(n^{1/2}))$ bound on their order. In a similar vein, our result implies an $\exp(\tilde{O}(n^{1/3}))$ 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 analysis relies on a new combinatorial structure theory we develop for PCCs. In particular, we demonstrate the presence of "asymptotically uniform clique geometries" on PCCs in a certain range of the parameters.