Coherent configurations associated with TI-subgroups

TL;DR

This paper characterizes when a coherent configuration corresponds to a transitive group with a TI-subgroup using intersection numbers, providing conditions that are both necessary and asymptotically sufficient, and generalizes related results.

Contribution

It establishes a new criterion based on intersection numbers for identifying TI-subgroups in transitive groups and extends existing results on association schemes.

Findings

Necessary condition for point stabilizer to be a TI-subgroup

Coherent configurations are determined by intersection numbers under this condition

Prime degree schemes with certain valency are associated with transitive groups

Abstract

Let X be a coherent configuration associated with a transitive group G. In terms of the intersection numbers of X, a necessary condition for the point stabilizer of G to be a TI-subgroup, is established. Furthermore, under this condition, X is determined up to isomorphism by the intersection numbers. It is also proved that asymptotically, this condition is also sufficient. More precisely, an arbitrary homogeneous coherent configuration satisfying this condition is associated with a transitive group, the point stabilizer of which is a TI-subgroup. As a byproduct of the developed theory, recent results on pseudocyclic and quasi-thin association schemes are generalized and improved. In particular, it is shown that any scheme of prime degree p and valency k is associated with a transitive group, whenever p>1+6k(k-1)^2.