On circulant ternary coherent configurations of prime degree

TL;DR

This paper investigates circulant ternary coherent configurations of prime degree, proving they are schurian except in specific algebraic cases involving automorphism groups.

Contribution

It establishes that all circulant ternary coherent configurations of prime degree are schurian, except for certain cases related to automorphism groups AGL1(p).

Findings

All circulant ternary coherent configurations of prime degree are schurian.

Exceptions occur when the automorphism group is AGL1(p) with p ≡ ±1 mod 8.

The paper characterizes configurations with automorphism groups as proper subgroups of AGL1(p).

Abstract

Ternary coherent configurations are, on the one hand, a special case of multidimensional coherent configurations introduced by L. Babai (2016), and, on the other hand, a natural generalization of association schemes on triples introduced by D. M. Mesner and P. Bhattacharya (1990). A ternary coherent configuration X is said to be circulant if the automorphism group Aut(X) of X has a regular cyclic subgroup, and schurian if the classes of X are the orbits of the componentwise action of the group Aut(X) on triples of points of X. It is proved that any circulant ternary coherent configuration X of prime degree p is schurian with the possible exception of the case when X is an association schemes on triples and either Aut(X) = AGL1(p) and p = +1, or -1 (mod 8), or Aut(X) is a proper subgrou of AGL1(p).