On pseudofrobenius imprimitive association schemes

TL;DR

This paper characterizes when imprimitive pseudofrobenius association schemes are Frobenius, providing conditions for their existence, and explores implications for Frobenius groups and the Weisfeiler-Leman dimension of certain circulant graphs.

Contribution

It establishes a necessary and sufficient condition for imprimitive pseudofrobenius schemes to be Frobenius and derives criteria for Frobenius groups with abelian kernels.

Findings

Necessary and sufficient condition for pseudofrobenius schemes to be Frobenius.

Strong necessary conditions for existence of non-Frobenius pseudofrobenius schemes.

Weisfeiler-Leman dimension of certain circulant graphs is 2, except for specific prime power cases.

Abstract

An (association) scheme is said to be Frobenius if it is the scheme of a Frobenius group. A scheme which has the same tensor of intersection numbers as some Frobenius scheme is said to be pseudofrobenius. We establish a necessary and sufficient condition for an imprimitive pseudofrobenius scheme to be Frobenius. We also prove strong necessary conditions for existence of an imprimitive pseudofrobenius scheme which is not Frobenius. As a byproduct, we obtain a sufficient condition for an imprimitive Frobenius group $G$ with abelian kernel to be determined up to isomorphism only by the character table of $G$. Finally, we prove that the Weisfeiler-Leman dimension of a circulant graph with $n$ vertices and Frobenius automorphism group is equal to $2$ unless $n\in \{p,p^2,p^3,pq,p^2q\}$, where $p$ and $q$ are distinct primes.