Separability of Schur rings over an abelian group of order 4p

TL;DR

This paper proves that all Schur rings over abelian groups of order 4p are separable with respect to abelian groups, leading to a bound on the Weisfeiler-Leman dimension of related Cayley graphs.

Contribution

It establishes the separability of Schur rings over abelian groups of order 4p, a new result linking algebraic isomorphisms to combinatorial ones.

Findings

All Schur rings over abelian groups of order 4p are separable.

The Weisfeiler-Leman dimension of Cayley graphs over these groups is at most 2.

Separable Schur rings simplify the isomorphism problem for associated Cayley graphs.

Abstract

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every its algebraic isomorphism to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. We prove that every Schur ring over an abelian group $G$ of order $4p$, where $p$ is a prime, is separable with respect to the class of abelian groups. This implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.