# On separable Schur rings over abelian groups

## TL;DR

This paper characterizes abelian groups that are weakly separable, showing they belong to specific known families, which advances understanding of algebraic isomorphisms in Schur rings.

## Contribution

It provides a classification of abelian weakly separable groups, identifying the explicit families to which they belong.

## Key findings

- Every abelian weakly separable group belongs to a specific family.
- Algebraic isomorphisms in Schur rings are induced by combinatorial isomorphisms in these groups.
- The classification narrows the scope of weakly separable groups in algebraic combinatorics.

## Abstract

A finite group is said to be weakly separable if every algebraic isomorphism between two $S$-rings over this group is induced by a combinatorial isomorphism. In the paper we prove that every abelian weakly separable group belongs to one of several explicitly given families only.