Coset closure of a circulant S-ring and schurity problem

TL;DR

This paper investigates the schurity problem for circulant S-rings, characterizing Galois-closed S-rings via coset closures and linking schurity to the consistency of associated modular linear systems.

Contribution

It introduces the concept of coset closure for circulant S-rings and establishes a criterion for schurity based on algebraic fusion and linear system consistency.

Findings

Schurity of circulant S-rings is characterized by algebraic fusion of coset closures.

Schurity problem reduces to checking the consistency of a modular linear system.

A circulant S-ring is Galois closed if and only if its dual is Galois closed.

Abstract

Let $G$ be a finite group. There is a natural Galois correspondence between the permutation groups containing $G$ as a regular subgroup, and the Schur rings (S-rings) over~$G$. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group $G$ is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Basing on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.