A family of non-Schurian $p$-Schur rings over groups of order $p^3$

TL;DR

This paper investigates the Schurity problem for non-commutative $p$-Schur rings over groups of order $p^3$, providing a new family of non-Schurian examples and advancing understanding in algebraic combinatorics.

Contribution

It introduces a family of non-Schurian $p$-Schur rings over groups of order $p^3$, addressing the Schurity problem in the non-commutative case.

Findings

Identified non-Schurian $p$-Schur rings over groups of order $p^3$

Extended the classification of $p$-Schur rings beyond the commutative case

Provided explicit examples of non-Schurian structures

Abstract

Recently, it was proved that every commutative $p$-Schur ring over a group of order $p^3$ is Schurian. In this article, we consider the Schurity problem of non-commutative $p$-Schur rings over groups of order $p^3$. In particular, it is given a family of non-Schurian $p$-Schur rings over groups of order $p^3$.