On multidimensional Schur rings of finite groups

TL;DR

This paper introduces a new class of Schur rings over direct powers of finite groups, linking algebraic structures with the Weisfeiler-Leman algorithm and group isomorphism problem, providing insights into group characterization.

Contribution

It defines multidimensional Schur rings over $G^m$, showing their ability to determine the group up to isomorphism for $m \,\geq 3$, and relates their limit to the automorphism group, connecting to the group isomorphism problem.

Findings

Schur rings over $G^m$ encode Weisfeiler-Leman partitions.

For $m\geq 3$, these rings determine the group $G$ up to isomorphism.

Finding the limit ring is polynomial-time equivalent to the group isomorphism problem.

Abstract

For any finite group $G$ and a positive integer $m$, we define andstudy a Schur ring over the direct power $G^m$, which gives an algebraic interpretation of the partition of $G^m$ obtained by the $m$-dimensional Weisfeiler-Leman algorithm. It is proved that this ring determines the group $G$ up to isomorphism if $m\ge 3$, and approaches the Schur ring associated with the group $Aut(G)$ acting on $G^m$ naturally if $m$ increases. It turns out that the problem of finding this limit ring is polynomial-time equivalent to the group isomorphism problem.