On multidimensional Schur rings of finite groups

Abstract For any finite group 𝐺 and a positive integer 𝑚, we define and study a Schur ring over the direct power G m G^{m} , which gives an algebraic interpretation of the partition of G m 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 ≥ 3 m\geq 3 , and approaches the Schur ring associated with the group Aut ⁡ ( G ) \operatorname{Aut}(G) acting on G m 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.