  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.
