On highly closed cellular algebras

We define and study m-closed cellular algebras (coherent configurations) and m-isomorphisms of cellular algebras which can be regarded as mth approximations of Schurian algebras (i.e. the centralizer algebras of permutation groups) and of strong isomorphisms (i.e. bijections of the point sets taking one algebra to the other) respectively. If m = 1 we come to arbitrary cellular algebras and their weak isomorphisms (i.e. matrix algebra isomorphisms preserving the Hadamard multiplication). On the other hand, the algebras which are m-closed for all m ≥ 1 are exactly Schurian ones whereas the weak isomorphisms which are m-isomorphisms for all m ≥ 1 are exactly ones induced by strong isomorphisms. We show that for any m there exist m-closed algebras on O(m) points which are not Schurian and m-isomorphisms of cellular algebras on O(m) points which are not induced by strong isomorphisms. This enables us to find for any m an edge colored graph with O(m) vertices satisfying the m-vertex condition and having non-Schurian adjacency algebra. On the other hand, we rediscover and explain from the algebraic point of view the Cai-Fürer-Immerman phenomenon that the m-dimensional Weisfeiler-Lehman method fails to recognize the isomorphism of graphs in an efficient way. ∗Research supported by RFFI, grants 96-15-96060 and 96-01-00676.