ON HIGH CLOSED CELLULAR ALGEBRASAND HIGH CLOSED ISOMORPHISMSSapienti

We study m-closed cellular algebras (coherent connguration) and m-isomorphisms of cellular algebras which can be regarded as mth approximations to Schurian algebras (i.e. the centralizer algebras of permutation groups) and to 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 non-Schurian algebras on O(m) points and m-isomorphisms of cellular algebras on O(m) points which are not induced by strong isomorphisms. One more result establishes the equivalence of the Schurian polynomial approximation schemes deened by the m-closure and the m-dimensional Weisfeiler-Lehman methods. As a consequence, on one hand this enables us to nd 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-F urer-Immerman phenomenon that the m-dimensional Weisfeiler-Lehman method fails to recognize in an eecient way the isomorphism of graphs.