Standard Character Condition for Table Algebras

It is well known that the complex adjacency algebra $A$ of an association scheme has a specific module, namely the standard module, that contains the regular module of $A$ as a submodule. The character afforded by the standard module is called the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the standard character condition if it admits the standard character. Among other results we acquire a necessary and sufficient condition for a table algebra to originate from an association scheme. Finally, we prove that given a C-algebra admits the standard character and its all degrees are integers if and only if so its dual.

On Highly Closed Cellular Algebras and Highly Closed Isomorphisms

We define and study $m$-closed cellular algebras (coherent configurations) and $m$-isomorphisms of cellular algebras which can be regarded as $m$th 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 $m\ge 1$ are exactly Schurian ones whereas the weak isomorphisms which are $m$-isomorphisms for all $m\ge 1$ 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.