Balanced sets andQ-polynomial association schemes

The notion of a balanced set of vectors is defined, and the classification of such sets suggested. A slightly stronger condition is considered for association schemes, with the following result. LetX be ad-class symmetric association scheme with Bose-Mesner algebraM and Krein parametersq _ij ^h , and let E = E_t (0 t d) be any primitive idempotent ofM. For eachx X letx _E denote the diagonal matrix withy, y entryE _xy(y X). Define the representation diagramD _E on the nodes 0, 1, ...,d by drawing an undirected arc between any nodesh, j for whichq _tj ^h > 0. CallD _E an augmented tree if it has a single loop, whose removal yields a tree. Writexy = i ifx, y X are ith associates (0 i d).