A symmetric (v, k, λ) design has a natural six-dimensional algebra on the flags. It is easy to show that this “flag algebra” (or “Hecke algebra”) is the algebra of an association scheme if and only if the design is a projective plane. However, if the symmetric design is either the point-hyperplane design of a projective space (of dimension at least three) or the complement of a projective space over a field of order two, then the flag algebra has a natural extension to a seven-dimensional algebra of an association scheme.
It is conjectured that these are the only cases where this natural extension has dimension seven. Several weak forms of the conjecture are proved: In Theorem 5.2 it is assumed that λ divides k − 1; the added hypothesis is “k = 2λ” in Theorem 5.4.
